Positive solutions of difference equations with two-point right focal boundary conditions
type: Item
instance of: scholarly article
title: Positive solutions of difference equations with two-point right focal boundary conditions
zbMATH Open document ID: 0921.39001
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-09-29T00:00:00Z
review text: The author considers the \(n\)th-order difference equation \[ (-1)^{n-p} \Delta^ny= F(k,y, \Delta y,\dots, \Delta^{n-1}y), \quad k\in[0,N], \tag{1} \] together with the two-point right focal boundary conditions \[ \Delta^i y(0)=0, \quad 0\leq i\leq p-1, \qquad \Delta^i y(N+1)=0, \quad p\leq i\leq n-1, \tag{2} \] where \(n,p,N\) are fixed integers with \(n\geq 2\), \(0\leq p\leq n-1\) and \(N\geq p\), and \(\Delta y(k)= y(k+1)- y(k)\) and \(\Delta^ny(k)= \Delta(\Delta^{n-1}y(k))\) for \(n\geq 2\). It is assumed that there exist continuous functions \(f,F\) and \(a,b: [0,N]\to \mathbb{R}\) such that \[ a(k)\leq \frac{F(k,u,u_1,\dots, u_{n-1})} {f(u)}\leq b(k), \quad u\in (0,+\infty), \] \(a(k)\geq 0\), \(k\in [0,N]\), \(a(k)\not\equiv 0\), \(k\in [p,N]\). By a positive solution of (1), (2) we mean a nontrivial \(y: [0,N+n]\to [0,+\infty)\) satisfying (1), (2). By defining an appropriate Banach space and cone, the author uses a fixed point theorem due to Krasnosel'skij, to give existence results for two positive solutions of (1), (2). Further, the author considers a special case of (1) for \(n=2\), \(p=1\), subject to some boundary conditions, and establishes upper and lower bounds for the positive solutions.
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1215753
zbMATH Keywords: difference equation
zbMATH Keywords: two-point right focal boundary conditions
zbMATH Keywords: positive solutions
zbMATH Keywords: upper and lower bounds
author: Patricia J. Y. Wong
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5983
OpenAlex ID: W2031957848
cites work: Q3732891
cites work: Q4033029
cites work: Q4387609
cites work: Oscillation theorems and existence of positive monotone solutions for second order nonlinear difference equations
cites work: On the existence of positive solutions of higher order difference equations
cites work: Stabilization of solutions of a degenerate nonlinear diffusion problem
cites work: Bounds for positive solutions for a focal boundary value problem
cites work: Nonlinear elliptic problems in annular domains
cites work: Semilinear elliptic problems in annular domains
cites work: AN UNEXPECTED STABILITY RESULT OF THE NEAR-EXTINCTION DIFFUSION FLAME FOR NON-UNITY LEWIS NUMBERS
cites work: Positive solutions for singular higher order nonlinear equations
cites work: Multiple Stable Solutions of Nonlinear Boundary Value Problems Arising in Chemical Reactor Theory
cites work: On the structure of solutions of an equation in catalysis theory when a parameter is large
cites work: Q4327739
cites work: Positive solutions for (n − 1, 1) conjugate boundary value problems
cites work: Multiple positive solutions of some boundary value problems
cites work: On the Existence of Positive Solutions of Ordinary Differential Equations
cites work: Q4764499
cites work: Eigenvalues of generalized Gelfand models
cites work: On the nonlinear equations Δ𝑢+𝑒^{𝑢}=0 and ∂𝑣/∂𝑡=Δ𝑣+𝑒^{𝑣}
cites work: Existence of positive radial solutions for semilinear elliptic equations in the annulus
cites work: Singular nonlinear boundary value problems for second-order ordinary differential equations
cites work: Q3274320
cites work: Q3994493
cites work: Comparison of eigenvalues for focal point problems for \(n\)-th order difference equations
cites work: Q4015775
cites work: Q4878954
cites work: Q5732050
cites work: Two positive solutions of a boundary value problem for difference equations
cites work: Q4862635
cites work: Q4862647
cites work: Positive solutions to certain classes of singular nonlinear second order boundary value problems
cites work: Solutions of a Differential Equation Arising in Chemical Reactor Processes
cites work: Boundary Value Problems for annth Order Linear Difference Equation
cites work: On the existence of positive solutions for semilinear elliptic equations in the annulus
cites work: Q4214726
cites work: Eigenvalues of boundary value problems for higher-order differential equations
cites work: Double positive solutions of \((n,p)\) boundary value problems for higher order difference equations
cites work: On the existence of solutions of singular boundary value problems for higher order difference equations
cites work: On the eigenvalues of boundary value problems for higher order difference equations
cites work: On two-point right focal eigenvalue problems
cites work: On eigenvalue intervals and twin eigenfunctions of higher-order boundary value problems
DOI: 10.1006/JMAA.1998.5983
description: scientific article; zbMATH DE number 1215753