Positive solutions to boundary value problems of fractional difference equation with nonlocal conditions

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Positive solutions to boundary value problems of fractional difference equation with nonlocal conditions Shugui Kang1* , Yan Li2 and Huiqin Chen1 * Correspondence: [email protected] 1 School of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, P.R. China Full list of author information is available at the end of the article

Abstract In this paper, we will use the Krasnosel’skii ﬁxed point theorem to investigate a discrete fractional boundary value problem of the form –ν y(t) = λh(t + ν – 1)f (y(t + ν – 1)), y(ν – 2) = (y), y(ν + b) = (y), where 1 < ν ≤ 2, t ∈ [0, b]N0 , f : [0, ∞) → [0, ∞) is a continuous function, h : [ν – 1, ν + b – 1]Nν –1 → [0, ∞), , : Rb+3 → R are given functionals, where , are linear functionals, and λ is a positive parameter. MSC: 26A33; 39A05; 39A12 Keywords: nonlocal conditions; positive solution; cone; ﬁxed point theorem

1 Introduction The theory of fractional diﬀerential equations and their applications has received intensive attention. However, the theory of fraction diﬀerence equations is still limited. But in the last few years, a number of papers on fractional diﬀerence equations have appeared [–]. Among them, Atici and Eloe [] introduced and developed properties of discrete fractional calculus. In [], Atici and Eloe studied a two-point boundary valve problem for a ﬁnite fractional diﬀerence equation. They obtained suﬃcient conditions for the existence of solutions for the following boundary value problem: –ν y(t) = f t + ν – , y(t + ν – ) ,

y(ν – ) =  = y(ν + b + ),

where t ∈ [, b + ]N ,  < ν ≤ , and f : [ν – , ν + b]Nν– × R → R is a continuous function. Goodrich [] deduced uniqueness theorems by means of the Lipschitz condition and deduced the existence of one or more positive solutions by using the cone theoretic techniques for this same boundary value problem. He showed that many of the classical existence and uniqueness theorems for second-order discrete boundary value problems extend to the fraction-order case. In [], Goodrich obtained the existence of positive solutions to another boundary value problem. Goodrich [] also considered a pair of discrete fractional boundary value problem of the form –ν y (t) = λ a (t + ν – )f y (t + ν – ), y (t + ν – ) , –ν y (t) = λ a (t + ν – )f y (t + ν – ), y (t + ν – ) ©2014 Kang 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.

Kang et al. Advances in Diﬀerence Equations 2014, 2014:7 http://www.advancesindifferenceequations.com/content/2014/1/7

y (ν – ) =  (y ),

y (ν – ) =  (y ),

y (ν + b) =  (y ),

y (ν + b) =  (y ),

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where t ∈ [, b]N , λi > , ai : R → [, ∞), νi ∈ (, ] for each i = , . i , i : Rb+ → R are given

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