Existence and multiplicity of solutions for nonlinear second order difference boundary value problems

In the present paper, the following nonlinear second order difference boundary value problems Δ[p(n−1)Δu(n−1)]+q(n)u(n)+f(n,u(n))=0,n∈Z(1,N),u(0)=u(N),p(0)Δu(0)=p(N)Δu(N) are studied by using the critical theory, and three new multiple results of solutions are obtained, where Z(1,N)={1,2,…,N},f:Z(1,N)×R→R is a continuous function in the second variable, p,q:Z(0,N)→R and p(n)≠0p(n)≠0.