Existence results for fractional differential inclusions with three-point fractional integral boundary conditions

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Existence results for fractional differential inclusions with three-point fractional integral boundary conditions Xi Fu* * Correspondence: [email protected] Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, P.R. China

Abstract This paper is concerned with fractional differential inclusions with three-point fractional integral boundary conditions. We consider the fractional differential inclusions under both convexity and nonconvexity conditions on the multivalued term. Some new existence results are obtained by using standard fixed point theorems. Two examples are given to illustrate the main results. MSC: 34A60; 26A33; 34B15 Keywords: fractional differential inclusions; boundary value problems; existence results; multivalued maps

1 Introduction Fractional differential equations have recently gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many physical phenomena [–]. For some recent developments on the existence results of fractional differential equations, we can refer, for instance, to [–] and the references therein. Differential inclusions arise in the mathematical modeling of certain problems in economics, optimal control, etc. and are widely studied by many authors, see [, ] and the references therein. For some recent works on differential inclusions of fractional order, we refer the reader to the references [, , –]. Motivated by the above papers, in this article, we study a new class of fractional boundary value problems, i.e., the following fractional differential inclusions with three-point fractional integral boundary conditions: 

Dα x(t) ∈ F(t, x(t), c Dβ x(t)), t ∈ [, ],  < α ≤ ,  < β < , x() = , aI γ x(η) + bx() = c,  < η < , c

()

where c Dp denotes the Caputo fractional derivative of order p, I q the Riemann-Liouville fractional integral of order q, F : [, ] × R → R is a multifunction and a, b, c are real constants with aη+γ = –b(γ + ). We remark that when b = –, c =  and third variable of the function F in () vanishes, problem () reduces to a three-point fractional integral boundary value problem (see [] with F = f a given continuous function). ©2013 Fu; 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.

Fu Advances in Difference Equations 2013, 2013:304 http://www.advancesindifferenceequations.com/content/2013/1/304

The rest of this paper is organized as follows. In Section  we present the notations, definitions and give some preliminary results that we need in the sequel, Section  is dedicated to the existence results of problem (), in the final Section , two examples are given to illustrate the main results.

2 Preliminaries In this section, we introduce notations, definitions and preliminary facts that

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