Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

Abstract

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

Solvability of Nonlocal Fractional Boundary Value Problems

“…Lemma 7 (see [17]). A real symmetric matrix is positive definite if there exists a real nonsingular matrix such that = † , where † is the transpose.…”

“…There are many literatures dealing with the discrete fractional difference equation subject to various boundary value conditions or initial value conditions. We refer to [8][9][10][11][12][13][14][15][16][17][18] and references therein. However, we note that these results were usually obtained by analytic techniques and various fixed point theorems.…”

“…For example, in [13][14][15][16], authors investigated the existence to some boundary value problems by fixed point theorems on a cone. In [17], we given the existence of multiple solutions for a fractional difference boundary value problem with parameter by establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory. As we know, the coincidence degree theory has played an important role in dealing with the existence and multiple solutions for differential equations, which include the boundary value problems.…”

Existence and uniqueness of solutions for nonlinear Caputo fractional difference equations

“…Then, after one year, Holm [7] explored (N − 1, 1) fractional BVPs in his Ph.D. Dissertation. Xie et al [10] studied multiple solutions for a fractional difference BVP by a variational approach. Sitthiwirattham et al [9] and Reunsumrit et al [8] considered BVP for fractional difference equations with three-point and four-point fractional sum boundary conditions, respectively.…”

Multiple Positive Solutions of Nonlinear Boundary Value Problem for Finite Fractional Difference

“…In [28], we established the existence conditions for a boundary value problem by using the coincidence degree theory. In [29], authors pointed out the existence of multiple solutions for a FBVP with parameter by establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory.…”