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RESEARCH

Open Access

Existence of positive solutions for a discrete fractional boundary value problem Jinhua Wang1† , Hongjun Xiang2*† and Fulai Chen1† *

Correspondence: [email protected] 2 The Editorial Department of Journal of Xiangnan University, East Wangxian Park, Chenzhou, 423000, China † Equal contributors Full list of author information is available at the end of the article

Abstract This paper is concerned with the existence of positive solutions to a discrete fractional boundary value problem. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established. Additionally, examples are provided to illustrate the effectiveness of the main results. MSC: 26A33; 39A10; 47H07 Keywords: existence; positive solution; discrete; fractional boundary value problem

1 Introduction Fractional differential equations have received increasing attention within the last ten years or so. The theory of fractional differential equations has been a new important mathematical branch due to its wide applications in different research areas and engineering, such as physics, chemistry, economics, control of dynamical etc. For more details, see [– ] and the references therein. On the other hand, accompanied with the development of the theory for fractional calculus, fractional difference equations have attracted increasing attention slowly but steadily in the past three years or so. Some research papers have appeared, see [–]. For example, Atici and Eloe [] analyzed the conjugate discrete fractional boundary value problem (FBVP) with delta derivative: 

–v y(t) = f (t + v – , y(t + v – )), t ∈ [, b]N , y(v – ) = y(v + b + ) = ,  < v ≤ .

Goodrich [] studied the discrete fractional boundary value problems: 

v y(t) = λf (t + v – , y(t + v – )), t ∈ [, T]Z ,   < v < . y(v – ) = y(v + T) + N i= F(ti , y(ti )),

In [], Lv discussed the existence of solutions for discrete fractional boundary value problems with a p-Laplacian operator: ⎧ β α ⎪ ⎨c [φp (c u)](t) = f (t + α + β – , u(t + α + β – )), t ∈ [, b]N , α c u(t)|t=β– + αc u(t)|t=β+b = , ⎪ ⎩ u(α + β – ) + u(α + β + b) = ,  < α, β ≤ ,  < α + β ≤ . © 2014 Wang et al.; 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.

Wang et al. Advances in Difference Equations 2014, 2014:253 http://www.advancesindifferenceequations.com/content/2014/1/253

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They obtained a series of excellent results of discrete fractional boundary value problems. Motivated by the aforementioned works, in this paper we consider a discrete fractional boundary value problem (FBVP):  v y(t) = f (t + v – , y(t + v – )), t ∈ [, b]N , y(v – ) = , y(v – ) = y(v + b – ),

(.)

where  < v ≤ , v denotes the Riemann-Liouville fractional difference operator, Na = {a, a + , a + , .