Advances in Difference Equations
Volume 2010 (2010), Article ID 674630, 8 pages
doi:10.1155/2010/674630
Research Article

A Note on a Semilinear Fractional Differential Equation of Neutral Type with Infinite Delay

Gisle M. Mophou1 and Gaston M. N'Guérékata2

1Département de Mathématiques et Informatique, Université des Antilles et de La Guyane, Campus Fouillole 97159 Pointe-à-Pitre Guadeloupe (FWI), France
2Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA

Received 28 November 2009; Accepted 21 January 2010

Academic Editor: A. Pankov

Copyright © 2010 Gisle M. Mophou and Gaston M. N'Guérékata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t)+Ax(t)=f(t,xt), t∈[0,T], x(t)=ϕ(t), t∈]−∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness) of solutions, assuming that −A is a linear closed operator which generates an analytic semigroup (T(t))t≥0 on a Banach space 𝕏 by means of the Banach's fixed point theorem. This generalizes some recent results.