Positive solutions for a second-order three-point discrete boundary value problem

Abstract

We establish the existence of at least three positive solutions for the second-order three-point discrete boundary value problem:

$$\Delta ^{2}y(k-1)+f(k,y(k))=0,\quad k\in \{1,\ldots ,T\},$$

$$y(0)=0,\quad y(T+1)=\alpha y(n),$$

where f is continuous, T≥3 and n∈{2,…,T−1} are two fixed positive integers, constant α>0 such that α n<T+1. Under suitable conditions, we accomplish this by using the property of the associate Green’s function and Leggett-Williams fixed point theorem.