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

Abstract

By using Krasnoselskii's fixed point theorem, we study the existence of positive solutions to the three-point summation boundary value problem ${\Delta }^{2}u(t-1)+a\left(t\right)f\left(u\right(t\left)\right)=0$, $t\in \{\mathrm{1,2},\dots ,T\}$, $u\left(0\right)=\beta {\sum }_{s=1}^{\eta }u\left(s\right)$, $u(T+1)=\alpha {\sum }_{s=1}^{\eta }u\left(s\right)$, where $f$ is continuous, $T\ge 3$ is a fixed positive integer, $\eta \in \{\mathrm{1,2},...,T-1\}$, , and $\Delta u(t-1)=u\left(t\right)-u(t-1)$. We show the existence of at least one positive solution if $f$ is either superlinear or sublinear.