Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay G.M. Mophou a , G.M. N’Guérékata b, * a Département de Mathématiques et Informatique, Université des Antilles et de La Guyane, Campus Fouillole 97159 Pointe-à-Pitre, Guadeloupe (FWI) b Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA article info Keywords: Fractional abstract differential equation Infinite delay Mild solution abstract We deal in this paper with the mild solution for fractional semilinear differential equations with infinite delay: D a xðtÞ¼ AxðtÞþ f ðt; x t ; BxðtÞÞ; t 2½0; T; xðtÞ¼ /ðtÞ; t 2  1; 0 with T > 0 and 0 < a < 1. We prove the existence (and uniqueness) of solutions, assuming that A generates an a-resolvent family ðS a ðtÞÞ tP0 on a complex Banach space X by means of clas- sical fixed points methods. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction Our aim in this paper is to investigate the existence and uniqueness of the mild solution for the fractional differential equation with infinite delay: D a xðtÞ¼ AxðtÞþ f ðt; x t ; BxðtÞÞ; t 2 I ¼½0; T ; xðtÞ¼ /ðtÞ; t 2  1; 0;  ð1Þ where T > 0; 0 < a < 1; A : DðAÞ X ! X is the infinitesimal generator of an a-resolvent family ðS a ðtÞÞ tP0 defined on a complex Banach space X. The term BxðtÞ is given by: BxðtÞ :¼ R t 0 Kðt; sÞxðsÞ ds, where K 2 CðD; R þ Þ, the set of all positive func- tions which are continuous on D :¼ fðt; sÞ2 R 2 : 0 6 s 6 t 6 T g and B  ¼ sup t2½0;T Z t 0 Kðt; sÞ ds < 1: ð2Þ The fractional derivative D a is understood here in the Riemann–Liouville sense. The function f is given and satisfies some conditions that will be specified later, / belongs to the phase space B with /ð0Þ¼ 0. For any function x defined on 1; T  and any t 2 I, we denote by x t the element of B defined by x t ðhÞ¼ xðt þ hÞ; h 2  1; 0: ð3Þ The function x t represents the history of the state from 1 up to the present time t. The theory of functional differential equations has emerged as an important branch of nonlinear analysis. It is worthwhile mentioning that several important problems of the theory of ordinary and delay differential equations lead to investigations of functional differential equations of various types (see the books of Hale and Verduyn Lunel [8], Wu [20], and the references therein). On the other hand the theory of fractional differential equations is also intensively studied and finds numerous 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.12.062 * Corresponding author. E-mail addresses: gmophou@univ-ag.fr (G.M. Mophou), Gaston.N’Guerekata@morgan.edu (G.M. N’Guérékata). Applied Mathematics and Computation 216 (2010) 61–69 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc