Existence of solutions for semilinear differential equations with nonlocal conditions in Banach spaces

Gang Li,Qixiang Dong

doi:10.14232/ejqtde.2009.1.47

Abstract

This paper is concerned with semilinear differential equations with nonlocal conditions in Banach spaces. Using the tools involving the measure of noncompactness and fixed point theory, existence of mild solutions is obtained without the assumption of compactness or equicontinuity on the associated linear semigroup.

Highlights

Introduction and preliminariesIn this paper we discuss the semilinear differential equation with nonlocal condition (1.1) d dt x(t) = Ax(t) +f (t, x(t)), t ∈ (0, b], (1.2)x(0) = x0 + g(x) where A is the infinitesimal generator of a strongly continuous semigroup {T (t) : t ≥ 0} of linear operators defined on a Banach space X, f : [0, b] × X → X and g : C([0, b]; X) → X are appropriate given functions.The theory of differential equations with nonlocal conditions was initiated by Byszewski and it has been extensively studied in the literature
X(0) = x0 + g(x) where A is the infinitesimal generator of a strongly continuous semigroup {T (t) : t ≥ 0} of linear operators defined on a Banach space X, f : [0, b] × X → X and g : C([0, b]; X) → X are appropriate given functions
In [5], Byszewski and Akca give the existence of semilinear functional differential equation when T (t) is compact, and g is convex and compact on a given ball of C([0, b]; X)

Summary

Introduction and preliminaries

In this paper we discuss the semilinear differential equation with nonlocal condition (1.1). X(0) = x0 + g(x) where A is the infinitesimal generator of a strongly continuous semigroup {T (t) : t ≥ 0} of linear operators defined on a Banach space X, f : [0, b] × X → X and g : C([0, b]; X) → X are appropriate given functions. By using the tools involving the measure of noncompactness and fixed point theory, we obtain existence of mild solution of semilinear differential equation with nonlocal conditions (1.1)-(1.2), and the compactness of solution set, without the assumption of compactness or equicontinuity on the associated semigroup. For more details of the semigroup theory we refer the readers to [15]

Measure of noncompactness

The existence of mild solution