Existence of a positive solution for an nth order boundary value problem for nonlinear difference equations. (1997) - Henderson, Johnny; Lauer, Susan D.
Abstract : Then th order eigenvalue problem:                                                                                  Δ n x ( t ) = ( − 1 ) n − k λ f ( t, x ( t ) ),                     t ∈ [ 0, T ], x ( 0 ) = x ( 1 ) = ⋯ = x ( k − 1 ) = x ( T + k + 1 ) = ⋯ = x ( T + n ) = 0, is considered, wheren ≥ 2 andk ∈ { 1, 2, …, n − 1 } are given. Eigenvaluesλ are determined forf continuous and the case where the limitsf 0 ( t ) = lim n → 0 + f ( t, u ) u andf ∞ ( t ) = lim n → ∞ f ( t, u ) u exist for allt ∈ [ 0, T ] . Guo's fixed point theorem is applied to operators defined on annular regions in a cone.