Let be a an open bounded domain in , with a smooth boundary Let and where is a positive real number Then is smooth and any point on satisfies the inside (and outside) strong sphere property (see ). For we denote its partial derivatives in the distributional sense (when they exist) by ,
In this paper, we study the following parabolic differential equation with a nonlocal initial condition
where is not necessarily continuous, but is such that for every fixed the function is measurable and is of bounded variations over compact interval in and nondecreasing, and is continuous; is a strongly elliptic operator given by
Discontinuous parabolic problems have been studied by many authors, see for instance [2–5]. Parabolic problems with integral conditions appear in the modeling of concrete problems, such as heat conduction [6–10] and in thermoelasticity .
In order to investigate problem (1.1), we introduce some notations, function spaces, and notions from set-valued analysis.
Let denote the Banach space of all continuous functions , equipped with the norm Let for each and for each For we say that is in if is measurable and in which case we define its norm by
Let and let denote the Sobolev space of functions having first generalized derivatives in and let be its corresponding dual space. Then and they form an evolution triple with all embeddings being continuous, dense, and compact (see [2, 12]). The Bochner space (see ) is the set of functions with generalized derivative For we define its norm by
Then is a separable reflexive Banach space. The embedding of into is continuous and the embedding is compact.
Now, we introduce some facts from set-valued analysis. For complete details, we refer the reader to the following books. [14–16]. Let and be Banach spaces. We will denote the set of all subsets, of having property by For instance, denotes the set of all nonempty subsets of ; means closed in when we have the bounded subsets of for convex subsets, for compact subsets and for compact and convex subsets. The domain of a multivalued map is the set is convex (closed) valued if is convex (closed) for each is bounded on bounded sets if is bounded in for all (i.e., is called upper semicontinuous (u.s.c.) on if for each the set is nonempty, and for each open subset of containing , there exists an open neighborhood of such that In terms of sequences, is usc if for each sequence , , and is a closed subset of such that then
The set-valued map is called completely continuous if is relatively compact in for every If is completely continuous with nonempty compact values, then is usc if and only if has a closed graph (i.e., , ). When then has a fixed point if there exists such that A multivalued map is called measurable if for every , the function defined by is measurable. denotes The Kuratowski measure of noncompactness (see [15, page 113]) of is defined by
The Kuratowski measure of noncompactness satisfies the following properties.
(i) if and only if is compact;
(v) where denotes the convex hull of .
Definition 1.1 (see ).
A function is called N- measurable on if for every measurable function the function is measurable.
Examples of N- measurable functions are Carathéodory functions, Baire measurable functions.
Let and Then (see [17, Proposition ]) the function is lower semicontinuous, that is, for every the set is open for any , and the function is upper semicontinuous, that is, for every the set is open for any . Moreover, the functions and are nondecreasing.
The multivalued function defined by for all is called N- measurable on if both functions and are N- measurable on .
The operator defined by
is called the Nemitskii operator of the multifunction
Since is an N- measurable and upper semicontinuous multivalued function with compact and convex values, we have the following properties for the operator (see [17, Corollary ]).
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:
where for all As pointed out in [15, Example page 5], this is the most general upper semicontinuous set-valued map with compact and convex values in .
Theorem 1.5 (see ).
Let be a Banach space and a condensing map. If the set for some is bounded, then has a fixed point.
We remark that a compact map is the simplest example of a condensing map.