Existence of solutions of abstract non-autonomous second order integro-differential equations

Hernán R Henríquez,Juan C Pozo

doi:10.1186/s13661-016-0675-7

Abstract

In this paper the existence of solutions of a non-autonomous abstract integro-differential equation of second order is considered. Assuming the existence of an evolution operator corresponding to the associate abstract non-autonomous Cauchy problem of second order, we establish the existence of a resolvent operator for the homogeneous integro-differential equation and the existence of mild solutions to the inhomogeneous integro-differential equation. Furthermore, we study the existence of classical solutions of the integro-differential equation. Finally, we apply our results to the study of the existence of solutions of a non-autonomous wave equation.

Highlights

Abstract integro-differential equations have been used to model various physical phenomena
This paper is devoted to the study of the existence of mild and classical solutions for initial value problems described as an abstract non-autonomous second order integrodifferential equation in Banach spaces
In this work we will be concerned with the existence of solutions to initial value problems that can be modeled as t x (t) = A(t)x(t) + P(t, s)x(s) ds + f (t), ≤ t ≤ a, ( . )

Summary

Introduction

Abstract integro-differential equations have been used to model various physical phenomena. This paper is devoted to the study of the existence of mild and classical solutions for initial value problems described as an abstract non-autonomous second order integrodifferential equation in Banach spaces. ) has been studied by several authors [ , , ] At this point, we will just say that the function x : [ , a] → X given by t x(t) = C(t, s)y + S(t, s)z + S(t, ξ )f (ξ ) dξ s is a mild solution of problem In Section , we are concerned with the existence of mild and classical solutions of the inhomogeneous non-autonomous integro-differential equation ). in Section we apply our results to the study of the existence of solutions to non-autonomous wave equations.

Using obtain t s t

Let x