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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Dirichlet problem in Banach spaces: the bound sets approach

Jan Andres1*, Luisa Malaguti2 and Martina Pavlačková1

Author Affiliations

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

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

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Boundary Value Problems 2013, 2013:25  doi:10.1186/1687-2770-2013-25

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/25


Received:19 October 2012
Accepted:16 January 2013
Published:11 February 2013

© 2013 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

The existence and localization result is obtained for a multivalued Dirichlet problem in a Banach space. The upper-Carathéodory and Marchaud right-hand sides are treated separately because in the latter case, the transversality conditions derived by means of bounding functions can be strictly localized on the boundaries of bound sets.

MSC: 34A60, 34B15, 47H04.

Keywords:
Dirichlet problem; bounding functions; solutions in a given set; condensing multivalued operators

1 Introduction

The Dirichlet problem and its special case with homogeneous boundary conditions, usually called the Picard problem, belong to the most frequently studied boundary value problems. A lot of results concerning the standard problem for scalar second-order ordinary differential equations were generalized in various directions.

In Euclidean spaces, besides many extensions to vector equations, vector inclusions were under consideration, e.g., in [1-4]. In abstract spaces, usually in Banach and Hilbert spaces, equations, e.g., in [5-11] and inclusions, e.g., in [9,12,13] were treated.

Sadovskii’s or Darbo’s fixed point theorems, jointly with the usage of a measure of noncompactness, were applied in [5,8,9,11]. Kakutani’s or Ky Fan’s fixed point theorems were applied with the upper and lower solutions technique in [9] and with a measure of noncompactness in [13]. On the other hand, continuation principles were employed in [2,4,7].

The main aim of our present paper is an extension of the finite-dimensional results in [2,4] into infinite-dimensional ones. We were also stimulated by the work of Jean Mawhin in [7], where degree arguments were applied to the Dirichlet problem in a Hilbert space probably for the first time, and in [14], where a bound sets approach was systematically developed. Hence, besides these two approaches, our extension consists in the consideration of differential inclusions in rather general Banach spaces and the usage of a measure of noncompactness. Similar results were already obtained in an analogous way by ourselves for Floquet problems in [15-18].

Besides the existence, the localization of solutions will be obtained in our main theorems (see Theorem 5.1 and Theorem 5.2). Unlike in [10], where the solutions belong to a positively invariant set, in our paper, some trajectories can escape from the prescribed set of candidate solutions. Moreover, the associated bound set need not be compact as in [10]. Similarly, the main difference between our results and those in [9,13] consists in the application of a continuation principle jointly with a bound sets approach, which allows us to check fixed point free boundaries of given bound sets. This, in particular, means that, unlike in [9,13], some trajectories can again escape from the prescribed set of candidate solutions in a transversal way.

Let E be a Banach space (with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M1">View MathML</a>) satisfying the Radon-Nikodym property (e.g., reflexivity) and let us consider the Dirichlet boundary value problem (b.v.p.)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M2">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3">View MathML</a> is an upper-Carathéodory mapping or a globally upper semicontinuous mapping with compact, convex values (for the related definitions, see Section 2).

The main purpose of the present paper is to prove the existence of a Carathéodory solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M4">View MathML</a> to problem (1) in a given set Q. This will be achieved by means of a suitable continuation principle. The crucial condition of the continuation principle described in Section 3 consists in guaranteeing the fixed point free boundary of Q w.r.t. an admissible homotopical bridge starting from (1) (see condition (v) in Proposition 3.1 below). This requirement will be verified by means of Lyapunov-like bounding functions, i.e., via a bound sets technique. That is also why the whole Section 4 is devoted to this technique applied to Dirichlet problem (1). We will distinguish two cases, namely when F is an upper-Carathéodory mapping and when F is globally upper semicontinuous (i.e., a Marchaud mapping). Unlike in the first case, the second one allows us to apply bounding functions which can be strictly localized at the boundaries of given bound sets.

2 Preliminaries

Let E be a Banach space having the Radon-Nikodym property (see, e.g., [[19], pp.694-695]), i.e., if for every finite measure space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M5">View MathML</a> and every vector measure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M6">View MathML</a> of bounded variation, which is absolutely continuous w.r.t. μ, we can find a Bochner integrable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M7">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M8">View MathML</a>

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M9">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M10">View MathML</a> be a closed interval. By the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M11">View MathML</a>, we will mean the set of all Bochner integrable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M12">View MathML</a>. For the definition and properties, see, e.g., [[19], pp.693-701].

The symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M13">View MathML</a> will denote the set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M14">View MathML</a> whose first derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15">View MathML</a> is absolutely continuous. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M16">View MathML</a> and the fundamental theorem of calculus (the Newton-Leibniz formula) holds (see, e.g., [[15], pp.243-244], [[19], pp.695-696]). In the sequel, we will always consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M17">View MathML</a> as a subspace of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18">View MathML</a>.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M20">View MathML</a>, the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M21">View MathML</a> will denote, as usually, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M22">View MathML</a>, where B is the open unit ball in E, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M23">View MathML</a>.

We will 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/2013/1/25/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M24">View MathML</a>) if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M25">View MathML</a>, a nonempty subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M26">View MathML</a> of Y is given. We associate with F its graph <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M27">View MathML</a>, the subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M28">View MathML</a>, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M29">View MathML</a>

A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M30">View MathML</a> is called upper semicontinuous (shortly, u.s.c.) if for each open subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M31">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M32">View MathML</a> is open in X.

A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M30">View MathML</a> is called compact if the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M34">View MathML</a> is contained in a compact subset of Y; it is called quasi-compact if it maps compact sets onto relatively compact sets; and completely continuous if it maps bounded sets onto relatively compact sets.

We say that a multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M35">View MathML</a> with closed values is a step multivalued mapping if there exists a finite family of disjoint measurable subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M37">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M38">View MathML</a> and F is constant on every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M36">View MathML</a>. A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M35">View MathML</a> with closed values is called strongly measurable if there exists a sequence of step multivalued mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M41">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M42">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M43">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M45">View MathML</a> stands for the Hausdorff distance.

It is well known that if Y is a Banach space, then a strongly measurable mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M35">View MathML</a> with compact values possesses a single-valued strongly measurable selection (see, e.g., [12,20]).

A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M47">View MathML</a> is called an upper-Carathéodory mapping if the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M48">View MathML</a> is strongly measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M25">View MathML</a>, the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M50">View MathML</a> is u.s.c. for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M51">View MathML</a> and the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M52">View MathML</a> is compact and convex for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M53">View MathML</a>.

Let us note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M54">View MathML</a> are Banach spaces, then an upper-Carathéodory mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M47">View MathML</a> is weakly superpositionally measurable, i.e., that for each continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M56">View MathML</a>, the composition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M57">View MathML</a> possesses a single-valued measurable selection (see, e.g., [12,20]).

A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M58">View MathML</a> is called Lipschitzian in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M59">View MathML</a> if there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M61">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M63">View MathML</a>.

For more details concerning multivalued analysis, see, e.g., [12,15,20,21].

In the sequel, the measure of noncompactness will also be employed.

Definition 2.1 Let N be a partially ordered set, E be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M64">View MathML</a> denote the family of all nonempty subsets of E. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M65">View MathML</a> is called a measure of noncompactness (m.n.c.) in E if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M66">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M67">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M68">View MathML</a> denotes the closed convex hull of Ω.

An m.n.c. β is called:

(i) monotone if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M69">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M70">View MathML</a>,

(ii) nonsingular if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M71">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73">View MathML</a>,

(iii) invariant with respect to the union with compact sets if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M74">View MathML</a> for every relatively compact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M75">View MathML</a> and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73">View MathML</a>,

(iv) regular when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M77">View MathML</a> if and only if Ω is relatively compact,

(v) algebraically semi-additive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M79">View MathML</a>.

Definition 2.2 An m.n.c. β with values in a cone of a Banach space has the semi-homogeneity property if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M80">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M81">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73">View MathML</a>.

It is obvious that an m.n.c. which is invariant with respect to the union with compact sets is also nonsingular.

The typical example of an m.n.c. is the Hausdorff measure of noncompactnessγ defined, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M73">View MathML</a>, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M84">View MathML</a>

The Hausdorff measure of noncompactness is monotone, nonsingular, algebraically semi-additive and has the semi-homogeneity property.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M85">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M87">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>, all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M89">View MathML</a> and suitable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M90">View MathML</a>, then (cf.[20])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M91">View MathML</a>

(2)

Moreover, for all subsets Ω of E (see, e.g., [18]),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M92">View MathML</a>

(3)

Let us now introduce the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M93">View MathML</a>

(4)

defined on the bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M94">View MathML</a>, where the ordering is induced by the positive cone in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M95">View MathML</a> and where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M96">View MathML</a> denotes the modulus of continuity of a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M97">View MathML</a>.a Such a μ is an m.n.c. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18">View MathML</a>, as shown in the following lemma (proven in [16]), where the properties of μ will be also discussed.

Lemma 2.1The functionμgiven by (4) defines an m.n.c. in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18">View MathML</a>; such an m.n.c. μis monotone, invariant with respect to the union with compact sets and regular.

The m.n.c. μ defined by (4) will be used in order to solve problem (1) (cf. Theorem 5.1).

Definition 2.3 Let E be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M103">View MathML</a>. A multivalued mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M104">View MathML</a> with compact values is called condensing with respect to an m.n.c.β (shortly, β-condensing) if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M105">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M106">View MathML</a>, it holds that Ω is relatively compact.

A family of mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M107">View MathML</a> with compact values is called β-condensing if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M109">View MathML</a>, it holds that Ω is relatively compact.

The following convergence result will be also employed.

Lemma 2.2 (cf. [[15], Lemma III.1.30])

LetEbe a Banach space and assume that the sequence of absolutely continuous functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M110">View MathML</a>satisfies the following conditions:

(i) the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M111">View MathML</a>is relatively compact for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>,

(ii) there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M114">View MathML</a>for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M116">View MathML</a>,

(iii) the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M117">View MathML</a>is weakly relatively compact for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>.

Then there exists a subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M119">View MathML</a> (for the sake of simplicity denoted in the same way as the sequence) converging to an absolutely continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M120">View MathML</a>in the following way:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M119">View MathML</a>converges uniformly toxin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M122">View MathML</a>,

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M123">View MathML</a>converges weakly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M124">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M125">View MathML</a>.

The following lemma is well known when the Banach spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M127">View MathML</a> coincide (see, e.g., [[22], p.88]). The present slight modification for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M128">View MathML</a> was proved in [23].

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M129">View MathML</a>be a compact interval, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M131">View MathML</a>be Banach spaces and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M132">View MathML</a>be a multivalued mapping satisfying the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M133">View MathML</a>has a strongly measurable selection for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M134">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M135">View MathML</a>is u.s.c. for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>,

(iii) the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M52">View MathML</a>is compact and convex for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M138">View MathML</a>.

Assume in addition that for every nonempty, bounded set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M139">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M140">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M141">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M142">View MathML</a>and every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M143">View MathML</a>. Let us define the Nemytskiǐ operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M144">View MathML</a>in the following way: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M145">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M146">View MathML</a>. Then, if sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M147">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M150">View MathML</a>, are such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M151">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M152">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M153">View MathML</a>weakly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M154">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M155">View MathML</a>.

3 Continuation principle

The proof of the main result (cf. Theorem 5.1 below) will be based on the combination of a bound sets technique together with the following continuation principle developed in [16].

Proposition 3.1Let us consider the general multivalued b.v.p.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M156">View MathML</a>

(5)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M157">View MathML</a>is an upper-Carathéodory mapping and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M158">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M159">View MathML</a>be an upper-Carathéodory mapping such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M160">View MathML</a>

(6)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M161">View MathML</a>. Moreover, assume that the following conditions hold:

(i) There exist a closed set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M162">View MathML</a>and a closed, convex set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M163">View MathML</a>with a nonempty interior IntQsuch that each associated problem

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M166">View MathML</a>, has a nonempty, convex set of solutions (denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M167">View MathML</a>).

(ii) For every nonempty, bounded set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M168">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M169">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M170">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M174">View MathML</a>.

(iii) The solution mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a>is quasi-compact andμ-condensing with respect to a monotone and nonsingular m.n.c. μdefined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18">View MathML</a>.

(iv) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a>, the set of solutions of the problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M178">View MathML</a>is a subset of IntQ, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M179">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a>.

(v) For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a>, the solution mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M182">View MathML</a>has no fixed points on the boundary∂QofQ.

Then the b.v.p. (5) has a solution inQ.

The proof of the continuation principle is based on the fact that the family <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a> of problems depending on two parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M166">View MathML</a> is associated to the original b.v.p. (5). This family is defined in such a way that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M186">View MathML</a> is its corresponding solution mapping, then all fixed points of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M187">View MathML</a> are solutions of (5) (see condition (6)).

4 Bound sets technique

The continuation principle formulated in Proposition 3.1 requires, in particular, the existence of a suitable set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M188">View MathML</a> of candidate solutions. The set Q should satisfy the transversality condition (v), i.e., it should have a fixed-point free boundary with respect to the solution mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a>. Since the direct verification of the transversality condition is usually a difficult task, we will devote this section to a bound sets technique which can be used for guaranteeing such a condition. For this purpose, we will define the set Q as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M190">View MathML</a>, where K is nonempty and open in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M191">View MathML</a> denotes its closure.

Hence, let us consider the Dirichlet boundary value problem (1) and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M192">View MathML</a> be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M193">View MathML</a>-function satisfying

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M194">View MathML</a>,

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M195">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M196">View MathML</a>.

Definition 4.1 A nonempty open set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M197">View MathML</a> is called a bound set for the b.v.p. (1) if every solution x of (1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M198">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> does not satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M200">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M201">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M202">View MathML</a> be the Banach space dual to E and let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M203">View MathML</a> the pairing (the duality relation) between E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M202">View MathML</a>, i.e., for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M206">View MathML</a>, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M207">View MathML</a>. The proof of the following proposition is quite analogous to the finite-dimensional case considered in [4]. Nevertheless, for the sake of completeness, we present it here, too.

Proposition 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M208">View MathML</a>be an open set such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M209">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M210">View MathML</a>be an upper-Carathéodory mapping. Assume that the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M211">View MathML</a>has a locally Lipschitzian Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M212">View MathML</a>and satisfies conditions (H1) and (H2). Suppose, moreover, that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M216">View MathML</a>, at least one of the following conditions:

(7)

(8)

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M219">View MathML</a>. ThenKis a bound set for the Dirichlet problem (1).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M220">View MathML</a> be a solution of problem (1). We assume, by a contradiction, that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M222">View MathML</a>. The point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M223">View MathML</a> must lie in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M224">View MathML</a> according to the Dirichlet boundary conditions and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M209">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M226">View MathML</a> is locally Lipschitzian, there exist a neighborhood U of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M227">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M228">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M229">View MathML</a> is Lipschitzian with a constant L. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M230">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M231">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M232">View MathML</a>.

In order to get the desired contradiction, let us define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M233">View MathML</a> as the composition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M234">View MathML</a>. According to the regularity properties of x and V, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M235">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M237">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M239">View MathML</a> is a local maximum point for g. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M240">View MathML</a>. Moreover, there exist points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M242">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M243">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M244">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M245">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M246">View MathML</a> is locally Lipschitzian and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M247">View MathML</a> is absolutely continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M249">View MathML</a> exists for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M250">View MathML</a>. Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M251">View MathML</a>

(9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M252">View MathML</a>

(10)

At first, let us assume that condition (7) holds and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M253">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M255">View MathML</a> exist. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M256">View MathML</a>

and so there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M257">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M258">View MathML</a>, such that for each h,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M259">View MathML</a>

Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M260">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M261">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M258">View MathML</a>, such that for each h,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M263">View MathML</a>

Consequently, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M264">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M265">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M266">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M267">View MathML</a>

Moreover, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M269">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M270">View MathML</a>

According to the Lipschitzianity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271">View MathML</a>, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M272">View MathML</a> is sufficiently small, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M273">View MathML</a>

where L denotes the local Lipschitz constant of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271">View MathML</a> in a neighborhood of x. It implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M275">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M276">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M277">View MathML</a>

according to assumption (7), it leads to a contradiction with inequality (9).

Secondly, let us assume that condition (8) holds and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M278">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M280">View MathML</a> exist. Then it is possible to show, using the same procedure as before, that according to assumption (8),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M281">View MathML</a>

which leads to a contradiction with inequality (10).

Therefore, we get the contradiction in case that at least one of conditions (7), (8) holds which completes the proof. □

If the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M282">View MathML</a> is globally u.s.c. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M283">View MathML</a>, then the transversality conditions can be localized directly on the boundary of K, as will be shown in the following proposition, whose proof is again quite analogous to the finite-dimensional case considered in [2].

Proposition 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M208">View MathML</a>be a nonempty open set such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M285">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3">View MathML</a>be an upper semicontinuous multivalued mapping with compact, convex values. Assume that there exists a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M287">View MathML</a>with a locally Lipschitzian Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M226">View MathML</a>which satisfies conditions (H1) and (H2). Suppose, moreover, that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M291">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M292">View MathML</a>

(11)

the following condition holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M293">View MathML</a>

(12)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M294">View MathML</a>. ThenKis a bound set for problem (1).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M220">View MathML</a> be a solution of problem (1). We assume, by a contradiction, that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M296">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M297">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M285">View MathML</a> and x satisfies Dirichlet boundary conditions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M299">View MathML</a>.

Let us define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M300">View MathML</a> as the composition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M301">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M303">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M304">View MathML</a>, i.e., there is a local maximum for g at the point 0, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M305">View MathML</a>. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M306">View MathML</a> satisfies condition (11).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M226">View MathML</a> is locally Lipschitzian, there exist a neighborhood U of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M308">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M228">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M229">View MathML</a> is Lipschitzian with a constant L.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M311">View MathML</a> be an arbitrary decreasing sequence of positive numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M312">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M314">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M315">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M317">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M318">View MathML</a>, there exists, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M320">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M321">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M260">View MathML</a>, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M323">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M324">View MathML</a>

(13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M325">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M326">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M327">View MathML</a>

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213">View MathML</a> be given. As a consequence of the regularity assumptions imposed on F and of the continuity of both x and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M329">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M330">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M331">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M332">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M333">View MathML</a>

Subsequently, according to the mean-value theorem (see, e.g., [[24], Theorem 0.5.3]), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M334">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M335">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M336">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M337">View MathML</a>

Since F has compact values and ε is arbitrary, we obtain that ζ is a relatively compact set. Thus, there exist a subsequence, for the sake of simplicity denoted as the sequence, of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M338">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M339">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M340">View MathML</a>

(14)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M341">View MathML</a> implying, for the arbitrariness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M343">View MathML</a>

As a consequence of the property (14), there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M345">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M313">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M347">View MathML</a>

(15)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M323">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M349">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M321">View MathML</a>, in view of (13) and (15),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M351">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M352">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M353">View MathML</a>, we have, according to (13), that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M354">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M353">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M356">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M341">View MathML</a>, it is possible to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M358">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M359">View MathML</a>, it holds that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M360">View MathML</a>. By means of the local Lipschitzianity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M359">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M363">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M364">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M365">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M366">View MathML</a>

(16)

If we consider, instead of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M367">View MathML</a>, an increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M368">View MathML</a> of negative numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M369">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M370">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M371">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M372">View MathML</a>, we are able to find, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M374">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M375">View MathML</a>. Therefore, using the same procedure as in the first part of the proof, we obtain, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M150">View MathML</a> sufficiently large, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M377">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M378">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M379">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M313">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M381">View MathML</a>.

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M382">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M365">View MathML</a>, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M384">View MathML</a>

(17)

Inequalities (16) and (17) are in a contradiction with condition (12), because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M385">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M386">View MathML</a> satisfies condition (11) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M387">View MathML</a>. □

Remark 4.1 One can readily check that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M388">View MathML</a>, inequalities (7) and (8), as well as (12), become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M389">View MathML</a>

with t, x, y, w as in Proposition 4.1 or in Proposition 4.2.

The typical case occurs when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M390">View MathML</a> is a Hilbert space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M391">View MathML</a> denotes the scalar product and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M392">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M393">View MathML</a>. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M394">View MathML</a> and it is not difficult to see that conditions (7) and (8), as well as (12), become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M395">View MathML</a>

with t, x, y and w as in Proposition 4.1 or in Proposition 4.2, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M396">View MathML</a>.

Definition 4.2 A <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M193">View MathML</a>-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M398">View MathML</a> with a locally Lipschitzian Fréchet derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M271">View MathML</a> which satisfies conditions (H1), (H2) and all assumptions in Proposition 4.1 or Proposition 4.2 is called a bounding function for problem (1).

5 Existence and localization results

Combining the continuation principle with the bound sets technique, we are ready to state the main result of the paper concerning the solvability and localization of a solution of the multivalued Dirichlet problem (1).

Theorem 5.1Consider the Dirichlet b.v.p. (1), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3">View MathML</a>is an upper-Carathéodory multivalued mapping. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M401">View MathML</a>is an open, convex set containing 0. Furthermore, let the following conditions be satisfied:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M403">View MathML</a>for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M404">View MathML</a>and each bounded<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M405">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M406">View MathML</a>andγis the Hausdorff measure of noncompactness in E.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407">View MathML</a>) For every nonempty, bounded set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M168">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M169">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M410">View MathML</a>

(18)

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M172">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M414">View MathML</a>.

Finally, let there exist a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M211">View MathML</a>with a locally Lipschitzian Fréchet derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M416">View MathML</a>satisfying conditions (H1), (H2), and at least one of conditions (7), (8) for a suitable<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M417">View MathML</a>, all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M418">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M422">View MathML</a>. Then the Dirichlet b.v.p. (1) admits a solution whose values are located in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M191">View MathML</a>.

Proof Let us define the closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M424">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M425">View MathML</a>

and let the set Q of candidate solutions be defined as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M426">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/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M166">View MathML</a>, consider still the associated fully linearized problem

and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a> a solution mapping which assigns to each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M431">View MathML</a> the set of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a>. We will show that the family of the above b.v.p.s <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a> satisfies all assumptions of Proposition 3.1.

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M434">View MathML</a> which, together with the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a>, ensures the validity of (6).

ad (i) In order to verify condition (i) in Proposition 3.1, we need to show that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M436">View MathML</a>, the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a> is solvable with a convex set of solutions. So, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M431">View MathML</a> be arbitrary and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M439">View MathML</a> be a strongly measurable selection of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M440">View MathML</a>. The homogeneous problem corresponding to b.v.p. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M442">View MathML</a>

(19)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M443">View MathML</a>

admits a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M444">View MathML</a> which is one of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a>. This is given, for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M447">View MathML</a>, where G is the Green function associated to the homogeneous problem (19). The Green function G and its partial derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M448">View MathML</a> are defined by (cf., e.g., [[12], pp.170-171])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M449">View MathML</a>

Thus, the set of solutions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a> is nonempty. The convexity of the solution sets follows immediately from the properties of a mapping F and the fact that problems <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M183">View MathML</a> are fully linearized.

ad (ii) Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M452">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M453">View MathML</a>, condition (ii) in Proposition 3.1 is ensured directly by assumption (5).

ad (iii) Since the verification of condition (iii) in Proposition 3.1 is technically the most complicated, it will be subdivided into two parts: (iii1) the quasi-compactness of the solution operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a>, (iii2) the condensity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a> w.r.t. the monotone and nonsingular (cf. Lemma 2.1) m.n.c. μ defined by (4).

ad (iii1) Let us firstly prove that the solution mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a> is quasi-compact. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M458">View MathML</a> is a metric space, it is sufficient to prove the sequential quasi-compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a>. Hence, let us consider the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M460">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M461">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M462">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M463">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M458">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M465">View MathML</a>. Moreover, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M466">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M89">View MathML</a>. Then there exists, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M469">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M470">View MathML</a>

(20)

and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M471">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M472">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M473">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M474">View MathML</a>, there exists a bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M475">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M476">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M478">View MathML</a>. Therefore, there exists, according to condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M169">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M481">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M483">View MathML</a>.

Moreover, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M486">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M487">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M488">View MathML</a> satisfies, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M491">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M492">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M493">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M494">View MathML</a>

Furthermore, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482">View MathML</a> and a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M497">View MathML</a>

Hence, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M498">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499">View MathML</a> are bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M500">View MathML</a> is uniformly integrable.

Since the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M501">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M502">View MathML</a> are converging, we obtain, in view of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M504">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M506">View MathML</a> is relatively compact.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M507">View MathML</a>, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M508">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/2013/1/25/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M509">View MathML</a>

(21)

Moreover, by means of (2), (3), (21) and the semi-homogeneity of the Hausdorff m.n.c.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M510">View MathML</a>

By similar reasonings, we can also get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M511">View MathML</a>

by which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M512">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M513">View MathML</a> are relatively compact for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M488">View MathML</a> satisfies for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M516">View MathML</a> equation (20), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M517">View MathML</a> is relatively compact for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>. Thus, according to Lemma 2.2, there exist a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499">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/2013/1/25/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M520">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M329">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M500">View MathML</a> converges weakly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M525">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M11">View MathML</a>. Therefore, the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a> is quasi-compact.

ad (iii2) In order to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a> is μ-condensing, where μ is defined by (4), we will prove that any bounded subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M529">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M530">View MathML</a> is relatively compact. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M531">View MathML</a> be a sequence such that

Then we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M533">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M534">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M535">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M537">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M539">View MathML</a>

(22)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M540">View MathML</a>

(23)

In view of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402">View MathML</a>), we have, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>,

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M544">View MathML</a> and Θ is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M18">View MathML</a>, by means of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407">View MathML</a>), we get the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M547">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M548">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M478">View MathML</a>. This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M551">View MathML</a> for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M552">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M478">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/2013/1/25/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M554">View MathML</a>, we have

According to (2), (3) and (22), we so obtain for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M557">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M558">View MathML</a>

By the similar reasonings, we can obtain that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M560">View MathML</a>

when starting from condition (23). Subsequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M561">View MathML</a>

yielding

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M562">View MathML</a>

(24)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M563">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M544">View MathML</a>, we so get

and, in view of (24) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413">View MathML</a>), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M567">View MathML</a>

Inequality (24) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M568">View MathML</a>

(25)

Now, we show that both the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M498">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M499">View MathML</a> are equi-continuous. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M571">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M572">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M573">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M112">View MathML</a>. Thus, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M576">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M577">View MathML</a> comes from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407">View MathML</a>), and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M579">View MathML</a> is uniformly integrable. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M580">View MathML</a> is equi-continuous. Moreover, according to (23), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M581">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M482">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>, implying that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M584">View MathML</a> is bounded; consequently, also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M585">View MathML</a> is equi-continuous. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M586">View MathML</a>

In view of (25), we have so obtained that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M587">View MathML</a>

Hence, also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M588">View MathML</a> and since μ is regular, we have that Θ is relatively compact. Therefore, condition (iii) in Proposition 3.1 holds.

ad (iv) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M165">View MathML</a>, the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M178">View MathML</a> has only the trivial solution. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M209">View MathML</a>, condition (iv) in Proposition 3.1 is satisfied.

ad (v) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M592">View MathML</a> be a solution of the b.v.p. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M593">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a>, i.e., a fixed point of the solution mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M175">View MathML</a>. In view of conditions (7), (8) (see Proposition 4.1), K is, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a>, a bound set for the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M597">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M598">View MathML</a>, which ensures condition (v) in Proposition 3.1. □

If the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M282">View MathML</a> is globally u.s.c. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M283">View MathML</a> (i.e., a Marchaud map), then we are able to improve Theorem 5.1 in the following way.

Theorem 5.2Consider the Dirichlet b.v.p. (1), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M3">View MathML</a>is an upper semicontinuous mapping with compact, convex values. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M401">View MathML</a>is an open, convex set containing 0. Moreover, let conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M402">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413">View MathML</a>) from Theorem 5.1 be satisfied.

Furthermore, let there exist a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M211">View MathML</a>with a locally Lipschitz Frechét derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M416">View MathML</a>satisfying (H1) and (H2). Moreover, let, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M608">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M609">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M291">View MathML</a>satisfying (11), condition (12) hold for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M612">View MathML</a>. Then the Dirichlet b.v.p. (1) admits a solution whose values are located in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M191">View MathML</a>.

Proof The verification is quite analogous as in Theorem 5.1 when just replacing the usage of Proposition 4.1 by Proposition 4.2. □

6 Illustrative example

Example 6.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M390">View MathML</a> be a Hilbert space and let us consider the Dirichlet b.v.p.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M615">View MathML</a>

(26)

where

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M616">View MathML</a> is an upper-Carathéodory multivalued mapping and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M617">View MathML</a> is completely continuous for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M619">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M621">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M623">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M624">View MathML</a> is a Carathéodory multivalued mapping such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M625">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M626">View MathML</a> is Lipschitzian for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> with the Lipschitz constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M628">View MathML</a>

Moreover, suppose that

(iii) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M393">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M213">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M631">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M632">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M634">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M636">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M637">View MathML</a>

Then the Dirichlet problem (26) admits, according to Theorem 5.1, a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M638">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M639">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>.

Indeed. The properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641">View MathML</a> guarantee that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641">View MathML</a> satisfies the inequality (cf., e.g., [20])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M643">View MathML</a>

(27)

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M404">View MathML</a> and every bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M645">View MathML</a>, where γ stands for the Hausdorff measure of noncompactness in H.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M646">View MathML</a> is completely continuous and thanks to the algebraic semi-additivity of γ, inequality (27) can be rewritten into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M647">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M648">View MathML</a> and every bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M645">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M650">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M651">View MathML</a> (cf. (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M413">View MathML</a>)).

Moreover, according to the Lipschitzianity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641">View MathML</a>, the following inequalities take place:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M654">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M621">View MathML</a>.

Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M623">View MathML</a>, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M659">View MathML</a>

i.e., (18) in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M407">View MathML</a>).

Finally, in view of Remark 4.1, we can define the bounding function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M661">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M662">View MathML</a>

and the bound set K as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M663">View MathML</a> in order to get a claim.

Remark 6.1 Consider again (26) in a Hilbert space H, but let this time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M664">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M641">View MathML</a> be globally u.s.c. mappings with compact, convex values (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M666">View MathML</a> is compact (cf., e.g., [[15], Proposition I.3.20]) and, in particular, bounded) such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M617">View MathML</a> is a completely continuous mapping for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M669">View MathML</a>

for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M621">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M672">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M673">View MathML</a>.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M674">View MathML</a> is a Lipschitzian mapping for a.a. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a> with the Lipschitz constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M676">View MathML</a>

(iiiusc) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M393">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M631">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M679">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M681">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M682">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M636">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M685">View MathML</a>

Applying now Theorem 5.2, by the analogous arguments as in Example 6.1, the Dirichlet problem (26) admits a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M638">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M687">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M44">View MathML</a>.

Remark 6.2 Since the solution derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15">View MathML</a> takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M690">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M691">View MathML</a>

and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M692">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M693">View MathML</a>, we obtain (under the above assumptions) the implicit inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M694">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M695">View MathML</a>.

Thus, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M696">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M697">View MathML</a>, and subsequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M698">View MathML</a>

Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M699">View MathML</a> is compact, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M700">View MathML</a>

holds with a suitable constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M701">View MathML</a>, and the following estimate holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M702">View MathML</a>

Because of the Dirichlet boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M703">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M704">View MathML</a>, there exists a zero point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M705">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M707">View MathML</a>, by which the same estimates can be also obtained without an explicit usage of the Green function above. Otherwise, it is not so easy to obtain such estimates, because Rolle’s theorem fails in general.

For obtaining the estimation of the solution derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M15">View MathML</a> in a Hilbert space H, one can also apply, under natural assumptions, the p-Nagumo condition derived in [7].

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 the grant PrF_2012_017. 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/2013/1/25/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M96">View MathML</a> is monotone, nonsingular and algebraically subadditive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M99">View MathML</a> (cf., e.g., [20]) and it is equal to zero if and only if all the elements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/25/mathml/M143">View MathML</a> are equi-continuous.

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