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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Scorza-Dragoni approach to Dirichlet problem in Banach spaces

Jan Andres1*, Luisa Malaguti2 and Martina Pavlačková1

Author Affiliations

1 Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 771 46, Czech Republic

2 Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, Via G. Amendola, 2 - pad. Morselli, Reggio Emilia, 42122, Italy

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Boundary Value Problems 2014, 2014:23  doi:10.1186/1687-2770-2014-23

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/23


Received:16 October 2013
Accepted:9 January 2014
Published:27 January 2014

© 2014 Andres et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Hartman-type conditions are presented for the solvability of a multivalued Dirichlet problem in a Banach space by means of topological degree arguments, bounding functions, and a Scorza-Dragoni approximation technique. The required transversality conditions are strictly localized on the boundaries of given bound sets. The main existence and localization result is applied to a partial integro-differential equation involving possible discontinuities in state variables. Two illustrative examples are supplied. The comparison with classical single-valued results in this field is also made.

MSC: 34A60, 34B15, 47H04.

Keywords:
Dirichlet problem; Scorza-Dragoni-type technique; strictly localized bounding functions; solutions in a given set; condensing multivalued operators

1 Introduction

In this paper, we will establish sufficient conditions for the existence and localization of strong solutions to a multivalued Dirichlet problem in a Banach space via degree arguments combined with a bound sets technique. More precisely, Hartman-type conditions (cf.[1]), i.e. sign conditions w.r.t. the first state variable and growth conditions w.r.t. the second state variable, will be presented, provided the right-hand side is a multivalued upper-Carathéodory mapping which is γ-regular w.r.t. the Hausdorff measure of non-compactness γ.

The main aim will be two-fold: (i) strict localization of sign conditions on the boundaries of bound sets by means of a technique originated by Scorza-Dragoni [2], and (ii) the application of the obtained abstract result (see Theorem 3.1 below) to an integro-differential equation involving possible discontinuities in a state variable. The first aim allows us, under some additional restrictions, to extend our earlier results obtained for globally upper semicontinuous right-hand sides and partly improve those for upper-Carathéodory right-hand sides (see [3]). As we shall see, the latter aim justifies such an abstract setting, because the problem can be transformed into the form of a differential inclusion in a Hilbert <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M1">View MathML</a>-space. Roughly speaking, problems of this type naturally require such an abstract setting. In order to understand in a deeper way what we did and why, let us briefly recall classical results in this field and some of their extensions.

Hence, consider firstly the Dirichlet problem in the simplest vector form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M2">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M3">View MathML</a> is, for the sake of simplicity allowing the comparison of the related results, a continuous function.

The first existence results, for a bounded f in (1), are due to Scorza-Dragoni [4,5]. Let us note that his name in the title is nevertheless related to the technique developed in [2] rather than to the existence results in [4,5].

It is well known (see e.g.[3,6-13]) that the problem (1) is solvable on various levels of generality provided:

(isign) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M4">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M5">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M6">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M7">View MathML</a>,

(iigrowth) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M10">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M11">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M12">View MathML</a>.

Let us note that the existence of the same constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13">View MathML</a> in (isign) and (iigrowth) can be assumed either explicitly as in [6,7,9,11,13] or it follows from the assumptions as those in [8,10,12].

(1) Hartmann [9] (cf. also [1]) generalized both conditions as follows:

(iH) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M4">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M15">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M19">View MathML</a>,

(iH) the well-known Bernstein-Nagumo-Hartman condition (for its definition and more details, see e.g.[1,14]).

Let us note that the strict inequality in (iH) can be replaced by a non-strict one (see e.g. [[1], Chapter XII,II,5], [[11], Corollary 6.2]).

(2) Lasota and Yorke [10] improved condition (isign) with suitable constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M21">View MathML</a> in the following way:

(iLY) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M22">View MathML</a>,

but for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M24">View MathML</a>, and replaced (iigrowth) by the Bernstein-Nagumo-Hartman condition.

Since (iLY) implies (cf.[10]) the existence of a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M25">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M26">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M27">View MathML</a>, the sign condition (iLY) is obviously more liberal than (isign) as well as than (iH), on the intersection of their domains.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M28">View MathML</a> in (iLY), then constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M29">View MathML</a> can be even equal to zero, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M30">View MathML</a>, in (iLY) (see e.g. [[7], Corollary V.26 on p.74]). Moreover, the related Bernstein-Nagumo-Hartman condition can only hold for x in a suitable convex, closed, bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M31">View MathML</a> (see again e.g. [7]).

(3) Following the ideas of Mawhin in [7,11,12], Amster and Haddad [6] demonstrated that an open, bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M31">View MathML</a>, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M33">View MathML</a>, need not be convex, provided it has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M34">View MathML</a>-boundary ∂D such that condition (iH) can be generalized as follows:

(iAH) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M36">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M37">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M38">View MathML</a> is the outer-pointing normal unit vector field, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M39">View MathML</a> denotes the tangent vector bundle and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M40">View MathML</a> stands for the second fundamental form of the hypersurface.

Since for the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13">View MathML</a>, we can have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M43">View MathML</a>

condition (iAH) is obviously more general than the original Hartman condition (iH).

Nevertheless, the growth condition takes there only the form (iigrowth), namely with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M12">View MathML</a> replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M45">View MathML</a>, where R denotes, this time, the radius of D.

For a convex, open, bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M33">View MathML</a>, the particular case of (iAH) can read as follows:

(iconv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M47">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M48">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M37">View MathML</a>,

which is another well-known generalization of (isign).

(4) In a Hilbert space H, for a completely continuous mapping f, Mawhin [12] has shown that, for real constants a, b, c such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M51">View MathML</a>, condition (isign) can be replaced in particular by

(iM) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M53">View MathML</a>,

and (iigrowth) by an appropriate version of the Bernstein-Nagumo-Hartman condition.

(5) In a Banach space E, Schmitt and Thompson [13] improved, for a completely continuous mapping f, condition (iconv) in the sense that the strict inequality in (iconv) can be replaced by a non-strict one. More concretely, if there exists a convex, open, bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M54">View MathML</a> of E with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M55">View MathML</a> such that

(iST) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M56">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M57">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M59">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M60">View MathML</a> denotes this time the pairing between E and its dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M61">View MathML</a>, jointly with the appropriate Bernstein-Nagumo-Hartman condition, then the problem (1) admits a solution whose values are located in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M62">View MathML</a> (see [[13], Theorem 4.1]).

In the Carathéodory case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M63">View MathML</a> in (1), for instance, the strict inequality in condition (isign) can be replaced, according to [[8], Theorem 6.1], by a non-strict one and the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M65">View MathML</a> can be replaced without the requirement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M9">View MathML</a>, but globally in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M67">View MathML</a>, by functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M69">View MathML</a> which are bounded on bounded sets. Moreover, system (1) can be additively perturbed, for the same goal, by another Carathéodory function which is sublinear in both states variables x and y.

On the other hand, the Carathéodory case brings about some obstructions in a strict localization of sign conditions on the boundaries of bound sets (see e.g.[3,15]). The same is also true for other boundary value problems (for Floquet problems, see e.g.[16-18]). Therefore, there naturally exist some extensions of classical results in this way. Further extensions concern problems in abstract spaces, functional problems, multivalued problems, etc. For the panorama of results in abstract spaces, see e.g.[19], where multivalued problems are also considered.

Nevertheless, let us note that in abstract spaces, it is extremely difficult (if not impossible) to avoid the convexity of given bound sets, provided the degree arguments are applied for non-compact maps (for more details, see [20]).

In this light, we would like to modify in the present paper the Hartman-type conditions (isign), (iigrowth) at least in the following way:

• the given space E to be Banach (or, more practically, Hilbert),

• the right-hand side to be a multivalued upper-Carathéodory mapping F which is γ-regular w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M70">View MathML</a> and either globally measurable or globally quasi-compact,

• the inequality in (isign) to hold w.r.t. x strictly on the boundary ∂D of a convex, bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M71">View MathML</a> (or, more practically, of the ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M72">View MathML</a>),

• condition (iigrowth) to be replaced by a suitable growth condition which would allow us reasonable applications (the usage of the Bernstein-Nagumo-Hartman-type condition will be employed in this context by ourselves elsewhere).

Hence, let E be a separable Banach space (with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M73">View MathML</a>) satisfying the Radon-Nikodym property (e.g. reflexivity, see e.g. [[21], pp.694-695]) and let us consider the Dirichlet boundary value problem (b.v.p.)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M74">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M75">View MathML</a> is an upper-Carathéodory multivalued mapping.

Let us note that in the entire paper all derivatives will be always understood in the sense of Fréchet and, by the measurability, we mean the one with respect to the Lebesgue σ-algebra in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M76">View MathML</a> and the Borel σ-algebra in E.

The notion of a solution will be understood in a strong (i.e. Carathéodory) sense. Namely, by a solution of problem (2) we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M77">View MathML</a> whose first derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M78">View MathML</a> is absolutely continuous and satisfies (2), for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>.

The solution of the b.v.p. (2) will be obtained as the limit of a sequence of solutions of approximating problems that we construct by means of a Scorza-Dragoni-type result developed in [22]. The approximating problems will be treated by means of the continuation principle developed in [19].

2 Preliminaries

Let E be as above and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M80">View MathML</a> be a closed interval. By the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M81">View MathML</a>, we shall mean the set of all Bochner integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M82">View MathML</a>. For the definition and properties of Bochner integrals, see e.g. [[21], pp.693-701]. The symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M83">View MathML</a> will be reserved for the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M82">View MathML</a> whose first derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M78">View MathML</a> is absolutely continuous. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M86">View MathML</a> and the fundamental theorem of calculus (the Newton-Leibniz formula) holds (see e.g. [[21], pp.695-696], [[23], pp.243-244]). In the sequel, we shall always consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M87">View MathML</a> as a subspace of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M88">View MathML</a> and by the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M89">View MathML</a> we shall mean the Banach space of all linear, bounded transformations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M90">View MathML</a> endowed with the sup-norm.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M92">View MathML</a>, the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M93">View MathML</a> will denote, as usually, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M94">View MathML</a>, where B is the open unit ball in E centered at 0, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M95">View MathML</a>. In what follows, the symbol μ will denote the Lebesgue measure on ℝ.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M61">View MathML</a> be the Banach space dual to E and let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M97">View MathML</a> the pairing (the duality relation) between E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M61">View MathML</a>, i.e., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M99">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M100">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M101">View MathML</a>.

We recall also the Pettis measurability theorem which will be used in Section 4 and which we state here in the form of proposition.

Proposition 2.1 [[24], p.278]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M102">View MathML</a>be a measure space, Ebe a separable Banach space. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M103">View MathML</a>is measurable if and only if for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M104">View MathML</a>the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M105">View MathML</a>is measurable with respect to Σ and the Borelσ-algebra in ℝ.

We shall also need the following definitions and notions from multivalued analysis. Let X, Y be two metric spaces. We say that F is a multivalued mapping from X to Y (written <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M106">View MathML</a>) if, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M107">View MathML</a>, a non-empty subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M108">View MathML</a> of Y is given. We associate with F its graph <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M109">View MathML</a>, the subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M110">View MathML</a>, defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M111">View MathML</a>.

A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M112">View MathML</a> is called upper semicontinuous (shortly, u.s.c.) if, for each open subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M113">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M114">View MathML</a> is open in X.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M115">View MathML</a> be a compact interval. A mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M116">View MathML</a>, where Y is a separable metric space, is called measurable if, for each open subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M117">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M118">View MathML</a> belongs to a σ-algebra of subsets of J.

A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M112">View MathML</a> is called compact if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M120">View MathML</a> is contained in a compact subset of Y and it is called quasi-compact if it maps compact sets onto relatively compact sets.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M115">View MathML</a> be a given compact interval. A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M122">View MathML</a>, where Y is a separable Banach space, is called an upper-Carathéodory mapping if the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M123">View MathML</a> is measurable, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M107">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M125">View MathML</a> is u.s.c., for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M126">View MathML</a>, and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M127">View MathML</a> is compact and convex, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M128">View MathML</a>.

The technique that will be used for proving the existence and localization result consists in constructing a sequence of approximating problems. This construction will be made on the basis of the Scorza-Dragoni-type result developed in [22] (cf. also [25]).

For more details concerning multivalued analysis, see e.g.[23,26,27].

Definition 2.1 An upper-Carathéodory mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M129">View MathML</a> is said to have the Scorza-Dragoni property if there exists a multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M130">View MathML</a> with compact, convex values having the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M131">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M132">View MathML</a>,

(ii) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M133">View MathML</a> are measurable functions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M134">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, then also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M136">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>,

(iii) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M138">View MathML</a>, there exists a closed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M139">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M141">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M142">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M143">View MathML</a> is u.s.c. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M144">View MathML</a>.

The following two propositions are crucial in our investigation. The first one is almost a direct consequence of the main result in [22] (cf.[25] and [[16], Proposition 2]). The second one allows us to construct a sequence of approximating problems of (2).

Proposition 2.2LetEbe a separable Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M145">View MathML</a>be an upper-Carathéodory mapping. IfFis globally measurable or quasi-compact, thenFhas the Scorza-Dragoni property.

Proposition 2.3 (cf. [[18], Theorem 2.2])

LetEbe a Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M146">View MathML</a>a non-empty, open, convex, bounded set such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M147">View MathML</a>. Moreover, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M138">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M149">View MathML</a>be a Fréchet differentiable function with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150">View MathML</a>Lipschitzian in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M151">View MathML</a>satisfying

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M152">View MathML</a>,

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M153">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M154">View MathML</a>,

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M155">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M156">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M157">View MathML</a>is given.

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M158">View MathML</a>and a bounded Lipschitzian function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M159">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M160">View MathML</a>, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M161">View MathML</a>.

Remark 2.1 Let us note that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M162">View MathML</a>, where ϕ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M163">View MathML</a> are the same as in Proposition 2.3, is Lipschitzian and bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M164">View MathML</a>. The symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M163">View MathML</a> denotes as usually the first Fréchet derivative of V at x.

Example 2.1 If V satisfies all the assumptions of Proposition 2.3, then it is easy to prove the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M166">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M167">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M168">View MathML</a>. Consequently, when E is an arbitrary Hilbert space, we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M169">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M170">View MathML</a>

which satisfies all the properties mentioned in Proposition 2.3.

Definition 2.2 Let N be a partially ordered set, E be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M171">View MathML</a> denote the family of all non-empty bounded subsets of E. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M172">View MathML</a> is called a measure of non-compactness (m.n.c.) in E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M173">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M174">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M175">View MathML</a> denotes the closed convex hull of Ω.

A m.n.c. β is called:

(i) monotone if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M176">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M177">View MathML</a>,

(ii) non-singular if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M178">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M179">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180">View MathML</a>.

If N is a cone in a Banach space, then a m.n.c. β is called:

(iii) semi-homogeneous if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M181">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M182">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180">View MathML</a>,

(iv) regular when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M184">View MathML</a> if and only if Ω is relatively compact,

(v) algebraically subadditive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M185">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M186">View MathML</a>.

The typical example of an m.n.c. is the Hausdorff measure of non-compactnessγ defined, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M188">View MathML</a>

The Hausdorff m.n.c. is monotone, non-singular, semi-homogeneous and regular. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180">View MathML</a>, then (see, e.g., [27])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M191">View MathML</a>

(3)

Let E be a separable Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M192">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M194">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196">View MathML</a> and suitable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M197">View MathML</a>, then (cf.[27])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M198">View MathML</a>

(4)

Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M199">View MathML</a> is L-Lipschitzian, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M200">View MathML</a>

(5)

for all bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180">View MathML</a>.

Furthermore, for all subsets Ω of E (see e.g.[17]),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M202">View MathML</a>

(6)

Let us now introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M203">View MathML</a>

(7)

defined on the bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M204">View MathML</a>, where the ordering is induced by the positive cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M205">View MathML</a> and where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M206">View MathML</a> denotes the modulus of continuity of a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M207">View MathML</a>.a It was proved in [19] that the function α given by (7) is an m.n.c. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M88">View MathML</a> that is monotone, non-singular and regular.

Definition 2.3 Let E be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M212">View MathML</a>. A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M213">View MathML</a> with compact values is called condensing with respect to an m.n.c.β (shortly, β-condensing) if, for every bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M214">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M215">View MathML</a>, we see that Ω is relatively compact.

A family of mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M216">View MathML</a> with compact values is called β-condensing if, for every bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M214">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M218">View MathML</a>, we see that Ω is relatively compact.

The proof of the main result (cf. Theorem 3.1 below) will be based on the following slight modification of the continuation principle developed in [19]. Since the proof of this modified version differs from the one in [19] only slightly in technical details, we omit it here.

Proposition 2.4Let us consider the b.v.p.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M219">View MathML</a>

(8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M220">View MathML</a>is an upper-Carathéodory mapping and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M221">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M222">View MathML</a>be an upper-Carathéodory mapping such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M223">View MathML</a>

Moreover, assume that the following conditions hold:

(i) There exist a closed set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M224">View MathML</a>and a closed, convex set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M225">View MathML</a>with a non-empty interior IntQsuch that each associated problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M226">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228">View MathML</a>, has a non-empty, convex set of solutions (denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M229">View MathML</a>).

(ii) For every non-empty, bounded set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M230">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M231">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M232">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M234">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228">View MathML</a>.

(iii) The solution mappingis quasi-compact andμ-condensing with respect to a monotone and non-singular m.n.c. μdefined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M88">View MathML</a>.

(iv) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227">View MathML</a>, the set of solutions of problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M239">View MathML</a>is a subset of IntQ, i.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M240">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227">View MathML</a>.

(v) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M242">View MathML</a>, the solution mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M243">View MathML</a>has no fixed points on the boundary∂QofQ.

Then the b.v.p. (8) has a solution inQ.

3 Main result

Combining the foregoing continuation principle with the Scorza-Dragoni-type technique (cf. Proposition 2.2), we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (2).

Theorem 3.1Consider the Dirichlet b.v.p. (2). Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M244">View MathML</a>is an upper-Carathéodory mapping which is either globally measurable or quasi-compact. Furthermore, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M146">View MathML</a>be a non-empty, open, convex, bounded subset containing 0 of a separable Banach spaceEsatisfying the Radon-Nikodym property. Let the following conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247">View MathML</a>) be satisfied:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M249">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M250">View MathML</a>and each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M251">View MathML</a>, and each bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M252">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M253">View MathML</a>andγis the Hausdorff m.n.c. inE.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>) For every non-empty, bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M180">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M256">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M257">View MathML</a>

(9)

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M259">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M261">View MathML</a>

Furthermore, let there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M138">View MathML</a>and a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M263">View MathML</a>, i.e. a twice continuously differentiable function in the sense of Fréchet, satisfying (H1)-(H3) (cf. Proposition 2.3) with Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150">View MathML</a>Lipschitzian in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M265">View MathML</a>.bLet there still exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M270">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M271">View MathML</a>

(10)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M272">View MathML</a>denotes the second Fréchet derivative ofVatxin the direction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M273">View MathML</a>. Finally, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M274">View MathML</a>

(11)

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M275">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M277">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M278">View MathML</a>.

Then the Dirichlet b.v.p. (2) admits a solution whose values are located in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M279">View MathML</a>. If, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M280">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M250">View MathML</a>, then the obtained solution is non-trivial.

Proof Since the proof of this result is rather technical, it will be divided into several steps. At first, let us define the sequence of approximating problems. For this purpose, let k be as in Proposition 2.3 and consider a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M282">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M283">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M284">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M285">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M286">View MathML</a>. According to Proposition 2.3 (see also Remark 2.1), the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M287">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M288">View MathML</a>

is well defined, continuous and bounded.

Since the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M289">View MathML</a> has, according to Proposition 2.2, the Scorza-Dragoni property, we are able to find a decreasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M290">View MathML</a> of subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M291">View MathML</a> and a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M292">View MathML</a> with compact, convex values such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M293">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M294">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M295">View MathML</a> is closed,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M296">View MathML</a> is u.s.c. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M297">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M298">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M295">View MathML</a> (cf.e.g.[2]).

If we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M300">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M141">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M303">View MathML</a>, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M296">View MathML</a> is u.s.c. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M305">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M298">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M307">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a>, let us define the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M309">View MathML</a> with compact, convex values by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M310">View MathML</a>

Let us consider the b.v.p.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M311">View MathML</a>

Now, let us verify the solvability of problems <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M312">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a> be fixed. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M143">View MathML</a> is globally u.s.c. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M316">View MathML</a> is measurable, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M70">View MathML</a>, and, due to the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M319">View MathML</a> is u.s.c., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M320">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M321">View MathML</a> is an upper-Carathéodory mapping. Moreover, let us define the upper-Carathéodory mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M322">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M323">View MathML</a>

Let us show that, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a> is sufficiently large, all assumptions of Proposition 2.4 (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M325">View MathML</a>) are satisfied.

For this purpose, let us define the closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M326">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M327">View MathML</a>

and let the set Q of candidate solutions be defined as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M328">View MathML</a>. Because of the convexity of K, the set Q is closed and convex.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228">View MathML</a>, consider still the associated fully linearized problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M331">View MathML</a>

and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a> the solution mapping which assigns to each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M333">View MathML</a> the set of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334">View MathML</a>.

ad (i) In order to verify condition (i) in Proposition 2.4, we need to show that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M333">View MathML</a>, the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334">View MathML</a> is solvable with a convex set of solutions. So, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M333">View MathML</a> be arbitrary and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M338">View MathML</a> be a measurable selection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M339">View MathML</a>, which surely exists (see, e.g., [[27], Theorem 1.3.5]). According to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M341">View MathML</a>, it is also easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M342">View MathML</a>. The homogeneous problem corresponding to b.v.p. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M344">View MathML</a>

(12)

has only the trivial solution, and therefore the single-valued Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M345">View MathML</a>

admits a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M346">View MathML</a> which is one of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334">View MathML</a>. This is given, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M349">View MathML</a>, where G is the Green function associated to the homogeneous problem (12). The Green function G and its partial derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M350">View MathML</a> are defined by (cf.e.g. [[28], pp.170-171])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M351">View MathML</a>

Thus, the set of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334">View MathML</a> is non-empty. The convexity of the solution sets follows immediately from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M341">View MathML</a> and the fact that problems <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M334">View MathML</a> are fully linearized.

ad (ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M355">View MathML</a> be bounded. Then, there exists a bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M356">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M357">View MathML</a> and, according to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M341">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M360">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M361">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M362">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M365">View MathML</a>

Therefore, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M366">View MathML</a> satisfies condition (ii) from Proposition 2.4.

ad (iii) Since the verification of condition (iii) in Proposition 2.4 is technically the most complicated, it will be split into two parts: (iii1) the quasi-compactness of the solution operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a>, (iii2) the condensity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a> w.r.t. the monotone and non-singular m.n.c. α defined by (7).

ad (iii1) Let us firstly prove that the solution mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a> is quasi-compact. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M370">View MathML</a> is a complete metric space, it is sufficient to prove the sequential quasi-compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a>. Hence, let us consider the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M373">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M374">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M375">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M377">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M370">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M379">View MathML</a>. Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M380">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196">View MathML</a>. Then there exists, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M383">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M384">View MathML</a>

(13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M385">View MathML</a>

(14)

and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M386">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M387">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M388">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M389">View MathML</a>, there exists a bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M390">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M391">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a>. Therefore, there exists, according to condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M231">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M396">View MathML</a>, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M399">View MathML</a>.

Moreover, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M402">View MathML</a>

(15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M403">View MathML</a>

(16)

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M404">View MathML</a> satisfies, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M407">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M408">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M409">View MathML</a>

Furthermore, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M412">View MathML</a>

Hence, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M413">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M414">View MathML</a> are bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M415">View MathML</a> is uniformly integrable.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, the properties of the Hausdorff m.n.c. yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M417">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M418">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M420">View MathML</a>, it follows from condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246">View MathML</a>) that, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M423">View MathML</a>

Since the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M162">View MathML</a> is Lipschitzian on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M151">View MathML</a> with some Lipschitz constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M426">View MathML</a> (see Remark 2.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M427">View MathML</a>

(17)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M377">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M429">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M430">View MathML</a>, we get, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M432">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M433">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M435">View MathML</a>, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M436">View MathML</a> is relatively compact as well since, according to the semi-homogeneity of the Hausdorff m.n.c.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M437">View MathML</a>

(18)

Moreover, by means of (4) and (18),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M438">View MathML</a>

By similar reasoning, we also get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M439">View MathML</a>

by which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M440">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M441">View MathML</a> are relatively compact, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>.

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M404">View MathML</a> satisfies for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a> (13), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M445">View MathML</a> is relatively compact, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>. Thus, according to [[23], Lemma III.1.30], there exist a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M414">View MathML</a>, for the sake of simplicity denoted in the same way as the sequence, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M448">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M414">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M450">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M451">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M415">View MathML</a> converges weakly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M453">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M81">View MathML</a>. According to the classical closure results (cf.e.g. [[27], Lemma 5.1.1]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M455">View MathML</a>, which implies the quasi-compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a>.

ad (iii2) In order to show that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M293">View MathML</a> sufficiently large, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M332">View MathML</a> is α-condensing with respect to the m.n.c. α defined by (7), let us consider a bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M459">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M460">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M461">View MathML</a> be a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M462">View MathML</a>

At first, let us show that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M463">View MathML</a> is bounded. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M464">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M465">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M467">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M468">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M469">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>.

Since Θ is bounded, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M471">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M472">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M465">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>. Hence, according to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M476">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M477">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>. Consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M479">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M480">View MathML</a>

Thus, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M481">View MathML</a> is bounded.

Moreover, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M482">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M483">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M484">View MathML</a> satisfying, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M485">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M486">View MathML</a>, such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M488">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M489">View MathML</a> are defined by (15) and (16), respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M490">View MathML</a> is defined by (14).

By similar reasoning as in the part ad (iii1), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M491">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>, and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M493">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M494">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M496">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a>, where Θ is a bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M370">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M499">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M500">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M501">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>. Hence, it follows from condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M504">View MathML</a>

(19)

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M505">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M506">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M393">View MathML</a>.

Moreover, by virtue of the semi-homogeneity of the Hausdorff m.n.c., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M508">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M509">View MathML</a>

Let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M510">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M511">View MathML</a>

According to (4) and (15) we thus obtain for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M513">View MathML</a>

By similar reasonings, we can see that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M515">View MathML</a>

when starting from condition (16). Subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M516">View MathML</a>

(20)

Since we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M460">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M518">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M519">View MathML</a>

Since we have, according to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M521">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M522">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M524">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M525">View MathML</a>

Therefore, we get, for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a>, the contradiction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M527">View MathML</a> which ensures the validity of condition (iii) in Proposition 2.4.

ad (iv) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M529">View MathML</a> coincides with the unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M530">View MathML</a> of the linear system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M531">View MathML</a>

According to (15) and (16), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M533">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M534">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M535">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M536">View MathML</a>

we have, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M538">View MathML</a>

(21)

Let us now consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M539">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M540">View MathML</a>. Then it follows from (21) that we are able to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M522">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M524">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M545">View MathML</a>. Therefore, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M308">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M524">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M548">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M227">View MathML</a>, which ensures the validity of condition (iv) in Proposition 2.4.

ad (v) The validity of the transversality condition (v) in Proposition 2.4 can be proven quite analogously as in [16] (see pp.40-43 in [16]) with the following differences:

– due to the Dirichlet boundary conditions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M550">View MathML</a> belongs to the open interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M551">View MathML</a>,

– since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M552">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M553">View MathML</a>.

In this way, we can prove that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M522">View MathML</a> such that every problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M555">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M556">View MathML</a>, satisfies all the assumptions of Proposition 2.4. This implies that every such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M555">View MathML</a> admits a solution, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M530">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M559">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>. By similar arguments as in [16], but with the expression <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M561">View MathML</a> replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M562">View MathML</a>, according to condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>), we can obtain the result that there exists a subsequence, denoted as the sequence, and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M564">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M565">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M566">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M430">View MathML</a> and also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M568">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M569">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M570">View MathML</a>. Thus, a classical closure result (see e.g. [[27], Lemma 5.1.1]) guarantees that x is a solution of (2) satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M571">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M79">View MathML</a>, and the sketch of proof is so complete. □

The case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M573">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M574">View MathML</a> to be completely continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M575">View MathML</a> to be Lipschitzian, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, represents the most classical example of a map which is γ-regular w.r.t. the Hausdorff measure of non-compactness γ. The following corollary of Theorem 3.1 can be proved quite analogously as in [[3], Example 6.1 and Remark 6.1].

Corollary 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M577">View MathML</a>be a separable Hilbert space and let us consider the Dirichlet b.v.p.:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M578">View MathML</a>

(22)

where

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M579">View MathML</a>is an upper-Carathéodory, globally measurable, multivalued mapping and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M580">View MathML</a>is completely continuous, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M582">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M584">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M585">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M586">View MathML</a>is an arbitrary constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M587">View MathML</a>, and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M588">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M589">View MathML</a>is a Carathéodory multivalued mapping such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M590">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M591">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M592">View MathML</a>is Lipschitzian, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, with the Lipschitz constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M594">View MathML</a>

Moreover, suppose that

(iii) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M596">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M597">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M598">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M599">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M600">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M601">View MathML</a>

Then the Dirichlet problem (22) admits, according to Theorem 3.1, a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M602">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M603">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>.

Remark 3.1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M605">View MathML</a>, the completely continuous mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M606">View MathML</a> allows us to make a comparison with classical single-valued results recalled in the Introduction. Unfortunately, our <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M607">View MathML</a> in (i) (see also (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>) in Theorem 3.1) is the only mapping which is (unlike in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions quite liberal growth restrictions were permitted) globally bounded w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M588">View MathML</a>. Furthermore, our sign condition in (iii) is also (unlike again in [[3], Example 6.1 and Remark 6.1], where under some additional restrictions the Hartman-type condition like (iH) in the Introduction was employed) the most restrictive among their analogies in [6-13]. On the other hand, because of multivalued upper-Carathéodory maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M610">View MathML</a> in a Hilbert space which are γ-regular, our result has still, as far as we know, no analogy at all.

4 Illustrative examples

The first illustrative example of the application of Theorem 3.1 concerns the integro-differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M611">View MathML</a>

(23)

involving discontinuities in a state variable. In this equation, the non-local diffusion term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M612">View MathML</a> replaces the classical diffusion behavior given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M613">View MathML</a>. In dispersal models such an integral term takes into account the long-distance interactions between individuals (see e.g.[29]). Moreover, when φ is linear in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M614">View MathML</a>, (23) can be considered as an alternative version of the classical telegraph equation (see e.g.[30] and the references therein), where the classical diffusivity is replaced by the present non-local diffusivity.

Telegraph equations appear in many fields such as modeling of an anomalous diffusion, a wave propagation phenomenon, sub-diffusive systems or modeling of a pulsate blood flow in arteries (see e.g.[31,32]).

For the sake of simplicity, we will discuss here only the case when φ is globally bounded w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M614">View MathML</a>. On the other hand, for non-strictly localized transversality conditions as in [3], for instance, a suitable linear growth estimate w.r.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M614">View MathML</a> can be permitted.

Example 4.1 Let us consider the integro-differential equation (23) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M617">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M618">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M619">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M620">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M621">View MathML</a>. We assume that

(a) φ is Carathéodory, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M622">View MathML</a> is measurable, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M623">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M624">View MathML</a> is continuous, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M626">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M627">View MathML</a>-Lipschitzian with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M628">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M629">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M630">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M623">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M632">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M633">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M634">View MathML</a>, for all a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M636">View MathML</a>,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M637">View MathML</a> and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M638">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>,

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M640">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M641">View MathML</a>,

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M642">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M643">View MathML</a>; and there can exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M644">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M645">View MathML</a> is continuous, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M646">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M645">View MathML</a> has discontinuities at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M648">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M649">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M650">View MathML</a>,

(e) f is L-Lipschitzian; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M651">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M652">View MathML</a>; and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M653">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M654">View MathML</a>,

(f) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M655">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M656">View MathML</a>.

Since the function p can have some discontinuities, a solution of (23) satisfying the Dirichlet conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M657">View MathML</a>

(24)

will be appropriately interpreted in the sense of Filippov. More precisely, let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M658">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M659">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M660">View MathML</a> is said to be a solution of (23), (24) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M661">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M630">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M663">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M664">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M665">View MathML</a> if it is a solution of the inclusion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M666">View MathML</a>

(25)

and if it satisfies (24).

If we further assume the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M13">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M668">View MathML</a>

(26)

and that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M669">View MathML</a>

(27)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M670">View MathML</a>

(28)

then the problem (23), (24) has a solution, in the sense of Filippov, satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M671">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>.

In fact, problem (25), (24) can be transformed into the abstract setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M673">View MathML</a>

(29)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M674">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M630">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M676">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M677">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M678">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M679">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M680">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M681">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M682">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M683">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M684">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M685">View MathML</a>.

Let us now examine the properties of F. According to (a), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M686">View MathML</a> is well defined. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M687">View MathML</a>, let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M688">View MathML</a> is measurable. For this purpose, let Ψ be an arbitrary element in the dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M689">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M690">View MathML</a>. Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M691">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M692">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M693">View MathML</a>, and consequently the composition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M694">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M695">View MathML</a>. Since φ is Carathéodory, it is globally measurable, and so the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M696">View MathML</a> is globally measurable as well. This implies that, according to the Fubini Theorem, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M697">View MathML</a> is measurable, too. Finally, since Ψ was arbitrary, according to the Pettis Theorem (see Proposition 2.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M688">View MathML</a> is measurable.

Furthermore, let us show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M699">View MathML</a> is u.s.c. For this purpose, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M700">View MathML</a> be fixed.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M701">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M702">View MathML</a>, then it is possible to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M157">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M704">View MathML</a> is single-valued, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M705">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M706">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M707">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M708">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M702">View MathML</a>. Since p is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M710">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M711">View MathML</a> is Lipschitzian, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M699">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M713">View MathML</a>.

(ii) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M714">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M715">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M716">View MathML</a> be open and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M717">View MathML</a>. Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M718">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M719">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M720">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M721">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M722','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M722">View MathML</a> is equal either to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M723">View MathML</a> or to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M724">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M721">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M726">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M705">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M728','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M728">View MathML</a>

which implies that it is possible to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M729">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M730">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M731">View MathML</a>. Similarly, we would obtain the same when assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M732">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M733">View MathML</a> then, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M734">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M735">View MathML</a>

which implies that also in this case it is possible to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M736','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M736">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M730">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M738','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M738">View MathML</a>.

Moreover, according to (a) and (c), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M739">View MathML</a> is a Carathéodory mapping such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M740">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M627">View MathML</a>-Lipschitzian, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a>, and K is well defined and 1-Lipschitzian. It can also be shown that, according to (d) and (e), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M699">View MathML</a> has compact and convex values. Therefore, the mapping F is globally measurable, and so has the Scorza-Dragoni property (cf. Proposition 2.2).

Let us now verify particular assumptions of Theorem 3.1.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M744">View MathML</a>. Then, according to (f),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M745">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M746','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M746">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M747">View MathML</a>

where m is defined by (28).

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M748">View MathML</a>

according to the Lipschitzianity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M711">View MathML</a> and property (6). For a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M135">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M751','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M751">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M752">View MathML</a>

and so condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M246">View MathML</a>) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M754">View MathML</a>. The obtained form of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M755">View MathML</a> together with assumption (27) directly guarantee the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M247">View MathML</a>). It can also be easily shown that properties of F ensure the validity of condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M254">View MathML</a>).

In order to verify conditions imposed on a bounding function, let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M758">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M759">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M760','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M760">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M761">View MathML</a> obviously satisfies (10), so it is only necessary to check condition (11). Thus, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M762">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M763">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M765">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M766','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M766">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M767">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M768">View MathML</a>

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M769">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M770">View MathML</a>, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M771','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M771">View MathML</a>

(30)

and since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M772','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M772">View MathML</a>

we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M773">View MathML</a>

(31)

The properties (a)-(f) together with the well-known Hölder inequality then yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M774','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M774">View MathML</a>

in view of condition (26), (30), and (31).

Hence, the Dirichlet problem (29) admits, according to Theorem 3.1, a solution y satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M775','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M775">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M275">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M777','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M777">View MathML</a>, then u is a solution of (24), (25) which is the Filippov solution of the original problem (23), (24).

Finally, we can sum up the above result in the form of the following theorem.

Theorem 4.1Let the assumptions (a)-(f) be satisfied. If still conditions (26), (27) hold, then the problem (23), (24) admits a non-trivial solutionuin the sense of Fillippov such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M778">View MathML</a>.

Remark 4.1 In [[13], Example 5.2], the following formally simpler integro-differential equation in ℝ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M779','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M779">View MathML</a>

with non-homogeneous Dirichlet conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M780">View MathML</a>

was solved provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M781','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M781">View MathML</a> is a positive kernel of the Hilbert-Schmidt-type and the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M782">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M783','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M783">View MathML</a> are finite.

After the homogenization of boundary conditions, the Dirichlet problem takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M784','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M784">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M785','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M785">View MathML</a>.

Thus, it can be naturally extended onto the infinite strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M786','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M786">View MathML</a>, into the form (23), (24), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M787','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M787">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M788','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M788">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M789">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M790">View MathML</a>.

The result in [[13], Example 5.2] cannot be, however, deduced from Theorem 3.1, because condition (b) in Example 4.1 cannot be satisfied in this way.

On the other hand, the linear term with coefficient b could not be implemented in their equation, because it is not completely continuous in (36) below, as required in [13].

In view of the arguments in Remark 4.1, we can conclude by the second illustrative example.

Example 4.2 Consider the following non-homogeneous Dirichlet problem in ℝ:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M791','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M791">View MathML</a>

(32)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M792">View MathML</a> is a positive kernel of the Hilbert-Schmidt-type such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M793">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M794">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M795','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M795">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M796','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M796">View MathML</a>.

Furthermore, let there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M797">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M798">View MathML</a>

(33)

The properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M799">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M800','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M800">View MathML</a> guarantee that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M801">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M802','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M802">View MathML</a>

(34)

We will show that, under (33) and (34), problem (32) is solvable, in the abstract setting, by means of Corollary 3.1.

Problem (32) can be homogenized as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M803','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M803">View MathML</a>

(35)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M804">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M805','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M805">View MathML</a>.

Since the Hilbert-Schmidt operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M806','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M806">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M807','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M807">View MathML</a> is well known to be completely continuous (cf. [[13], Example 5.2]) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M808','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M808">View MathML</a> is, according to (33), L-Lipschitzian with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M797">View MathML</a>, conditions (i), (ii) in Corollary 3.1 can be easily satisfied, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M810','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M810">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M811">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M812','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M812">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M813','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M813">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M814','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M814">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M815','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M815">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M816','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M816">View MathML</a>

In this setting, problem (35) takes the abstract form as (22), namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M817','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M817">View MathML</a>

(36)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M818','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M818">View MathML</a> holds, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M811">View MathML</a> (see [[13], Example 5.2]) one can check that the strict inequality in (iii) in Corollary 3.1 can be easily satisfied, for (32), whenever

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M820','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M820">View MathML</a>

(37)

Hence, applying Corollary 3.1, problem (36) admits a solution, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M821','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M821">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M822','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M822">View MathML</a>

where R satisfies (37), and subsequently the same is true for (32), i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M823','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M823">View MathML</a>

(38)

as claimed.

After all, we can sum up the sufficient conditions for the existence of a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M824','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M824">View MathML</a> of (32) satisfying (38) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M825','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M825">View MathML</a> is a positive kernel of the Hilbert-Schmidt operator with the finite norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M826','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M826">View MathML</a>

• there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M827','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M827">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M797">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M829','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M829">View MathML</a>, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M796','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M796">View MathML</a>,

• condition (37) holds.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Acknowledgements

The first and third authors were supported by grant ‘Singularities and impulses in boundary value problems for non-linear ordinary differential equations’. The second author was supported by the national research project PRIN ‘Ordinary Differential Equations and Applications’.

End notes

  1. The m.n.c. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M206">View MathML</a> is a monotone, non-singular and algebraically subadditive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M430">View MathML</a> (cf.e.g.[27]) and it is equal to zero if and only if all the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M210">View MathML</a> are equi-continuous.

  2. Since a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M34">View MathML</a>-function V has only a locally Lipschitzian Fréchet derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150">View MathML</a> (cf.e.g.[21]), we had to assume explicitly the global Lipschitzianity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M150">View MathML</a> in a non-compact set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/23/mathml/M269">View MathML</a>.

References

  1. Hartman, P: Ordinary Differential Equations, Willey, New York (1964)

  2. Scorza-Dragoni, G: Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un’altra variabile. Rend. Semin. Mat. Univ. Padova. 17, 102–106 (1948)

  3. Andres, J, Malaguti, L, Pavlačková, M: Dirichlet problem in Banach spaces: the bound sets approach. Bound. Value Probl.. 2013, (2013) Article ID 25

  4. Scorza-Dragoni, G: Sui sistemi di equazioni integrali non lineari. Rend. Semin. Mat. Univ. Padova. 7, 1–35 (1936)

  5. Scorza-Dragoni, G: Sul problema dei valori ai limiti per i systemi di equazioni differenziali del secondo ordine. Boll. Unione Mat. Ital.. 14, 225–230 (1935)

  6. Amster, P, Haddad, J: A Hartman-Nagumo type conditions for a class of contractible domains. Topol. Methods Nonlinear Anal.. 41(2), 287–304 (2013)

  7. Gaines, RE, Mawhin, J: Coincidence Degree, and Nonlinear Differential Equations, Springer, Berlin (1977)

  8. Granas, A, Guenther, RB, Lee, JW: Some existence principles in the Carathéodory theory of nonlinear differential system. J. Math. Pures Appl.. 70, 153–196 (1991)

  9. Hartman, P: On boundary value problems for systems of ordinary differential equations. Trans. Am. Math. Soc.. 96, 493–509 (1960). Publisher Full Text OpenURL

  10. Lasota, A, Yorke, JA: Existence of solutions of two-point boundary value problems for nonlinear systems. J. Differ. Equ.. 11(3), 509–518 (1972). Publisher Full Text OpenURL

  11. Mawhin, J: Boundary value problems for nonlinear second order vector differential equations. J. Differ. Equ.. 16, 257–269 (1974). Publisher Full Text OpenURL

  12. Mawhin, J: Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces. Tohoku Math. J.. 32, 225–233 (1980). Publisher Full Text OpenURL

  13. Schmitt, K, Thompson, RC: Boundary value problems for infinite systems of second-order differential equations. J. Differ. Equ.. 18(2), 277–295 (1975). Publisher Full Text OpenURL

  14. Mawhin, J: The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations. In: Farkas M (ed.) Budapest. (1981)

  15. Pavlačková, M: A Scorza-Dragoni approach to Dirichlet problem with an upper-Carathéodory right-hand side. Topol. Methods Nonlinear Anal. (to appear)

  16. Andres, J, Malaguti, L, Pavlačková, M: A Scorza-Dragoni approach to second-order boundary value problems in abstract spaces. Appl. Math. Inform. Sci.. 6(2), 177–192 (2012)

  17. Andres, J, Malaguti, L, Taddei, V: On boundary value problems in Banach spaces. Dyn. Syst. Appl.. 18, 275–302 (2009)

  18. Cecchini, S, Malaguti, L, Taddei, V: Strictly localized bounding functions and Floquet boundary value problems. Electron. J. Qual. Theory Differ. Equ.. 2011, (2011) Article ID 47

  19. Andres, J, Malaguti, L, Pavlačková, M: On second-order boundary value problems in Banach spaces: a bound sets approach. Topol. Methods Nonlinear Anal.. 37(2), 303–341 (2011)

  20. Andres, J, Väth, M: Coincidence index for noncompact mappings on nonconvex sets. Nonlinear Funct. Anal. Appl.. 7(4), 619–658 (2002)

  21. Papageorgiou, NS, Kyritsi-Yiallourou, ST: Handbook of Applied Analysis, Springer, Berlin (2009)

  22. Rzezuchowski, T: Scorza-Dragoni type theorem for upper semicontinuous multivalued functions. Bull. Acad. Pol. Sci., Sér. Sci. Math.. 28(1-2), 61–66 (1980)

  23. Andres, J, Górniewicz, L: Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic, Dordrecht (2003)

  24. Pettis, BJ: On the integration in vector spaces. Trans. Am. Math. Soc.. 44(2), 277–304 (1938). Publisher Full Text OpenURL

  25. Bader, R, Kryszewski, W: On the solution set of differential inclusions and the periodic problem in Banach spaces. Nonlinear Anal.. 54(4), 707–754 (2003). Publisher Full Text OpenURL

  26. Hu, S, Papageorgiou, NS: Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic, Dordrecht (1997)

  27. Kamenskii, MI, Obukhovskii, VV, Zecca, P: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter, Berlin (2001)

  28. Deimling, K: Multivalued Differential Equations, de Gruyter, Berlin (1992)

  29. Jin, Y, Zhao, X: Spatial dynamics of a periodic population model with dispersal. Nonlinearity. 22, 1167–1189 (2009). Publisher Full Text OpenURL

  30. Alonso, JM, Mawhin, J, Ortega, R: Bounded solutions of second order semilinear evolution equations and applications to the telegraph equation. J. Math. Pures Appl.. 78, 49–63 (1999). Publisher Full Text OpenURL

  31. Jeffrey, A: Advanced Engineering Mathematics, Harcourt Academic Press, Burlington (2002)

  32. Pozar, DM: Microwave Engineering, Adison-Wesley, New York (1990)