Positive Solutions to a Generalized Second-Order Difference Equation with Summation Boundary Value Problem

Abstract EN

By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem Δ2u(t-1)+a(t)f(u(t))=0, t∈{1,2,…,T}, u(0)=β∑s=1ηu(s), u(T+1)=α∑s=1ηu(s), where f is continuous, T≥3 is a fixed positive integer, η∈{1,2,...,T-1}, 0<α<(2T+2)/η(η+1), 0<β<(2T+2-αη(η+1))/η(2T-η+1), and Δu(t-1)=u(t)-u(t-1).

We show the existence of at least one positive solution if f is either superlinear or sublinear.