Asymptotically Almost Periodic Solutions for Abstract Partial Neutral Integro-Differential Equation

José Paulo C Dos Santos,Marcos N Rabelo,Sandro M Guzzo

doi:10.1155/2010/310951

Abstract

The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.

Highlights

In this paper, we study the existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations modelled in the form d dt xt f t, xt Ax t tB t − s x s ds g t, xt, 1.1 where A : D A ⊂ X → X and B t : D B t ⊂ X → X, t ≥ 0, are closed linear operators; X, · is a Banach space; the history xt : −∞, 0 → X, xt θ x t θ, belongs to some abstract phase space B defined axiomatically f, g : I × B → X are appropriated functions.The study of abstract neutral equations is motivated by different practical applications in different technical fields
For abstract integro-differential equations described on infinite dimensional spaces, we cite the Pruss book 17 and the papers 18–20, Da Prato et al 21, and Lunardi 9
To the best of our knowledge, the study of the existence of asymptotically almost periodic solutions of neutral integro-differential equations with unbounded delay described in the abstract form 1.1 is an untreated topic in the literature and this is the main motivation of this article

Summary

Introduction

We study the existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations modelled in the form d dt xt f t, xt. For abstract integro-differential equations described on infinite dimensional spaces, we cite the Pruss book and the papers 18–20 , Da Prato et al 21, , and Lunardi 9, To finish this short description of the related literature, we cite the papers 24–26 where some of the above topics for the case of abstract neutral integro-differential equations with unbounded delay are treated. To the best of our knowledge, the study of the existence of asymptotically almost periodic solutions of neutral integro-differential equations with unbounded delay described in the abstract form 1.1 is an untreated topic in the literature and this is the main motivation of this article. C0 0, ∞ , Z represents the subspace of Cb 0, ∞ , Z formed by the functions which vanish at infinity

Preliminaries

Resolvent Operators

12 The following estimate:

Existence Result of Asymptotically Almost Periodic Solutions

Applications