On boundary value problems for degenerate differential inclusions in Banach spaces

Valeri Obukhovskii,Pietro Zecca

doi:10.1155/s108533750330301x

Valeri Obukhovskii, Pietro Zecca

Open Access

https://doi.org/10.1155/s108533750330301x

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Abstract

We consider the applications of the theory of condensing set-valued maps, the theory of set-valued linear operators, and the topological degree theory of the existence of mild solutions for a class of degenerate differential inclusions in a reflexive Banach space. Further, these techniques are used to obtain the solvability of general boundary value problems for a given class of inclusions. Some particular cases including periodic problems are considered.

Highlights

In the last decades the theory of degenerate differential equations in Banach spaces attracted the attention of a large number of researchers
One of the main reasons is that many partial differential equations arising in mathematical physics and in applied sciences may be naturally presented in this form
We introduce a class of degenerate differential inclusions in a reflexive Banach space and define the notion of mild solution for such inclusion

Summary

Introduction

In the last decades the theory of degenerate differential equations in Banach spaces attracted the attention of a large number of researchers (see, e.g., Favini and Yagi [5], Showalter [13], and [6] and the references therein). We introduce a class of degenerate differential inclusions in a reflexive Banach space and define the notion of mild solution for such inclusion. It should be noted that, starting from the paper of Zecca and Zezza [15], nonlinear boundary value problems for nondegenerate differential inclusions in Banach spaces were studied in a number of papers (see, e.g., [4, 9, 10, 11, 12]) under compactness conditions on the evolution operator generated by the linear part of the problem.

Preliminaries

Existence results