Positive solutions of boundary value problem for singular positone and semi-positone third-order difference equations
Abstract: This article studies the boundary value problems for the third-order nonlinear singular difference equations
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 1 publication
References 17 publications
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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
