Abstract
By using a new fixed-point theorem introduced by Avery and Peterson (2001), we obtain sufficient conditions for the existence of at least three positive solutions for the equationΔ2x(k−1)+q(k)f(k,x(k),Δx(k))=0, fork∈{1,2,…,n−1}, subject to the following two boundary conditions:x(0)=x(n)=0orx(0)=Δx(n−1)=0, wheren≥3.
Highlights
The second-order differential and difference boundary value problems arise in many branches of both applied and basic mathematics and have been extensively studied in the literature
The main tools used in the above works are fixed-point theorems
Bai et al [5] have applied this theorem to prove the existence of three positive solutions for the secondorder differential equation x (t) + q(t) f (t,x(t),x (t)) = 0, 0 < t < 1
Summary
The second-order differential and difference boundary value problems arise in many branches of both applied and basic mathematics and have been extensively studied in the literature. We refer the reader to [1–4] for some recent results for second-order nonlinear two-point boundary value problems. Bai et al [5] have applied this theorem to prove the existence of three positive solutions for the secondorder differential equation x (t) + q(t) f (t,x(t),x (t)) = 0, 0 < t < 1. The aim of this work is to establish the existence of three positive solutions for the second-order difference equation. We will depend on an application of a fixed-point theorem due to Avery and Peterson, which deals with fixed points of a cone-preserving operator defined on an ordered Banach space to obtain our main results, and an example to illustrate the main results in this paper
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