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

Abstract

Thenth 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, wheren≥2andk∈{1,2,…,n−1}are given. Eigenvaluesλare determined forfcontinuous and the case where the limitsf0(t)=limn→0+f(t,u)uandf∞(t)=limn→∞f(t,u)uexist for allt∈[0,T]. Guo's fixed point theorem is applied to operators defined on annular regions in a cone.