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RESEARCH

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Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay Fang Li* * Correspondence: [email protected] School of Mathematics, Yunnan Normal University, Kunming, 650092, P.R. China

Abstract Of concern is the existence of mild solutions to delay fractional differential equations with almost sectorial operators. Combining the techniques of operator semigroup, noncompact measures and fixed point theory, we obtain a new existence theorem without the assumptions that the nonlinearity f satisfies a Lipschitz-type condition, and the resolvent operator associated with A is compact. An example is presented. MSC: 34A08; 34K30; 47D06; 47H10 Keywords: fractional differential equations; mild solution; infinite delay; measure of noncompactness; fixed point theorem

1 Introduction Fractional differential equations have been increasingly used for many mathematical models in probability, engineering, physics, astrophysics, economics, etc., so the theory of fractional differential equations has in recent years been an object of investigations with increasing interest [–]. Most of the previous research on the fractional differential equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, a compact semigroup, or an analytic semigroup, or is a Hille-Yosida operator (see, e.g., [, , , , ]). However, as presented in Example . and Example . in [], the resolvent operators do not satisfy the required estimate to be a sectorial operator. In [], W. von Wahl first introduced examples of almost sectorial operators which are not sectorial. To the author’s knowledge, there are few papers about the fractional evolution equations with almost sectorial operators. Moreover, equations with delay are often more useful to describe concrete systems than those without delay. So, the study of these equations has attracted so much attention (cf., e.g., [, , –] and references therein). In this paper, we pay our attention to the investigation of the existence of mild solutions to the following fractional differential equations with almost sectorial operators and infinite delay on a separable complex Banach space X: c

  q Dt u(t) = Au(t) + f t, u(t), ut ,

u = φ ∈ P ,

t ∈ (, T],

(.)

©2013Li; 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.

Li Advances in Difference Equations 2013, 2013:327 http://www.advancesindifferenceequations.com/content/2013/1/327

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where T > ,  < q < . The fractional derivative is understood here in the Caputo sense. P is a phase space that will be defined later (see Definition .). A is an almost sectorial operator to be introduced later. Here, f : [, T] × X × P → X, and ut : (–∞, ] → X is defined by