Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function

Abstract

In this article, we consider a discrete nonlinear third-order boundary value problem Δ3u(k−1)=λa(k)f(k,u(k)),k∈[1,N−2]Z,Δ2u(η)=αΔu(N−1),Δu(0)=−βu(0),u(N)=0,\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\Delta }^{3}u\left(k-1)=\lambda a\left(k)f\left(k,u\left(k)),\hspace{1em}k\in {\left[1,N-2]}_{{\mathbb{Z}}},\hspace{1.0em}\\ {\Delta }^{2}u\left(\eta )=\alpha \Delta u\left(N-1),\Delta u\left(0)=-\beta u\left(0),\hspace{1em}u\left(N)=0,\hspace{1.0em}\end{array}\right. where N>4N\gt 4 is an integer, λ>0\lambda \gt 0 is a parameter. f:[1,N−2]Z×[0,+∞)→[0,+∞)f:{\left[1,N-2]}_{{\mathbb{Z}}}\times \left[0,+\infty )\to \left[0,+\infty ) is continuous, a:[1,N−2]Z→(0,+∞)a:{\left[1,N-2]}_{{\mathbb{Z}}}\to \left(0,+\infty ), α∈0,1N−1\alpha \in \left[0,\frac{1}{N-1}\right), β∈0,2(1−α(N−1))N(2−α(N−1))\beta \in \left[0,\frac{2\left(1-\alpha \left(N-1))}{N\left(2-\alpha \left(N-1))}\right), and η∈N−22+1,N−2Z\eta \in {\left[\left[\frac{N-2}{2}\right]+1,N-2\right]}_{{\mathbb{Z}}}. With the sign-changing Green’s function, we obtain not only the existence of positive solutions but also the multiplicity of positive solutions to this problem.

Keywords

• third-order difference equation
• three-point boundary conditions
• sign-changing green’s function
• guo-krasnoselskii’s fixed-point theorem
• 39a10
• 39a12