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RESEARCH

Open Access

Fractional neutral evolution equations with nonlocal conditions Hamdy M Ahmed* *

Correspondence: [email protected] Higher Institute of Engineering, El-Shorouk Academy, P.O. 3, El-Shorouk City, Cairo, Egypt

Abstract In the present paper, we deal with the fractional neutral differential equations involving nonlocal initial conditions. The existence of mild solutions are established. The results are obtained by using the fractional power of operators and the Sadovskii’s fixed point theorem. An application to a fractional partial differential equation with nonlocal initial condition is also considered. MSC: 26A33; 34K30; 34K37; 34K40 Keywords: fractional calculus; semilinear neutral differential equations; semigroups; nonlocal conditions; mild solutions; Sadovskii fixed-point theorem

1 Introduction The nonlocal condition, which is a generalization of the classical condition, was motivated by physical problems. The pioneering work on nonlocal conditions is due to Byszewski (see [–]). Existence results for semilinear evolution equations with nonlocal conditions were investigated in [–]. Neutral differential equations arises in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for neutral functional differential equations is the Hale book []. Fractional differential equations describe many practical dynamical phenomena arising in engineering, physics, economy and science. In particular, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, seepage flow in porous media and in fluid dynamic traffic models (see [–]). The result obtained is a generalization and a continuation of some results reported in [–]. The main purpose of this paper is to study the existence of mild solutions of semilinear neutral fractional differential equations with nonlocal conditions in the following form c

      Dα x(t) + F t, x(t), x b (t) , . . . , x bm (t) + Ax(t)      = G t, x(t), x a (t) , . . . , x an (t) , t ∈ J = [, b],

x() + g(x) = x ,

(.)

where –A is the infinitesimal generator of an analytic semigroup and the functions F, G and g are given functions to be defined later. The fractional derivative c Dα ,  < α <  is understood in the Caputo sense. © 2013 Ahmed; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ahmed Advances in Difference Equations 2013, 2013:117 http://www.advancesindifferenceequations.com/content/2013/1/117

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2 Preliminaries Throughout this paper, X will be a Banach space with the norm  ·  and –A : D(A) → X is the infinitesimal generator of an analytic compact semigroup of uniformly bounded linear operators {S(t), t ≥ }. This means that there exists a M ≥  such that  S(t) ≤ M. We assume without