## Existence results for fractional differential equations with three-point boundary conditions

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RESEARCH

Open Access

Existence results for fractional differential equations with three-point boundary conditions Xi Fu* *

Correspondence: [email protected] Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, P.R. China

Abstract In this paper, we study three-point boundary value problems of nonlinear fractional diﬀerential equations. Existence and uniqueness results are obtained by using standard ﬁxed point theorems. Some examples are given to illustrate the results. MSC: 34A60; 26A33; 34B15 Keywords: fractional diﬀerential equations; boundary value problems; existence results

1 Introduction Fractional diﬀerential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, economics and biology, etc. [– ]. For some developments on the existence results of fractional diﬀerential equations, we can refer to [–] and the references therein. In recent years, there has been a great deal of research on the questions of existence and uniqueness of solutions to boundary value problems for diﬀerential equations of fractional order. For example, Ahmad and Nieto [] investigated the existence and uniqueness of solutions for an anti-periodic fractional boundary value problem 

Dα x(t) = f (t, x(t)), t ∈ [, T],  < α ≤ , T > , c γ D x() = –c Dγ x(T),  < γ < , x() = –x(T),

c

()

where c Dα denotes the Caputo fractional derivative of order α, f is a given continuous function. In [], the author discussed the existence of solutions for the following nonlinear fractional diﬀerential equations with anti-periodic-type fractional boundary conditions 

Dα x(t) = f (t, x(t), c Dβ x(t)), t ∈ [, T],  < α ≤ ,  < β ≤ , c γ D x() + μ c Dγ x(T) = σ ,  < γ < , x() + μ x(T) = σ ,

c

()

where c Dq denotes the Caputo fractional derivative of order q, μ = –, μ = , σ , σ are real constants, and f is 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, 2013:257 http://www.advancesindifferenceequations.com/content/2013/1/257

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Fractional diﬀerential equations with three-point integral boundary conditions of the following form were considered in [] by Ahmad et al. 

Dα x(t) = f (t, x(t)), t ∈ [, ],  < α ≤ , η x() = , x() = a  x(s) ds,  < η < ,

c

()

where c Dα denotes the Caputo fractional derivative of order α, f is a given continuous function, and a ∈ R with aη = . By a simple computation, we observed that c Dγ x() =  in equations () and (). This implies that the boundary conditions c Dγ x() = –c Dγ x(T) in () and c Dγ x() + μ c Dγ x(T) = σ in () actually are equivalent to the bou