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 Δ 3 u ( k − 1 ) = λ a ( k ) f ( k , u ( k ) ) , k ∈ [ 1 , N − 2 ] Z , Δ 2 u ( η ) = α Δ u ( N − 1 ) , Δ u ( 0 ) = − β u ( 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 > 4 N\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 , 1 N − 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 − 2 2 + 1 , N − 2 Z \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.

Positive Solutions for Periodic Boundary Value Problems of Fractional Differential Equations with Sign-Changing Nonlinearity and Green’s Function

“…Hence, a natural question is what would happen if the nonlinearity or the Green's function is sign-changing. Several papers have considered the positive solutions for BVPs with sign-changing nonlinearity and sign-changing Green's function [28][29][30][31][32][33][34]. Ma [29] studied the BVP with sign-changing Green's function:…”