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Properties of the solutions set for a class of nonlinear evolution inclusions with nonlocal conditions

Jingrui Zhang1*, Yi Cheng23, Changqin Yuan2 and Fuzhong Cong2

Author Affiliations

1 School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

2 Fundamental Department, Aviation University of Air Force, Changchun, 130022, People’s Republic of China

3 Institute of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

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Boundary Value Problems 2013, 2013:15  doi:10.1186/1687-2770-2013-15

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/15


Received:23 October 2012
Accepted:18 January 2013
Published:5 February 2013

© 2013 Zhang 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

In this paper, we consider the nonlocal problems for nonlinear first-order evolution inclusions in an evolution triple of spaces. Using techniques from multivalued analysis and fixed point theorems, we prove existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term with nonlocal conditions. Also, we prove the existence of extremal solutions and a strong relaxation theorem. Some examples are presented to illustrate the results.

MSC: 34B15, 34B16, 37J40.

Keywords:
evolution inclusions; nonlocal conditions; Leray-Schauder alternative theorem; extremal solutions

1 Introduction

In this paper, we examine the following nonlinear nonlocal problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M2">View MathML</a> is a nonlinear map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M3">View MathML</a> is a bounded linear map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M4">View MathML</a> is a continuous map and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M5">View MathML</a> is a multifunction to be given later. Concerning the function φ, appearing in the nonlocal condition, we mention here four remarkable cases covered by our general framework, i.e.:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M6">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M7">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M8">View MathML</a>;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M9">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M10">View MathML</a> are arbitrary, but fixed and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M11">View MathML</a>.

Many authors have studied the nonlocal Cauchy problem because it has a better effect in the applications than the classical initial condition. We begin by mentioning some of the previous work done in the literature. As far as we know, this study was first considered by Byszewski. Byszewski and Lakshmikantham [1,2] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when F is a single-valued function satisfying Lipschitz-type conditions. The fully nonlinear case was considered by Aizicovici and Lee [3], Aizicovici and McKibben [4], Aizicovici and Staicu [5], García-Falset [6], García-Falset and Reich [7], and Paicu and Vrabie [8]. All these studies were motivated by the practical interests of such nonlocal Cauchy problems. For example, the diffusion of a gas through a thin transparent tube is described by a parabolic equation subjected to a nonlocal initial condition very close to the one mentioned above, see [9]. For the nonlocal problems of evolution equations, in [10], Ntouyas and Tsamatos studied the case with compactness conditions. Subsequently, Byszewski and Akca [11] established the existence of a solution to functional-differential equations when the semigroup is compact and φ is convex and compact on a given ball. In [12], Fu and Ezzinbi studied neutral functional-differential equations with nonlocal conditions. Benchohra and Ntouyas [13] discussed second-order differential equations under compact conditions. For more details on the nonlocal problem, we refer to the papers of [14-18] and the references therein.

It is worth mentioning that many of these documents assume that a nonlocal function meets certain conditions of compactness and A is a strongly continuous semigroup of operators or accretive operators in studying the evolution equations or inclusions with nonlocal conditions. However, one may ask whether there are similar results without the assumption on the compactness or equicontinuity of the semigroup. This article will give a positive answer to this question. The works mentioned above mainly establish the existence of mild solutions for evolution equations or inclusions with nonlocal conditions. However, in the present paper, we consider the cases of a convex and of a nonconvex valued perturbation term in the evolution triple of spaces (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M12">View MathML</a>). We assume the nonlinear time invariant operator A to be monotone and the perturbation term to be multivalued, defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M13">View MathML</a> with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M14">View MathML</a> (not in H). We will establish existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term, which is new for nonlocal problems. Our approach will be based on the techniques and results of the theory of monotone operators, set-valued analysis and the Leray-Schauder fixed point theorem.

We pay attention to the existence of extreme solutions [19] that are not only the solutions of a system with a convexified right-hand side, but also they are solutions of the original system. We prove that, under appropriate hypotheses, such a solution set is dense and codense in the solution set of a system with a convexified right-hand side (‘bang-bang’ principle). Our results extend those of [20] and are similar to those of [21] in an infinite dimensional space. Furthermore, the process of our proofs is much shorter, and our conditions are more general. Finally, some examples are also given to illustrate the effectiveness of our results.

The paper is divided into five parts. In Section 2, we introduce some notations, definitions and needed results. In Section 3, we present some basic assumptions and main results, the proofs of the main results are given based on the Leray-Schauder alternative theorem. In Section 4, the existence of extremal solutions and a relaxation theorem are established. Finally, two examples are presented for our results in Section 5.

2 Preliminaries

In this section we recall some basic definitions and facts from multivalued analysis which we will need in what follows. For details, we refer to the books of Hu and Papageorgiou [22] and Zeidler [23]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M16">View MathML</a> be the Lebesgue measurable space and X be a separable Banach space. Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M17">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M19">View MathML</a>, then the distance from x to A is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M20">View MathML</a>. A multifunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M21">View MathML</a> is said to be measurable if and only if, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M22">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M23">View MathML</a> is measurable. A multifunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M24">View MathML</a> is said to be graph measurable if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M25">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M26">View MathML</a> being the Borel σ-field of X. On <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M27">View MathML</a> we can define a generalized metric, known in the literature as the ‘Hausdorff metric’, by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M28">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M29">View MathML</a>.

It is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M30">View MathML</a> is a complete metric space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M31">View MathML</a> is a closed subset of it. When Z is a Hausdorff topological space, a multifunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M32">View MathML</a> is said to be h-continuous if it is continuous as a function from Z into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M30">View MathML</a>.

Let Y, Z be Hausdorff topological spaces and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M34">View MathML</a>. We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M35">View MathML</a> is ‘upper semicontinuous (USC)’ (resp., ‘lower semicontinuous (LSC)’) if for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M36">View MathML</a> nonempty closed, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M37">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M38">View MathML</a>) is closed in Y. A USC multifunction has a closed graph in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M39">View MathML</a>, while the converse is true if G is locally compact (i.e., for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M40">View MathML</a>, there exists a neighborhood U of y such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M41">View MathML</a> is compact in Z). A multifunction which is both USC and LSC is said to be ‘continuous’. If Y, Z are both metric spaces, then the above definition of LSC is equivalent to saying that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M43">View MathML</a> is upper semicontinuous as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M44">View MathML</a>-valued function. Also, lower semicontinuity is equivalent to saying that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M45">View MathML</a> in Y as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M47">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M15">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M49">View MathML</a>, we denote the Lebesgue-Bochner space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M50">View MathML</a> equipped with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M52">View MathML</a>. A set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M53">View MathML</a> is said to be ‘decomposable’ if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M54">View MathML</a> and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M55">View MathML</a> measurable, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M56">View MathML</a>.

Let H be a real separable Hilbert space, V be a dense subspace of H having structure of a reflexive Banach space, with the continuous embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M57">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M14">View MathML</a> is the topological dual space of V. The system model considered here is based on this evolution triple. Let the embedding be compact. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M59">View MathML</a> denote the pairing of an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M60">View MathML</a> and an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M61">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M62">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M63">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M64">View MathML</a> is the inner product on H. The norm in any Banach space X will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M65">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M66">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M67">View MathML</a>. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M68">View MathML</a> by X. Then the dual space of X is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M69">View MathML</a> and is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. For p, q satisfying the above conditions, from reflexivity of V that both X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> are reflexive Banach spaces (see Zeidler [23], p.411]).

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M72">View MathML</a>, where the derivative in this definition should be understood in the sense of distribution. Furnished with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M73">View MathML</a>, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M74">View MathML</a> becomes a Banach space which is clearly reflexive and separable. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a> embeds into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a> continuously (see Proposition 23.23 of [23]). So, every element in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a> has a representative in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a>. Since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M79">View MathML</a> is compact, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M80">View MathML</a> is also compact (see Problem 23.13 of [23]). The pairing between X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M82">View MathML</a>. By ‘⇀’ we denote the weakly convergence. The following lemmas are still needed in the proof of our main theorems.

Lemma 2.1 (see [24])

IfXis a Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M83">View MathML</a>is nonempty, closed and convex with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M84">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M85">View MathML</a>is an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements are true:

(i) the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M86">View MathML</a>is unbounded;

(ii) the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M35">View MathML</a>has a fixed point, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M88">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M89">View MathML</a>.

Let X be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M90">View MathML</a> be the Banach space of all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M91">View MathML</a> which are Bochner integrable. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M92">View MathML</a> denotes the collection of nonempty decomposable subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M90">View MathML</a>. Now, let us state the Bressan-Colombo continuous selection theorem.

Lemma 2.2 (see [25])

LetXbe a separable metric space and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M94">View MathML</a>be a lower semicontinuous multifunction with closed decomposable values. ThenFhas a continuous selection.

Let X be a separable Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M95">View MathML</a> be the Banach space of all continuous functions. A multifunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M96">View MathML</a> is said to be Carathéodory type if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M98">View MathML</a> is measurable, and for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M100">View MathML</a> is h-continuous (i.e., it is continuous from X to the metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M30">View MathML</a>, where h is a Hausdorff metric).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M102">View MathML</a>, a multifunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M96">View MathML</a> is called integrably bounded on M if there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M104">View MathML</a> such that for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M106">View MathML</a>. A nonempty subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M107">View MathML</a> is called σ-compact if there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M108">View MathML</a> of compact subsets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M109">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M110">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M111">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M112">View MathML</a> is dense in M and σ-compact. The following continuous selection theorem in the extreme point case is due to Tolstonogov [26].

Lemma 2.3 (see [26])

Let the multifunction<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M113">View MathML</a>be Carathéodory type and integrably bounded. Then there exists a continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M114">View MathML</a>such that for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M115">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M116">View MathML</a> , then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M117">View MathML</a>, and if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M118">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M119">View MathML</a>.

3 Main results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M15">View MathML</a>, consider the following evolution inclusions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M121">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M2">View MathML</a> is a nonlinear map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M3">View MathML</a> is a bounded linear map, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M4">View MathML</a> is a continuous map and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M5">View MathML</a> is a multifunction satisfying some conditions mentioned later.

Definition 3.1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M126">View MathML</a> is called a solution to the problem (3.1) iff

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M127">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M129">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M130">View MathML</a> and almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>.

We will need the following hypotheses on the data problem (3.1).

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M132">View MathML</a> is an operator such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M133">View MathML</a> is measurable;

(ii) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M135">View MathML</a> is uniformly monotone and hemicontinuous, that is, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M136">View MathML</a> (independent of t) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M137">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M138">View MathML</a>, and the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M139">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M140">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M141">View MathML</a>;

(iii) there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M142">View MathML</a>, a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M143">View MathML</a> and a nondecreasing continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M144">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M145">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M146">View MathML</a>, a.e. on I;

(iv) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M148">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M149">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M150">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M151">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M152">View MathML</a> is a multifunction such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153">View MathML</a> is graph measurable;

(ii) for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M155">View MathML</a> is LSC;

(iii) there exist a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M156">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M157">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M158">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159">View MathML</a>.

(H3)

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M160">View MathML</a> is a bounded linear self-adjoint operator such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M161">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M146">View MathML</a>, a.e. on I;

(ii) there exists a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M163">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M164">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M165">View MathML</a>.

It is convenient to rewrite the system (3.1) as an operator equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M97">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M168">View MathML</a>

It follows from Theorem 30.A of Zeidler [23] that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M169">View MathML</a> is bounded, monotone, hemicontinuous and coercive. By using the same technique, one can show that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M170">View MathML</a> is bounded and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M171">View MathML</a>

for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M172">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173">View MathML</a>.

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M174">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M175">View MathML</a> stands for the generalized derivative of u, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M176">View MathML</a>

For the proofs of main results, we need the following lemma.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M177">View MathML</a>be an evolution triple and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M178">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M179">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M180">View MathML</a>. Then the linear operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M181">View MathML</a>defined by (3.2) is maximal monotone.

Proof In the sequel we will show that L is maximal monotone. To prove this, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M182">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M183">View MathML</a>

We have to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M185">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M186">View MathML</a>. Due to the arbitrariness of u, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M187">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M190">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M191">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M192">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M193">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M194">View MathML</a>

By the arbitrariness of z, one has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M195">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M196">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M197">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M198">View MathML</a>. It remains to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M184">View MathML</a>. Using the integration by parts formula for functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a> (see Zeidler [23], Proposition 23.23), we obtain from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M201">View MathML</a>

(3.3)

Choose a set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M202">View MathML</a> in H such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M203">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M205">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M206">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M207">View MathML</a>. By (3.3), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M208">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M184">View MathML</a>. This completes the proof. □

Theorem 3.1If hypotheses (H1), (H2) and (H3) hold, the problem (3.1) has at least one solution.

Proof The process of proof is divided into four parts.

Step 1. We claim that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M211">View MathML</a>

(3.4)

has only one solution.

Firstly, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M214">View MathML</a>, we claim that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M215">View MathML</a>

(3.5)

has only one solution. By (H1) and (H3), it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M216">View MathML</a> is bounded, monotone, hemicontinuous and coercive. Moreover, by Lemma 3.1, L is a linear maximal monotone operator. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M217">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M218">View MathML</a> is surjective (see [23], p.868]). The uniqueness is clear. Hence, for the Cauchy problem (3.5) has a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M219">View MathML</a>. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M220">View MathML</a>, then the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M221">View MathML</a> is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M222">View MathML</a>

By (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M223">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M224">View MathML</a>. Take an inner product over (3.5) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M225">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M226">View MathML</a>

By (H1)(ii), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M227">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M228">View MathML</a>

(3.6)

Invoking the Banach fixed point theorem, the operator P has only one fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M229">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M230">View MathML</a> is the uniform solution of (3.4).

Therefore, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M231">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M233">View MathML</a>. By Step 1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M234">View MathML</a> is one-to-one and surjective, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M235">View MathML</a> is well defined.

Step 2.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M236">View MathML</a> is completely continuous.

We only need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M237">View MathML</a> is continuous and maps a bounded set into a relatively compact set. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M238">View MathML</a> is continuous. In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M239">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M241">View MathML</a>. From (H1)(ii) and (H3), we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M244">View MathML</a>, a.e. I as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M246">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M247">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M248">View MathML</a> is continuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M249">View MathML</a> be a bound set, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M250">View MathML</a>, there is a priori bound in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a> for the possible solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M252">View MathML</a> of (3.4). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M253">View MathML</a>

(3.7)

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M254">View MathML</a>

(3.8)

By (H1)(iv),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M255">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M256">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257">View MathML</a>. But

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M258">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M259">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M260">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257">View MathML</a>. So, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M262">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M263">View MathML</a>. Because of the boundedness of operators A, B, we obtain that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M264">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M265">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M266">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M267">View MathML</a>. Therefore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M268">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a> is compactly embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M268">View MathML</a> is relatively compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M274">View MathML</a> be a multivalued Nemitsky operator corresponding to F and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M275">View MathML</a> was defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M276">View MathML</a> a.e. on I.

Step 3.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a> has nonempty, closed, decomposable values and is LSC.

The closedness and decomposability of the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a> are easy to check. For the nonemptiness, note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173">View MathML</a>, by the hypothesis (H2)(i), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153">View MathML</a> is graph measurable, so we apply Aumann’s selection theorem and obtain a measurable map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M281">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M282">View MathML</a> a.e. on I. By the hypothesis (H2)(iii), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M283">View MathML</a>. Thus, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M285">View MathML</a>. To prove the lower semicontinuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a>, we only need to show that every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M288">View MathML</a> is a USC <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M44">View MathML</a>-valued function. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M290">View MathML</a>

(see Hiai and Umegaki [27] Theorem 2.2). We will show that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M291">View MathML</a>, the superlevel set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M292">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M294">View MathML</a> and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. By passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297">View MathML</a> a.e. on I as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>. By (H2)(ii), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M299">View MathML</a> is an upper semicontinuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M44">View MathML</a>-valued function. So, via Fatou’s lemma, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M301">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M302">View MathML</a> and this proves the LSC of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a>. By Lemma 2.2, we obtain a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M304">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M305">View MathML</a>. To finish our proof, we need to solve the fixed point problem: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M306">View MathML</a>.

Since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M307">View MathML</a> is compact, the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M308">View MathML</a> is compact. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M309">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. By using the above relation and the continuity of f, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M313">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M317">View MathML</a> is compact.

Step 4. We claim that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M318">View MathML</a> is bounded.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M319">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M320">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M321">View MathML</a>

By (H1)(iv) and (H3)(i), one has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M322">View MathML</a>

(3.9)

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M323">View MathML</a>

(3.10)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257">View MathML</a>. By using the integration by parts formula, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M325">View MathML</a>

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M326">View MathML</a>. By (3.9), (3.10) and (3.11), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M328">View MathML</a>

(3.12)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M257">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M331">View MathML</a>

(3.13)

Thus, by virtue of the inequalities (3.12) and (3.13), we can find a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M262">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M263">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M334">View MathML</a>. From the boundedness of operators A, B and f, and the continuous embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M335">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M336">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M338">View MathML</a> for some constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M340">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M341">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M319">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M343">View MathML</a>

(3.14)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M319">View MathML</a>.

It follows from (3.14) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M345">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M346">View MathML</a>. Hence, Γ is a bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. So, Γ is a bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M308">View MathML</a> is compact.

Invoking the Leray-Schauder theorem, one has that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M350">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M351">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M230">View MathML</a> is a solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M353">View MathML</a>

(3.15)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M354">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M355">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M356">View MathML</a>, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M357">View MathML</a> which is a solution of the following equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M358">View MathML</a>

(3.16)

By Step 3, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M359">View MathML</a> is uniformly bounded. By the boundedness of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360">View MathML</a>, it follows that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M361">View MathML</a> is uniformly bounded and passing to subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M362">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. Evidently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M364">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. Since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M367">View MathML</a> is compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. Hence, from the hypothesis (H2)(ii), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M313">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M371">View MathML</a>. Since the operator A is hemicontinuous and monotone and B is a continuous linear operator, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M373">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>. Therefore, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M377">View MathML</a> a.e. on I. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M163">View MathML</a> is continuous, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M381">View MathML</a>

Hence, x is a solution of (3.1). The proof is completed. □

Next, we consider the convex case, the assumption on F is as follows:

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M382">View MathML</a> is a multifunction such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153">View MathML</a> is graph measurable;

(ii) for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M155">View MathML</a> has a closed graph; and (H2)(iii) hold.

Theorem 3.2If hypotheses (H1), (H3) and (H4) hold, the problem (3.1) has at least one solution; moreover, the solution set is weakly compact in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>.

Proof The proof is as that of Theorem 3.1. So, we only present those particular points where the two proofs differ.

In this case, the multivalued Nemistsky operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M387">View MathML</a> has nonempty closed, convex values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> and is USC from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> furnished with the weak topology (denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391">View MathML</a>). The closedness and convexity of the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a> are clear. To prove the nonemptiness, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M394">View MathML</a> be a sequence of step functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M395">View MathML</a> in H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M396">View MathML</a> a.e. on I. Then by virtue of the hypothesis (H4)(i), for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M397">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M398">View MathML</a> admits a measurable selector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M399">View MathML</a>. From the hypothesis (H4)(iii), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M400">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M401">View MathML</a> is uniformly integrable. So, by the Dunford-Pettis theorem, and by passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M402">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. Then from Theorem 3.1 in [28], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M404">View MathML</a>

the last inclusion being a consequence of the hypothesis (H4)(ii). So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M405">View MathML</a>, which means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a> is nonempty.

Next, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a> is USC from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391">View MathML</a>. Let Ξ be a nonempty and weakly closed subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. Obviously, it is sufficient to show that the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M411">View MathML</a>

is closed. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M412">View MathML</a> and assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. Passing to a subsequence, we can get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297">View MathML</a> a.e. on I. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M416">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M397">View MathML</a>. Then by virtue of the hypothesis (H4)(iii), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M418">View MathML</a>

So, by the Dunford-Pettis theorem, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M419">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391">View MathML</a>. As before, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M421">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M422">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M423">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. This proves the upper semicontinuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M277">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M391">View MathML</a>.

We consider the following fixed point problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M428">View MathML</a>

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M429">View MathML</a> is completely continuous, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M430">View MathML</a> is USC and maps bounded sets into relatively compact sets. We easily check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M431">View MathML</a>

is bounded, as a proof of Theorem 3.1. Invoking the Leray-Schauder fixed point theorem, one has that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M350">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M433">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M230">View MathML</a> is a solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M435">View MathML</a>

(3.17)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M354">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M355">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M356">View MathML</a>, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M357">View MathML</a> which is a solution of the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M440">View MathML</a>

(3.18)

By Step 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M359">View MathML</a> is uniformly bounded. By the boundedness of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360">View MathML</a>, it follows that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M361">View MathML</a> is uniformly bounded and, passing to subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M444">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M373">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>. Evidently, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M450">View MathML</a>, by virtue of the hypothesis (H4)(iv), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M451">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M452">View MathML</a> is uniformly integrable. So, by the Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M453">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. Therefore, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M377">View MathML</a> a.e. on I. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M163">View MathML</a> is continuous, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M460">View MathML</a>

Hence, evidently x is a solution of (3.1). As in the proof of Theorem 3.1, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M461">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M346">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M463">View MathML</a> is uniformly bounded. So, by the Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. As before, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M466">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M468">View MathML</a>. Thus, S is weakly compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. □

4 Relaxation theorem

Now, we prove the existence of extremal solutions and a strong relaxation theorem. Consider the extremal problem of the following evolution inclusion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M470">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M471">View MathML</a> denotes the extremal point set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M472">View MathML</a>. We need the following hypothesis:

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M473">View MathML</a> is a multifunction such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M153">View MathML</a> is graph measurable;

(ii) for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M155">View MathML</a> is h-continuous; and (H2)(iii) holds.

Theorem 4.1If hypotheses (H1), (H3) and (H5) hold, then the problem (4.1) has at least one solution.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M477">View MathML</a>, as in the proof of Theorem 3.1, we obtain a priori bound for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M478">View MathML</a>. We know that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M479">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M480">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M481">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M482">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M483">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M484">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M485">View MathML</a>. We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M486">View MathML</a> a.e. on I for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M487">View MathML</a>. By Theorem 3.1, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M488">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M490">View MathML</a> is well defined. So, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M491">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M492">View MathML</a> is a compact convex subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M494">View MathML</a> is convex. We only need to show the compactness. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M495">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M496">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M497">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M498">View MathML</a>. By the definition of W, W is uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499">View MathML</a>. By the Dunford-Pettis theorem, passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M500">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M502">View MathML</a>. From the definition of W, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M503">View MathML</a>

Therefore, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360">View MathML</a> is bounded. Because of the compactness of the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M505">View MathML</a>, we have that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M506">View MathML</a> is relatively compact. So, by passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. Moreover, by the boundedness of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M360">View MathML</a>, it follows that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M361">View MathML</a> is uniformly bounded and, passing to subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M444">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a>. Since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M220">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M505">View MathML</a> is compact, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M311">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M240">View MathML</a> in H for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M520">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M521">View MathML</a> (m being the Lebesgue measure on R). Since A is hemicontinuous and monotone and B is a continuous linear operator, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M372">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M373">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M70">View MathML</a> and as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M46">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M526">View MathML</a> a.e. on I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M128">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M528">View MathML</a>

(4.2)

Taking the inner product above with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M529">View MathML</a> and integrating from 0 to T, one can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M530">View MathML</a>

(4.3)

By the hypothesis (H1)(iii), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M531">View MathML</a>

(4.4)

So, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M532">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M533">View MathML</a>

Using the integration by parts formula for functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a> (see Zeidler [26], Proposition 23.23), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M536">View MathML</a>

(4.5)

By (4.5), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M537">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M297">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M540">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M502">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M542">View MathML</a> is compact. From Lemma 2.3, we can find a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M543">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M544">View MathML</a> a.e. on I for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M545">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M546">View MathML</a> is a compact operator. On applying the Schauder fixed point theorem, there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M545">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M548">View MathML</a>. This is a solution of (4.1), and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M549">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. □

For the relation theorem of the problem (4.1), we need the following definition and hypotheses.

Definition 4.1 A Carathéodory function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M551">View MathML</a> is said to be a Kamke function if it is integrally bounded on the bounded sets, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M552">View MathML</a> and the unique solution of the differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M553">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M554">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M555">View MathML</a>.

(H6) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, there exists a Kamke function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M557">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M558">View MathML</a>

and (H5) hold.

Theorem 4.2If hypotheses (H1), (H3) and (H6) hold, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M559">View MathML</a>, where the closure is taken in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M468">View MathML</a>, then there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M562">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M563">View MathML</a> a.e. on I such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M564">View MathML</a>

(4.6)

As before, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M565">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M492">View MathML</a> is a compact convex subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M568">View MathML</a>, we define the multifunction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M569">View MathML</a>

Clearly, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M571">View MathML</a>, and it is graph measurable. On applying Aumann’s selection theorem, we get a measurable function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M572">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M573">View MathML</a> almost everywhere on I. So, we define the multifunction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M574">View MathML</a>

We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M575">View MathML</a> has nonempty and decomposable values. It follows from Theorem 3 of [29] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M576">View MathML</a> is LSC. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M577">View MathML</a> is LSC and has closed and decomposable values. So, we apply Lemma 2.2 to get a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M578">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M579">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M568">View MathML</a>. Invoking II-Theorem 8.31 of [22] (in [22], p.260]), we can find a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M581">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M582">View MathML</a> almost everywhere on I, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M583">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M584">View MathML</a>. Now, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M585">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M586">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M587">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M588">View MathML</a> a.e. on I with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M589">View MathML</a>, so we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M590">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499">View MathML</a>. We consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M592">View MathML</a>

(4.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M593">View MathML</a>. We see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M594">View MathML</a> is a compact operator and by the Schauder fixed point theorem, we obtain a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M595">View MathML</a> of (4.1). We see that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M596">View MathML</a> is uniformly bounded. So, by passing to a subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M597">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M75">View MathML</a>. From the proof of Theorem 4.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M599">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M601">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M602">View MathML</a>. So, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M603">View MathML</a>

However,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M604">View MathML</a>

(4.8)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M605">View MathML</a>

(4.9)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M590">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M499">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M599">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M271">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M610">View MathML</a>

(4.10)

Hence, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M611">View MathML</a>, one has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M612">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M613">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M614">View MathML</a>

(4.11)

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M615">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M616">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M617">View MathML</a>. By (4.7), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M618">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M585">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M620">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M621">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M622">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M623">View MathML</a>, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M624">View MathML</a>. Also, S is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M76">View MathML</a> (see the proof of Theorem 3.2), thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M626">View MathML</a>. □

5 Examples

As an application of the previous results, we introduce two examples. Let Ω be a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M627">View MathML</a> with smooth boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M628">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M629">View MathML</a>. Firstly, consider the following nonlinear evolution equation with a discontinuous right-hand side:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M630">View MathML</a>

(5.1)

The p-Laplacian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M631">View MathML</a> arises in many applications such as Finsler geometry and non-Newtonian fluids. In [30], Liu showed the existence of anti-periodic solutions to the problem (5.1) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M632">View MathML</a> is continuous.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M632">View MathML</a> is not continuous, the problem (5.1) need not have solutions. To obtain an existence theorem for (5.1), we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M632">View MathML</a>. So, we introduce the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M635">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M636">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M637">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M638">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M639">View MathML</a>

Then, instead of (5.1), we study the following multivalued nonlinear evolution inclusion:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M640">View MathML</a>

(5.2)

The hypotheses on the data of this problem (5.1) are the following:

(H7)

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M641">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M480">View MathML</a>) are Nemitsky-measurable, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M643">View MathML</a> for all measurable, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M644">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M480">View MathML</a>) is measurable;

(ii) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M646">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M647">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M648">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M159">View MathML</a>.

In this case, the evolution triple is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M650">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M651">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M652">View MathML</a>. From the Sobolev embedding theorem, we see that all embeddings are compact. Let us define the following operator on V:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M653">View MathML</a>

By the monotone property of p-Laplacian, it is easy to verify that A satisfies our hypothesis (H1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M654">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M655">View MathML</a>

The hypothesis (H7) implies that (H4) is satisfied. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M656">View MathML</a> is lower semicontinuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M657">View MathML</a> is upper semicontinuous, and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M658">View MathML</a> is USC (see [22], Example 2.8, p.371]). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M659">View MathML</a>, it is easy to check that φ satisfies our hypothesis (H3)(ii). Then, we rewrite equivalently (5.1) as (3.1) , with A and F as above. Finally, we can apply Theorem 3.2 to the problem (5.1) and obtain the following.

Theorem 5.1If the hypothesis (H7) holds, then the problem (5.1) has a nonempty set of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M660">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M661">View MathML</a>.

Secondly, we present an example of a quasilinear distributed parameter control system, with a priori feedback (i.e., state dependent control constraint set). So, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M662">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M663">View MathML</a> be a bounded domain with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M664">View MathML</a>-boundary Γ. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M665">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M666">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M667">View MathML</a>. We consider the following control system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M668">View MathML</a>

(5.3)

The hypotheses on the data (5.3) are the following:

(H8) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M669">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M670">View MathML</a> are functions such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M671">View MathML</a> is measurable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M672">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M673">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M674">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M675">View MathML</a> for all almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M676">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M677">View MathML</a> with a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M678">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M679">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M680">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M681">View MathML</a> for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M680">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M683">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M676">View MathML</a>.

(H9) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M685">View MathML</a> satisfies the following:

(i) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M687">View MathML</a> is measurable;

(ii) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M689">View MathML</a> is continuous;

(iii) for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M692">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M693">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M694">View MathML</a>.

(H10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M695">View MathML</a> is a multifunction such that

(i) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M697">View MathML</a> is measurable;

(ii) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M699">View MathML</a> is h-continuous;

(iii) for almost all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M688">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M686">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M702">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M703">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M704">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M705">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M706">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M707">View MathML</a> is an evolution triple with compact embeddings. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M708">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M160">View MathML</a> be the operators defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M710">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M711">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M712">View MathML</a>.

Evidently, using the hypothesis (H8) , it is straightforward to check that A, B, φ satisfy hypotheses (H1), (H3). Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M713">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M714">View MathML</a>

Using hypotheses (H9) and (H10), it is straightforward to check that F satisfies the hypothesis (H5).

Rewrite the problem (5.3) in the following equivalent evolution inclusion form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M715">View MathML</a>

(5.4)

It is easy to get the following theorem by applying Theorem 4.1 to the problem (5.3).

Theorem 5.2If hypotheses (H8)-(H10) hold, then the problem (5.3) has one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M716">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/15/mathml/M717">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JZ and YC carried out the main part of this manuscript. CY participated in the discussion and corrected the main theorem. FC provided all examples for our results. All authors read and approved the final manuscript.

Acknowledgements

The authors are in debt to the anonymous referees whose comments helped them to improve the final version of this paper. This work is partially supported by the National Natural Science Foundation of China (No. 11172036, 11171350, 10902125) and the Natural Science Foundation of Jilin Province Grants 201115133.

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