Abstract
Using a variational approach and critical point theory, we investigate the existence of solutions for a fractional difference boundary value problem. MSC:26A33, 35A15, 39A12, 44A55.
Highlights
In this work, using variational methods and critical point theory, we study the fractional difference boundary value problem ⎧ ⎨T νt– (t νν– x(t)) = f (x(t + ν – )),⎩x(ν – ) = [t νν– x(t)] t=T =, t ∈ [, T]N, ( . )where ν ∈ (, ), t ν ν–and ν t are, respectively, the left fractional difference and the right fractional difference operators, t ∈ [, T]N := {, . . . , T}, and f : R → R is continuous.Fractional calculus has a long history, and there is renewed interest in the study of both fractional calculus and fractional difference equations
Using a variational approach and critical point theory, we investigate the existence of solutions for a fractional difference boundary value problem
In this work, using variational methods and critical point theory, we study the fractional difference boundary value problem νt– (t νν– x(t)) = f (x(t + ν – ))
Summary
In this work, using variational methods and critical point theory, we study the fractional difference boundary value problem. Atici and Eloe [ ] considered the existence of positive solutions for the following twopoint boundary value problem for a nonlinear finite fractional difference equation:. In [ ], the authors used the mountain pass theorem, a linking theorem, and Clark’s theorem to establish the existence of multiple solutions for a fractional difference boundary value problem with a parameter. We note that there are many papers in the literature [ – ] which discuss discrete problems via variational and critical point theory. In [ ], Tian and Henderson studied the nth order nonlinear difference equation n r(t – n) nx(t – n) + f t, x(t) = , t ∈ Z, and established some existence results for anti-periodic solutions under various assumptions on the nonlinearity. Assuming an Ambrosetti-Rabinowitz type condition, we show that problem ( . ) has many solutions if the nonlinearity is odd
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