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 Jm by
Jm(u)=∑k=1mA(k−1,Δu(k−1))−λ∑k=1mF(k,u(k))
and establish that a critical point of Jm 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

p(k)-Laplace; multiple solutions; discrete inclusions; three critical points theorem; locally Lipschitz continuous functions