Multiplicity of solutions to discrete inclusions with the \(p(k)\)-Laplace type equations
This paper is concerned with the existence and multiplicity of solutions to discrete inclusions with the \(p(k)\)-Laplace type equations. They begin by giving some basic definitions and preliminary results where they define the generalized gradient of a function, coercive and anti-coercive function. Then they introduce three critical points theorem for locally Lipschitz functionals. With this theorem, the authors prove the existence of three \(m\)-periodic solutions and show that at least two of which are non-trivial. For this, they define a functional \(J_m\) by \[ J_{m}(u) = \sum_{k=1}^{m}A\big(k-1,\Delta u(k-1)\big) - \lambda \sum_{k=1}^{m}F\big(k,u(k)\big) \] and establish that a critical point of \(J_m\) is a solution of there equations. By using three critical points theorem, they show that the problem has at least three solutions, at least two of which are necessarily non-zero.
scientific article; zbMATH DE number 6881274
scientific article