Existence of solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions

Shengli Xie

doi:10.1186/1687-2770-2012-100

Abstract

Using the Mönch fixed point theorem, this article proves the existence of mild solutions for nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in Banach spaces. Some restricted conditions on a priori estimation and measure of noncompactness estimation have been deleted, and compactness conditions of evolution operators or compactness conditions on a nonlinear term f(t, X r , X r , X r ) have been weakened. Our results extend and improve many known results.MSC:34G20, 34K30.

Highlights

Let (X, · ) be a Banach space, C[J, X] = {x : J = [, a] → X, x(t) is continuous in J} with the norm x C = supt∈J x(t)
In this paper, using the Mönch fixed point theorem, we investigate the existence of mild solutions of IVP ( . )-( . )
Our results extend and improve some corresponding results in papers [, – ]

Summary

Introduction

Let (X, · ) be a Banach space, C[J, X] = {x : J = [ , a] → X, x(t) is continuous in J} with the norm x C = supt∈J x(t) . Consider the following nonlinear mixed type integro-differential functional evolution equations with nonlocal conditions in a Banach space X(IVP), t x (t) = A x(t) + F(t – s)x(s) ds + f t, x σ (t) , (Kxσ )(t), (Hxσ )(t) , t ∈ J, A is the generator of a strongly continuous semigroup in the Banach space X, and F(t) is a bounded operator for t ∈ J, x ∈ X, f ∈ C[J × X , X], g : C[J, X] → X, k ∈ C[ × X, X],

Results

Conclusion