Existence of Solutions for Nonlinear Mixed Type Integrodifferential Functional Evolution Equations with Nonlocal Conditions

Shengli Xie

doi:10.1155/2012/913809

Abstract

Using Mönch fixed point theorem, this paper proves the existence and controllability of mild solutions for nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions in Banach spaces, some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve many known results. As an application, we have given a controllability result of the system.

Highlights

Since spectral radius σ B 0 of linear Volterra integral operator Bx t t 0 k t, s x s ds, in order to obtain the existence of solutions for nonlinear Volterra integrodifferential equations in abstract spaces by using fixed point theory, usually, some restricted conditions on a priori estimation and measure of noncompactness estimation will not be used even if the infinitesimal generator A 0
In this paper, using Monch fixed point theorem, we investigate the existence and controllability of mild solution of nonlinear Volterra-Fredholm integrodifferential system 1.1, some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve the corresponding results in papers
Let X, · be a real Banach space and let C a, b, X be a Banach space of all continuous X-valued functions defined on a, b with norm x a,b supt∈ a,b x t for x ∈ C a, b, X

Summary

Introduction

This paper related to the existence and controllability of mild solutions for the following nonlinear mixed type integrodifferential functional evolution equations with nonlocal conditions in Banach space X:. In order to obtain existence and controllability of mild solutions in these study papers, usually, some restricted conditions on a priori estimation and compactness conditions of evolution operator are used. Since spectral radius σ B 0 of linear Volterra integral operator Bx t t 0 k t, s x s ds, in order to obtain the existence of solutions for nonlinear Volterra integrodifferential equations in abstract spaces by using fixed point theory, usually, some restricted conditions on a priori estimation and measure of noncompactness estimation will not be used even if the infinitesimal generator A 0. In this paper, using Monch fixed point theorem, we investigate the existence and controllability of mild solution of nonlinear Volterra-Fredholm integrodifferential system 1.1 , some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, our results extend and improve the corresponding results in papers

Preliminaries

Existence Result

An Example

An Application