Abstract
In this paper, we give existence results for viable solutions in the so-called fully constrained set for functional differential inclusions in Banach spaces for a non-autonomous set-valued mapping with convex compact values. We study also the time dependent case of these invariance sets.
Highlights
IntroductionThe existence of viable solutions for such problems with memory has been studied by several authors ([4], [5], [10], [12], [13], [14])
Let E be a separable Banach space, C0 = C([−r, 0], E) the space of continuous functions from [−r, 0] to E
The purpose of this paper is to show the existence of viable solutions in a set more natural introduced by [11] for a functional differential equations and called the fully constrained set: ED = {φ ∈ C0, φ(s) ∈ D, ∀s ∈ [−r, 0]}
Summary
The existence of viable solutions for such problems with memory has been studied by several authors ([4], [5], [10], [12], [13], [14]). In [13], the existence of solutions in the invariance set E0 = {φ ∈ C0, φ(0) ∈ D} where D is closed convex nonempty in E, has been established. The following simple example (see [11]) shows that conditions of invariance for sets E0 and ED, may be different for the same problem. −r≤s≤0 we study the case of variable constraint D = Γ(t), where Γ is a set-valued function with closed graph.
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