中文摘要：

随着分数微积分理论的发展,分数阶差分方程边值问题的研究越来越受到关注,涌现了不少分数阶差分方程边值问题的文献,但主要都是集中在有限差分及两点边值问题,目前未见文献涉及带无穷边值条件的分数差分方程边值问题.本文应用Leray-Schauder非线性选择定理,讨论了一类无穷分数差分方程三点边值问题,获得了该边值问题正解存在的一些充分条件,并举例说明了所得结果的有效性.

英文摘要：

Accompany with the development of the theory for fractional calculus, boundary value problems of fractional difference equations have attracted increasing attention in recent years. Some research papers on the discrete fractional boundary value problems have appeared, but the works are almost concerned with two-point boundary value problems and finite boundary value conditions. Few papers consider the infinite boundary value condition at present. In this paper, by using the Leray-Schauder nonlinear alternative theorem, it is concerned with the three-point boundary value problems for infinite fractional difference equation. Sufficient conditions for the existence of positive solution are established. Additionally, an example is given to guarantee the main results.