Existence of solutions of Sobolev-type semilinear mixed integrodifferential inclusions in Banach spaces

M Kanakaraj,K Balachandran

doi:10.1155/s104895330300011x

Abstract

The existence of mild solutions of Sobolev-type semilinear mixed integrodifferential inclusions in Banach spaces is proved using a fixed point theorem for multivalued maps on locally convex topological spaces.

Highlights

The problem of proving the existence of mild solutions for differential and integrodifferential equations in abstract spaces has been studied by several authors [2, 4, 11, 12, 13]
Balachandran and Uchiyama [3] established the existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces
Benchohra [6] studied the existence of mild solutions on infinite intervals for a class of differential inclusions in Banach spaces

Summary

Introduction

The problem of proving the existence of mild solutions for differential and integrodifferential equations in abstract spaces has been studied by several authors [2, 4, 11, 12, 13]. Balachandran and Uchiyama [3] established the existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. Benchohra [6] studied the existence of mild solutions on infinite intervals for a class of differential inclusions in Banach spaces. For the existence results of differential inclusions on compact intervals, one can refer to the papers of Avgerinos and Papageorgiou [1], and Papageorgiou [14, 15]. Our method is to reduce the problem (1.1) to a fixed point problem of a suitable multivalued map in the Frechet space C(I, X) and we make use of a fixed point theorem due to Ma [10] for multivalued maps in locally convex topological spaces

Preliminaries

Main Result

Nonlocal Initial Conditions