Abstract and Applied Analysis
Volume 2 (1997), Issue 3-4, Pages 271-279
doi:10.1155/S1085337597000390
Existence of a positive solution for an nth order boundary value problem for nonlinear difference equations
1Department of Mathematics, Auburn University, Auburn, Alabama 38649, USA
2Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088, USA
Received 26 October 1997
Copyright © 1997 Johnny Henderson and Susan D. Lauer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.