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.
Highlights
Let Ω be a an open bounded domain in RN, N ≥ 2, with a smooth boundary ∂Ω
We study the following parabolic differential equation with a nonlocal initial condition
K x, t, u x, t dt, x ∈ Ω, Boundary Value Problems where f : QT × R → R is not necessarily continuous, but is such that for every fixed u ∈ R the function x, t → f x, t, u is measurable and u → f x, t, u is of bounded variations over compact interval in R and nondecreasing, and k : QT × R → R is continuous; L is a strongly elliptic operator given by
Summary
Let Ω be a an open bounded domain in RN, N ≥ 2, with a smooth boundary ∂Ω. Let QT Ω × 0, T and ΓT ∂Ω × 0, T where T is a positive real number. R is called upper semicontinuous u.s.c. on X if for each z ∈ X the set R z ∈ Pcl Y is nonempty, and for each open subset Λ. The set-valued map R is called completely continuous if R A is relatively compact in Y for every A ∈ Pb X. Since F is an N-measurable and upper semicontinuous multivalued function with compact and convex values, we have the following properties for the operator NF see 17, Corollary 1.1. NF is N-measurable, compact and convex-valued, upper semicontinuous and maps bounded sets into precompact sets. As pointed out in 15, Example 1.3 page 5 , this is the most general upper semicontinuous set-valued map with compact and convex values in R. We remark that a compact map is the simplest example of a condensing map
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