Discontinuous Parabolic Problems with a Nonlocal Initial Condition

Boundary Value Problems20102011:965759

DOI: 10.1155/2011/965759

Received: 28 February 2010

Accepted: 13 June 2010

Published: 5 July 2010

Abstract

We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Green's function, integral representation of solutions, the method of upper and lower solutions, and fixed point theorems for multivalued operators.

1. Introduction

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq1_HTML.gif be a an open bounded domain in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq3_HTML.gif with a smooth boundary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq4_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq6_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq7_HTML.gif is a positive real number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq8_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq9_HTML.gif is smooth and any point on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq10_HTML.gif satisfies the inside (and outside) strong sphere property (see [1]). For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq11_HTML.gif we denote its partial derivatives in the distributional sense (when they exist) by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq13_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq14_HTML.gif

In this paper, we study the following parabolic differential equation with a nonlocal initial condition
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ1_HTML.gif
(11)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq15_HTML.gif is not necessarily continuous, but is such that for every fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq16_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq17_HTML.gif is measurable and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq18_HTML.gif is of bounded variations over compact interval in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq19_HTML.gif and nondecreasing, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq20_HTML.gif is continuous; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq21_HTML.gif is a strongly elliptic operator given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ2_HTML.gif
(12)

Discontinuous parabolic problems have been studied by many authors, see for instance [25]. Parabolic problems with integral conditions appear in the modeling of concrete problems, such as heat conduction [610] and in thermoelasticity [11].

In order to investigate problem (1.1), we introduce some notations, function spaces, and notions from set-valued analysis.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq22_HTML.gif denote the Banach space of all continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq23_HTML.gif , equipped with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq24_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq25_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq27_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq28_HTML.gif For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq29_HTML.gif we say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq30_HTML.gif is in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq31_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq32_HTML.gif is measurable and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq33_HTML.gif in which case we define its norm by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ3_HTML.gif
(13)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq34_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq35_HTML.gif denote the Sobolev space of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq36_HTML.gif having first generalized derivatives in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq37_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq38_HTML.gif be its corresponding dual space. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq39_HTML.gif and they form an evolution triple with all embeddings being continuous, dense, and compact (see [2, 12]). The Bochner space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq40_HTML.gif (see [13]) is the set of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq41_HTML.gif with generalized derivative http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq42_HTML.gif For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq43_HTML.gif we define its norm by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ4_HTML.gif
(14)

Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq44_HTML.gif is a separable reflexive Banach space. The embedding of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq45_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq46_HTML.gif is continuous and the embedding http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq47_HTML.gif is compact.

Now, we introduce some facts from set-valued analysis. For complete details, we refer the reader to the following books. [1416]. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq49_HTML.gif be Banach spaces. We will denote the set of all subsets, of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq50_HTML.gif having property http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq51_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq52_HTML.gif For instance, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq53_HTML.gif denotes the set of all nonempty subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq54_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq55_HTML.gif means http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq56_HTML.gif closed in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq57_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq58_HTML.gif we have the bounded subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq59_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq60_HTML.gif for convex subsets, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq61_HTML.gif for compact subsets and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq62_HTML.gif for compact and convex subsets. The domain of a multivalued map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq63_HTML.gif is the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq64_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq65_HTML.gif is convex (closed) valued if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq66_HTML.gif is convex (closed) for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq67_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq68_HTML.gif is bounded on bounded sets if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq69_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq70_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq71_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq72_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq73_HTML.gif is called upper semicontinuous (u.s.c.) on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq74_HTML.gif if for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq75_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq76_HTML.gif is nonempty, and for each open subset http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq77_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq78_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq79_HTML.gif , there exists an open neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq80_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq81_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq82_HTML.gif In terms of sequences, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq83_HTML.gif is usc if for each sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq85_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq86_HTML.gif is a closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq87_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq88_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq89_HTML.gif

The set-valued map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq90_HTML.gif is called completely continuous if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq91_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq92_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq93_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq94_HTML.gif is completely continuous with nonempty compact values, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq95_HTML.gif is usc if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq96_HTML.gif has a closed graph (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq98_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq99_HTML.gif ). When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq100_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq101_HTML.gif has a fixed point if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq102_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq103_HTML.gif A multivalued map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq104_HTML.gif is called measurable if for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq105_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq106_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq107_HTML.gif is measurable. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq108_HTML.gif denotes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq109_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq110_HTML.gif The Kuratowski measure of noncompactness (see [15, page 113]) of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq111_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ5_HTML.gif
(15)

The Kuratowski measure of noncompactness satisfies the following properties.

(i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq112_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq113_HTML.gif is compact;

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq114_HTML.gif

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq115_HTML.gif

(iv) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq117_HTML.gif ;

(v) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq118_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq119_HTML.gif denotes the convex hull of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq120_HTML.gif .

Definition 1.1 (see [17]).

A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq121_HTML.gif is called N- measurable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq122_HTML.gif if for every measurable function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq123_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq124_HTML.gif is measurable.

Examples of N- measurable functions are Carathéodory functions, Baire measurable functions.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq126_HTML.gif Then (see [17, Proposition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq127_HTML.gif ]) the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq128_HTML.gif is lower semicontinuous, that is, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq129_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq130_HTML.gif is open for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq131_HTML.gif , and the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq132_HTML.gif is upper semicontinuous, that is, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq133_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq134_HTML.gif is open for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq135_HTML.gif . Moreover, the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq137_HTML.gif are nondecreasing.

Definition 1.2.

The multivalued function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq138_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq139_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq140_HTML.gif is called N- measurable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq141_HTML.gif if both functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq143_HTML.gif are N- measurable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq144_HTML.gif .

Definition 1.3.

The operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq145_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq146_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ6_HTML.gif
(16)

is called the Nemitskii operator of the multifunction http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq147_HTML.gif

Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq148_HTML.gif is an N- measurable and upper semicontinuous multivalued function with compact and convex values, we have the following properties for the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq149_HTML.gif (see [17, Corollary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq150_HTML.gif ]).

Lemma 1.4.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq151_HTML.gif is N-measurable, compact and convex-valued, upper semicontinuous and maps bounded sets into precompact sets.

We will consider solutions of problem (1.1) as solutions of the following parabolic problem with multivalued right-hand side:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ7_HTML.gif
(17)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq152_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq153_HTML.gif As pointed out in [15, Example http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq154_HTML.gif page 5], this is the most general upper semicontinuous set-valued map with compact and convex values in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq155_HTML.gif .

Theorem 1.5 (see [18]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq156_HTML.gif be a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq157_HTML.gif a condensing map. If the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq158_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq159_HTML.gif is bounded, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq160_HTML.gif has a fixed point.

We remark that a compact map is the simplest example of a condensing map.

2. The Linear Problem

We will assume throughout this paper that the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq161_HTML.gif are Hölder continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq162_HTML.gif and moreover, there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq163_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq164_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ8_HTML.gif
(21)
Given a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq165_HTML.gif the linear parabolic problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ9_HTML.gif
(22)

is well known and completely solved (see the books [1, 19, 20]).

The linear homogeneous problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ10_HTML.gif
(23)

has only the trivial solution. There exists a unique function, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq166_HTML.gif called Green's function corresponding to the linear homogeneous problem. This function satisfies the following (see [1, 20]):

(i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq167_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq168_HTML.gif

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq170_HTML.gif

(iv) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq171_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq172_HTML.gif

(v) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq174_HTML.gif are continuous functions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq175_HTML.gif

(vi) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq176_HTML.gif for some positive constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq177_HTML.gif (see [19]);

(vii)for any Hölder continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq178_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq179_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq180_HTML.gif , given for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq181_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq182_HTML.gif is the unique classical solution, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq183_HTML.gif of the nonhomogeneous problem (2.2).

It is clear from property (vi) above that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq184_HTML.gif Also, the integral representation in (vii) implies that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq185_HTML.gif is continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq186_HTML.gif

Lemma 2.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq187_HTML.gif then (2.2) has a unique weak solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq188_HTML.gif Moreover, there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq189_HTML.gif , depending only on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq191_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ11_HTML.gif
(24)

Proof.

Consider the following representation (see property (vii) above):
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ12_HTML.gif
(25)
Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq192_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ13_HTML.gif
(26)
Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq193_HTML.gif is a bounded linear operator with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ14_HTML.gif
(27)
Then for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq194_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ15_HTML.gif
(28)
This implies that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq195_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ16_HTML.gif
(29)
Minkowski's inequality leads to
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ17_HTML.gif
(210)

3. Problem with a Discontinuous Nonlinearity

In this section, we investigate the multivalued problem (1.7). We define the notion of a weak solution.

Definition 3.1.

A solution of (1.7) is a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq196_HTML.gif such that

(i)there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq197_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq198_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq199_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq200_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq201_HTML.gif

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq202_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq203_HTML.gif

We introduce the notion of lower and upper solutions of problem (1.7).

Definition 3.2.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq204_HTML.gif is a weak lower solution of (1.7) if

(i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq205_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq206_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq207_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq208_HTML.gif

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq209_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq210_HTML.gif

Definition 3.3.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq211_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq212_HTML.gif is a weak upper solution of (1.7) if

(j) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq213_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq214_HTML.gif

(jj) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq215_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq216_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq217_HTML.gif

(jjj) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq218_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq219_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq220_HTML.gif

We will assume that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq221_HTML.gif , generating the multivalued function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq222_HTML.gif , is N- measurable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq223_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq224_HTML.gif is an N- measurable, upper semicontinuous multivalued function with nonempty, compact, and convex values. In addition, we will need the following assumptions:

(H1)there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq225_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq226_HTML.gif    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq227_HTML.gif

(H2)there exist a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq228_HTML.gif and an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq229_HTML.gif of (1.7) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq230_HTML.gif ;

(H3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq231_HTML.gif is continuous, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq232_HTML.gif is nondecreasing with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq233_HTML.gif

We state and prove our main result.

Theorem 3.4.

Assume that (H1), (H2), and (H3) are satisfied. Then the multivalued problem (1.7) has at least one solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq234_HTML.gif

Proof.

First, it is clear that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq235_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ18_HTML.gif
(31)
is continuous and uniformly bounded. Consider the modified problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ19_HTML.gif
(32)
We show that possible solutions of (3.2) are a priori bounded. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq236_HTML.gif be a solution of (3.2). It follows from the definition and the representation (2.5) that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq237_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ20_HTML.gif
(33)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq238_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq239_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq240_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq241_HTML.gif is uniformly bounded there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq242_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq243_HTML.gif Also, assumption (H1) implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq244_HTML.gif The relation (3.3) together with Lemma 2.1 yields
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ21_HTML.gif
(34)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq245_HTML.gif depends only on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq246_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq247_HTML.gif

It is clear that solutions of (3.2) are fixed point of the multivalued operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq248_HTML.gif , defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ22_HTML.gif
(35)
Here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq249_HTML.gif is a single-valued operator defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ23_HTML.gif
(36)
and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq250_HTML.gif is a multivalued operator defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ24_HTML.gif
(37)

Claim 1.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq251_HTML.gif is compact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq252_HTML.gif . Since the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq253_HTML.gif is continuous and the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq254_HTML.gif is uniformly bounded http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq255_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq256_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq257_HTML.gif Also, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq258_HTML.gif is continuous and has no singularity for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq259_HTML.gif . It follows that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq260_HTML.gif is continuous and there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq261_HTML.gif  depending only on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq262_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq263_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq264_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq265_HTML.gif is uniformly bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq266_HTML.gif Since the embedding http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq267_HTML.gif is compact it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq268_HTML.gif is compact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq269_HTML.gif

Claim 2.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq270_HTML.gif is also compact in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq271_HTML.gif . This follows from the continuity of the Green's function and the properties of the Nemitski operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq272_HTML.gif See Lemma 1.4.

Claim 3.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq273_HTML.gif that is, it is a condensing multifunction http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq274_HTML.gif We have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq275_HTML.gif

Also Lemma 1.4 implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq276_HTML.gif has nonempty, compact, convex values. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq277_HTML.gif is single-valued, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq278_HTML.gif has nonempty compact and convex values. We show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq279_HTML.gif has a closed graph. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq280_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq281_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq282_HTML.gif We show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq283_HTML.gif Now, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq284_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq285_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq286_HTML.gif It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq287_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq288_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq289_HTML.gif We can use the last part of Lemma http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq290_HTML.gif in [13] to conclude that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq291_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq292_HTML.gif which, in turn, implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq293_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq294_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq295_HTML.gif This will imply that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq296_HTML.gif is upper semicontinuous.

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq297_HTML.gif is condensing. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq298_HTML.gif t remains to show that the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq299_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq300_HTML.gif is bounded; but this is a consequence of inequality (3.4). Theorem 1.5 implies that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq301_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq302_HTML.gif which is a solution of (3.2).

We, now, show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq303_HTML.gif We prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq304_HTML.gif It follows from the definition of a solution of (3.2) that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq305_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq306_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq307_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ25_HTML.gif
(38)
On the other hand, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq308_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ26_HTML.gif
(39)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq309_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq310_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq311_HTML.gif Then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ27_HTML.gif
(310)

Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq312_HTML.gif and the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq313_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq314_HTML.gif are nondecreasing, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq315_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq316_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq317_HTML.gif We can show in a similar way that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq318_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq319_HTML.gif In this case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq320_HTML.gif , and (3.2) reduces to (1.7). Therefore, problem (1.7) has a solution, and consequently, (1.1) has a solution.

4. Example

Consider the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ28_HTML.gif
(41)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq321_HTML.gif It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq322_HTML.gif is a classical solution of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ29_HTML.gif
(42)
and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq323_HTML.gif is a classical solution of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ30_HTML.gif
(43)

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq324_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq325_HTML.gif is a solution of the problem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq326_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq327_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq328_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq329_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq330_HTML.gif is an upper solution of problem (4.1) provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq331_HTML.gif

Similarly, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq332_HTML.gif be a solution of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq333_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq334_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq335_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq336_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq337_HTML.gif is a lower solution of problem (4.1) provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq338_HTML.gif

Declarations

Acknowledgments

This work is a part of a research project FT-090001. The author is grateful to King Fahd University of Petroleum and Minerals for its constant support. Also, he would like to thank the reviewers for comments that led to the improvement of the original manuscript.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals

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Copyright

© Abdelkader Boucherif. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.