On the Existence of Solutions for Boundary Value Problems in Banach Spaces

Abstract

In this paper, by applying the theory of condensing multimaps and the topological degree, we deal with the existence of solutions for boundary value problems with second order differential inclusions in different cases where the underlying space is a Banach space. Indeed, we investigate the existence of solutions for the BVP ( x ′′(t) ∈ F(t, x(t)) t ∈ I = [0,1], x(0) = x(1) = 0, where X is a real Banach space and the multifunction F : I ×X ⊸ K(X), in one case, has convex values and in another case has non-convex values (K(X) denotes compact subsets of X). Moreover, some results on the existence of solutions for the extended version of BVP ( u ′′(t) ∈ Q(u) t ∈ I, u(0) = u(1) = 0, are presented, where Q :C(I,X) ⊸C(L 2 ) is a multimap satisfying some appropriate conditions. Finally, we show how the results can be used to study periodic feedback control systems