A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative
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Research Article A Survey on Semilinear Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative Ravi P. Agarwal,1 Mohammed Belmekki,2 and Mouffak Benchohra2 1 2
Department of Mathematical Sciences, Florida Institute of Technology, Melboune, FL 32901-6975, USA Laboratoire de Math´ematiques, Universit´e de Sidi Bel Abb`es, BP 89, 22000 Sidi Bel Abb`es, Algeria
Correspondence should be addressed to Ravi P. Agarwal, [email protected] Received 16 July 2008; Revised 4 December 2008; Accepted 5 February 2009 Recommended by Alberto Cabada We establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the C0 -semigroups theory combined with some suitable fixed point theorems. Copyright q 2009 Ravi P. Agarwal et al. 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.
1. Introduction Differential equations and inclusions of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, including some applications and recent results, see the monographs of Kilbas et al. 1, Kiryakova 2, Miller and Ross 3, Podlubny 4 and Samko et al. 5, and the papers of Agarwal et al. 6, Diethelm et al. 7, 8, El-Sayed 9–11, Gaul et al. 12, Glockle and Nonnenmacher 13, Lakshmikantham and Devi 14, Mainardi 15, Metzler et al. 16, Momani et al. 17, 18, Podlubny et al. 19, Yu and Gao 20 and the references therein. Some classes of evolution equations have been considered by El-Borai 21, 22, Jaradat et al. 23 studied the existence and uniqueness of mild solutions for a class of initial value problem for a semilinear integrodifferential equation involving the Caputo’s fractional derivative.
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Advances in Difference Equations
In this survey paper, we give existence results for various classes of initial value problems for fractional semilinear functional differential equations and inclusions, both cases of finite and infinite delay are considered. More precisely the paper is organized as follows. In the second section we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. In the third section we will be concerned with semilinear functional differential equations with finite as well infinite delay. In the forth section, we consider semilinear functional differential equation of neutral type for the both cases of finite and infinite delay. Section 5 is devoted to the study of functional differential inclusions, we examine the case when the right-hand side is convex valued as well as nonconvex valued. In Section 6, we
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