Existence of a positive solution for an nth order boundary value problem for nonlinear difference equations

Abstract

The nth order eigenvalue problem:                                          Δnx(t)=(−1)n−kλf(t,x(t)),          t∈[0,T],x(0)=x(1)=⋯=x(k−1)=x(T+k+1)=⋯=x(T+n)=0, is considered, where n≥2 and k∈{1,2,…,n−1} are given. Eigenvalues λ are determined for f continuous and the case where the limits f0(t)=limn→0+f(t,u)u and f∞(t)=limn→∞f(t,u)u exist for all t∈[0,T]. Guo's fixed point theorem is applied to operators defined on annular regions in a cone.