Open Access

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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq3_HTML.gif with a smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq4_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq6_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq7_HTML.gif is a positive real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq8_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq9_HTML.gif is smooth and any point on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq13_HTML.gif https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ1_HTML.gif
(11)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq16_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq17_HTML.gif is measurable and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq19_HTML.gif and nondecreasing, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq20_HTML.gif is continuous; https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq23_HTML.gif , equipped with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq24_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq25_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq26_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq27_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq28_HTML.gif For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq29_HTML.gif we say that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq30_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq31_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq32_HTML.gif is measurable and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ3_HTML.gif
(13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq34_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq35_HTML.gif denote the Sobolev space of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq36_HTML.gif having first generalized derivatives in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq37_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq38_HTML.gif be its corresponding dual space. Then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq41_HTML.gif with generalized derivative https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq42_HTML.gif For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq43_HTML.gif we define its norm by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ4_HTML.gif
(14)

Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq45_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq46_HTML.gif is continuous and the embedding https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq48_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq50_HTML.gif having property https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq51_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq52_HTML.gif For instance, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq54_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq55_HTML.gif means https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq56_HTML.gif closed in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq57_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq58_HTML.gif we have the bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq59_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq60_HTML.gif for convex subsets, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq61_HTML.gif for compact subsets and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq63_HTML.gif is the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq64_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq65_HTML.gif is convex (closed) valued if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq66_HTML.gif is convex (closed) for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq67_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq68_HTML.gif is bounded on bounded sets if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq69_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq70_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq71_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq72_HTML.gif    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq74_HTML.gif if for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq75_HTML.gif the set https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq77_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq78_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq79_HTML.gif , there exists an open neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq80_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq81_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq82_HTML.gif In terms of sequences, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq83_HTML.gif is usc if for each sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq85_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq86_HTML.gif is a closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq87_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq88_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq89_HTML.gif

The set-valued map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq90_HTML.gif is called completely continuous if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq91_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq92_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq93_HTML.gif If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq95_HTML.gif is usc if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq96_HTML.gif has a closed graph (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq98_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq99_HTML.gif ). When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq100_HTML.gif then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq102_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq103_HTML.gif A multivalued map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq104_HTML.gif is called measurable if for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq105_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq106_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq107_HTML.gif is measurable. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq108_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq109_HTML.gif https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq111_HTML.gif is defined by
https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq112_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq113_HTML.gif is compact;

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

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

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

(v) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq118_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq119_HTML.gif denotes the convex hull of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq121_HTML.gif is called N- measurable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq122_HTML.gif if for every measurable function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq123_HTML.gif the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq126_HTML.gif Then (see [17, Proposition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq127_HTML.gif ]) the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq129_HTML.gif the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq130_HTML.gif is open for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq131_HTML.gif , and the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq133_HTML.gif the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq134_HTML.gif is open for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq135_HTML.gif . Moreover, the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq136_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq138_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq139_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq140_HTML.gif is called N- measurable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq141_HTML.gif if both functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq143_HTML.gif are N- measurable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq144_HTML.gif .

Definition 1.3.

The operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq145_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq146_HTML.gif defined by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq147_HTML.gif

Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq149_HTML.gif (see [17, Corollary https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq150_HTML.gif ]).

Lemma 1.4.

https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ7_HTML.gif
(17)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq152_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq153_HTML.gif As pointed out in [15, Example https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq155_HTML.gif .

Theorem 1.5 (see [18]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq156_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq157_HTML.gif a condensing map. If the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq158_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq159_HTML.gif is bounded, then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq161_HTML.gif are Hölder continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq162_HTML.gif and moreover, there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq163_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq164_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ8_HTML.gif
(21)
Given a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq165_HTML.gif the linear parabolic problem
https://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
https://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, https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq167_HTML.gif

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

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

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

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

(vi) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq176_HTML.gif for some positive constants https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq178_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq179_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq180_HTML.gif , given for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq181_HTML.gif by https://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, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq185_HTML.gif is continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq186_HTML.gif

Lemma 2.1.

If https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq188_HTML.gif Moreover, there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq189_HTML.gif , depending only on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq190_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq191_HTML.gif such that
https://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):
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ12_HTML.gif
(25)
Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq192_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ13_HTML.gif
(26)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq193_HTML.gif is a bounded linear operator with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ14_HTML.gif
(27)
Then for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq194_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ15_HTML.gif
(28)
This implies that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq195_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ16_HTML.gif
(29)
Minkowski's inequality leads to
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq196_HTML.gif such that

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

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

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

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

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

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

Definition 3.3.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq211_HTML.gif https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq213_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq214_HTML.gif

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

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

We will assume that the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq221_HTML.gif , generating the multivalued function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq222_HTML.gif , is N- measurable on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq223_HTML.gif , which implies that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq225_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq226_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq227_HTML.gif

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

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq231_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq232_HTML.gif is nondecreasing with https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq235_HTML.gif defined by
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq237_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ20_HTML.gif
(33)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq238_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq239_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq240_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq241_HTML.gif is uniformly bounded there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq242_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq243_HTML.gif Also, assumption (H1) implies that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ21_HTML.gif
(34)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq245_HTML.gif depends only on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq246_HTML.gif Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq248_HTML.gif , defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ22_HTML.gif
(35)
Here, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ23_HTML.gif
(36)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq250_HTML.gif is a multivalued operator defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ24_HTML.gif
(37)

Claim 1.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq251_HTML.gif is compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq252_HTML.gif . Since the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq253_HTML.gif is continuous and the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq254_HTML.gif is uniformly bounded https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq255_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq256_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq257_HTML.gif Also, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq259_HTML.gif . It follows that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq260_HTML.gif is continuous and there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq261_HTML.gif  depending only on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq262_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq263_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq264_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq265_HTML.gif is uniformly bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq266_HTML.gif Since the embedding https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq267_HTML.gif is compact it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq268_HTML.gif is compact in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq269_HTML.gif

Claim 2.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq270_HTML.gif is also compact in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq272_HTML.gif See Lemma 1.4.

Claim 3.

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq274_HTML.gif We have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq275_HTML.gif

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

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq297_HTML.gif is condensing. https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq299_HTML.gif for some https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq301_HTML.gif has a fixed point https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq303_HTML.gif We prove that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq305_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq306_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq307_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ25_HTML.gif
(38)
On the other hand, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq308_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ26_HTML.gif
(39)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq309_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq310_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq311_HTML.gif Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ27_HTML.gif
(310)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq312_HTML.gif and the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq313_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq314_HTML.gif are nondecreasing, it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq315_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq316_HTML.gif for a.e. https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq318_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq319_HTML.gif In this case https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ28_HTML.gif
(41)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq321_HTML.gif It is clear that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ29_HTML.gif
(42)
and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_Equ30_HTML.gif
(43)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq324_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq325_HTML.gif is a solution of the problem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq326_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq327_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq328_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq329_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq331_HTML.gif

Similarly, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq332_HTML.gif be a solution of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq333_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq334_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq335_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F965759/MediaObjects/13661_2010_Article_69_IEq336_HTML.gif and https://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 https://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

References

  1. Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, USA; 1964:xiv+347.
  2. Carl S, Grossmann Ch, Pao CV: Existence and monotone iterations for parabolic differential inclusions. Communications on Applied Nonlinear Analysis 1996, 3(1):1–24.MathSciNet
  3. Pavlenko VN, Ul'yanova OV: The method of upper and lower solutions for equations of parabolic type with discontinuous nonlinearities. Differential Equations 2002, 38(4):520–527. 10.1023/A:1016311716130View ArticleMathSciNet
  4. Pisani R: Problemi al contorno per operatori parabolici con non linearita discontinua. Rendiconti dell'Istituto di Matemàtica dell'Universitá di Trieste 1982, 14: 85–98.MathSciNet
  5. Rauch J: Discontinuous semilinear differential equations and multiple valued maps. Proceedings of the American Mathematical Society 1977, 64(2):277–282. 10.1090/S0002-9939-1977-0442453-6View ArticleMathSciNet
  6. Cannon JR: The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics 1963, 21: 155–160.MathSciNet
  7. Ionkin NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differential Equations 1977, 13: 204–211.
  8. Chegis RYu: Numerical solution of a heat conduction problem with an integral condition. Litovskiĭ Matematicheskiĭ Sbornik 1984, 24(4):209–215.MathSciNet
  9. Olmstead WE, Roberts CA: The one-dimensional heat equation with a nonlocal initial condition. Applied Mathematics Letters 1997, 10(3):89–94. 10.1016/S0893-9659(97)00041-4View ArticleMathSciNet
  10. Sapagovas MP, Chegis RYu: Boundary value problems with nonlocal conditions. Differential Equations 1988, 23: 858–863.
  11. Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983, 41(4):468–475.
  12. Zeidler E: Nonlinera Functional Analysis and Its Applications. Volume IIA. Springer, Berlin, Germany; 1990.View Article
  13. Fürst T: Asymptotic boundary value problems for evolution inclusions. Boundary Value Problems 2006, 2006:-12.
  14. Aubin JP, Cellina A: Differential Inclusions, Fundamental Principles of Mathematical Sciences. Springer, Berlin, Germany; 1984:xiii+342.View Article
  15. Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260.
  16. Hu S, Papageorgiou NS: Handbook of Multivalued Analysis. Vol. I, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xvi+964.
  17. Chang KC: The obstacle problem and partial differential equations with discontinuous nonlinearities. Communications on Pure and Applied Mathematics 1980, 33(2):117–146. 10.1002/cpa.3160330203View ArticleMathSciNet
  18. Martelli M: A Rothe's type theorem for non-compact acyclic-valued maps. 1975, 11(3):70–76.
  19. Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow, Russia; 1967.
  20. Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, NY, USA; 1992:xvi+777.

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.