Structure of the solution set for a partial differential inclusion

Yi Cheng,Donal O’Regan,Ravi P Agarwal,Afif Ben Amar

doi:10.1186/s13662-015-0723-0

Abstract

In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compact $R_{\delta}$ if the perturbation term of the related partial differential inclusion is convex, and its solution set is path-connected if the perturbation term is nonconvex.

Highlights

1 Introduction In this paper, we examine the following biharmonic problem of the partial differential inclusion:
Boundary value problems involving partial differential equations with discontinuous nonlinearities which may be reduced to boundary value problems for partial differential inclusions were studied by Carl-Heikkilä [, ] and Chang [, ]
We show that the resulting problem always has a solution (‘extremal solutions’) and the solution set is dense in the solution set of the convexified version of the problem (‘strong relaxation theorem’)

Summary

Introduction

In this paper, we obtain the Rδ-structure of the solution set for a biharmonic differential inclusion based on the space variable x ∈ . Let X be a Banach Space with the weak topology, and D ⊆ X a weakly compact, convex subset of X, any weakly sequentially upper semicontinuous map F : D → Pwkc(D) has a fixed point, i.e., there exists x ∈ D, such that x ∈ F(x). It follows from the Krein-Šmulian theorem that D is a weakly compact convex set.

Results

Conclusion