Semilinear Volterra integrodifferential equations with nonlocal initial conditions

Sergiu Aizicovici,Mark A Mckibben

doi:10.1155/s1085337599000123

Abstract

We establish the global existence of mild solutions to a class of nonlocal Cauchy problems associated with semilinear Volterra integrodifferential equations in a Banach space.

Highlights

We investigate the global existence of solutions to integrodifferential equations with nonlocal conditions of the general form t u (t) + a(t − s)Au(s) ds = F (u)(t), 0 < t < T, (1.1)
The conclusion of Theorem 5.2 follows readily

Summary

Introduction

Byszewski [6, 7] initiated the work concerning abstract nonlocal semilinear initial-value problems. He used fixedpoint methods to prove the existence and uniqueness of mild solutions to the Cauchy. The paper most closely related to the present one is by Lin and Liu [15] They developed an existence theory for the nonlocal integrodifferential equation t u (t) + A u(t) + a(t − s)u(s) ds = f t, u(t) , 0 < t < T ,.

Preliminaries

The case of a Lipschitz continuous nonlinearity

Examples