Solutions of 2nth-Order Boundary Value Problem for Difference Equation via Variational Method

Abstract

The variational method and critical point theory are employed to investigate the existence of solutions for 2nth-order difference equation Δn(pk−nΔnyk−n)+(−1)n+1f(k,yk)=0 for k∈[1,N] with boundary value condition y1−n=y2−n=⋯=y0=0,  yN+1=⋯=yN+n=0 by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.