By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results. Recently, fractional differential and difference "operators" are found themselves in concrete applications, and hence attention has to be paid to associated fractional difference 2 Abstract and Applied Analysis and differential equations under various boundary or side conditions. For example, a recent paper by Atici and Eloe 12 explores some of the theories of a discrete conjugate fractional BVP. Similarly, in 13 , a discrete right-focal fractional BVP is analyzed. Other recent advances in the theory of the discrete fractional calculus may be found in 14, 15 . In particular, an interesting recent paper by Atici and Şengül 16 addressed the use of fractional difference equations in tumor growth modeling. Thus, it seems that there exists some promise in using fractional difference equations as mathematical models for describing physical problems in more accurate manners.In order to handle the existence problem for fractional BVPs, various methods among which are some standard fixed-point theorems can be used. In this paper, however, we show that variational methods can also be applied. A good reason for picking such an approach is that, in Atici and Şengül 16 , some basic fractional calculuses are developed and a simple variational problem is demonstrated, and hence advantage can be taken in obvious manners. We remark, however, that fractional difference operators can be approached in different manners and one by means of operator convolution rings can be found in the book by Cheng 17, Chapter 3 published in 2003.More specifically, in this paper, we are interested in the existence of multiple solutions for the following 2ν-order fractional difference boundary value problem

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In this paper, the concepts of a right fractional sum and right fractional difference operators are introduced. Some basic properties of a right fractional sum and right fractional difference operators are proved. According to these properties of a right fractional sum and right fractional difference operators, we studied an initial problem and a boundary value problem with two-point boundary conditions. We hope that the present work will facilitate solving a fractional difference equation with right fractional difference operators.