Infinitely many solutions for perturbed difference equations

Abstract

Using variational methods and critical point theory, the existence of infinitely many solutions for perturbed nonlinear difference equations with discrete Dirichlet boundary conditions is ensured.

A Variational Approach to Perturbed Discrete Anisotropic Equations

“…On the other hand, in recent years some researchers have studied the existence and multiplicity of solutions for equations involving the discrete -Laplacian operator by using various fixed point theorems, lower and upper solutions method, critical point theory and variational methods, Morse theory, and the mountain-pass theorem. For background and recent results, we refer the reader to [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. For example, Atici and Guseinov in [3] investigated the existence of positive periodic solutions for nonlinear difference equations with periodic coefficients by employing a fixed point theorem in cone.…”

“…In [4] Bian et al by using critical point theory studied a class of discrete -Laplacian periodic boundary value problems; some results were obtained for the existence of two positive solutions, three solutions, and multiple pairs of solutions of the problem when the parameter lies in some suitable infinite or finite intervals. In [15], by using variational methods and critical point theory, the existence of infinitely many solutions for perturbed nonlinear difference equations with discrete Dirichlet boundary conditions was discussed.…”

Existence results to a nonlinear p(k)-Laplacian difference equation

“…In this context, anisotropic discrete nonlinear problems involving p(k)-Laplacian operator seem to have attracted a great deal of attention due to its usefulness of modelling some more complicated phenomenon such us fluid dynamics and nonlinear elasticity. We refer the reader to [1,2,3,4,5,6,7,9,12,13,14,16,17,18,19,20,21,22,24] and references therein, where they could find the detailed background as well as many different approaches and techniques applied in the related area.…”

Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator

“…This method has been applied for (1) [23,26], the authors considered Eq. (1) in the special case, w.k/ D 1 and p.x/ D p. In [21], the authors applying variational methods, studied the existence of solutions of the following Kirchhoff-type discrete boundary value problems…”