Multiple solutions for \(2m\text{th}\)-order Sturm-Liouville boundary value problems
type: Item
instance of: scholarly article
title: Multiple solutions for \(2m\text{th}\)-order Sturm-Liouville boundary value problems
zbMATH Open document ID: 0958.34018
DOI: 10.1016/S0898-1221(00)00156-5
author: Chuan J. Chyan
author: Johnny Henderson
publication date: 2000-11-13T00:00:00Z
review text: The authors consider the \(2m\)th-order ordinary differential equation \[ (-1)^{m}y^{(2m)} = f(t, y), \qquad 0\leq t\leq 1,\tag{1} \] with \(m\geq 1\) and satisfying the boundary conditions \[ \alpha_{i+1}y^{(2i)}(0) - \beta_{i+1}y^{(2i+1)}(0) = 0, \quad 0\leq i\leq m-1, \] \[ \gamma_{i+1}y^{(2i)}(1) - \delta_{i+1}y^{(2i+1)}(1) = 0, \quad 0\leq j\leq m-1, \tag{2} \] where \(f:\;[0, 1]\times [0, \infty) \to [0, \infty)\) is continuous, \(f(t, y) \neq 0\) on any subinterval of \([0, 1]\), and for all \(0<y<\infty \), and for each \(1\leq i\leq m\), \(\alpha_{i}, \beta_{i}, \gamma_{i}, \delta_{i} \geq 0\) such that \[ \rho_{i}=\gamma_{i} \beta_i +\alpha_i \gamma_i +\alpha_i \delta_i>0. \tag{3} \] The techniques for obtaining multiple solutions are involved in applications of Krasnosel'skii fixed-point theorem and properties of Green's function. The authors found by classical technique growth restrictions on \(f\) which yield the existence of positive and multiple solutions. Also, they show that any number of positive solutions to (1), (2) can be obtained when appropriate combinations of the proposed four assumptions are imposed on \(f\).
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1528550
zbMATH Keywords: boundary value problem
zbMATH Keywords: multiple solutions
zbMATH Keywords: growth conditions
MaRDI profile type: Publication
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1528550