Existence and multiplicity results for nonlinear periodic boundary value problems
type: Item
instance of: scholarly article
title: Existence and multiplicity results for nonlinear periodic boundary value problems
zbMATH Open document ID: 1195.34033
author: Xinan Hao
author: Lishan Liu
author: Yonghong Wu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-03-29T00:00:00Z
review text: Consider the following periodic boundary value problems \[ \begin{aligned} -u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \;\;}0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \;\;}u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\end{aligned} \] and \[ \begin{aligned} u^{\prime \prime }+a\left( t\right) u &=\lambda f\left( t,u\right) ,\text{ \;\;}0\leq t\leq 2\pi , \\ u\left( 0\right) &=u\left( 2\pi \right) ,\text{ \;\;}u^{\prime }\left( 0\right) =u^{\prime }\left( 2\pi \right) ,\end{aligned} \] where \(a\in L_{1}\left( 0,2\pi \right) \), \(f:\left[ 0,2\pi \right] \times [ 0,+\infty )\rightarrow [ 0,+\infty )\)\ is continuous, \(\lambda \) is a positive parameter. The criteria for the existence, nonexistence and multiplicity of positive solutions are established by using the Global continuation theorem, fixed point index theory and approximate method. The results obtained herein generalize and complement some previous findings of [\textit{J. R. Graef, L. Kong} and \textit{H. Wang}, J. Differ. Equations 245, No.~5, 1185--1197 (2008; Zbl 1203.34028)] and some other known results.
reviewed by: Kh. R. Mamedov
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5687526
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: positive solution
zbMATH Keywords: global continuation theorem
zbMATH Keywords: fixed point index
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
MaRDI profile type: Publication
full work available at URL: j.na.2009.12.044
OpenAlex ID: W2040231375
cites work: An existence result about periodic boundary value problems of second order differential equations
cites work: Extremal solutions of second order nonlinear periodic boundary value problems
cites work: The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems
cites work: A singular nonlinear second-order periodic boundary value problem
cites work: Multiplicity of positive periodic solutions to superlinear repulsive singular equations
cites work: Existence of two positive solutions of a singular nonlinear periodic boundary value problem
cites work: Positive solution of a singular nonlinear third-order periodic boundary value problem
cites work: Positive solutions of higher-order periodic boundary value problems
cites work: On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations
cites work: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem
cites work: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem
cites work: Positive solutions for some second-order four-point boundary value problems
cites work: Existence and multiplicity results for nonlinear boundary value problems
cites work: Q3706043
cites work: Q3994493
cites work: Q4716265
DOI: 10.1016/J.NA.2009.12.044
Recommended article: Global structure of positive solutions for superlinear second-order periodic boundary value problems
Recommended article: Q3378401
Recommended article: Exact solution of one boundary-value problem
Recommended article: First order ordinary differential equations with several periodic solutions
Recommended article: Periodic boundary value problems for integro-differential equations of volterra type
Recommended article: Q4842467
Recommended article: A Knobloch-type result for \(p(t\))-Laplacian systems.
Recommended article: Q3319071
Recommended article: Q3185084
Recommended article: Multiplicity of positive periodic solutions to nonlinear boundary value problems with a parameter
description: scientific article; zbMATH DE number 5687526