Multiple positive solutions of second-order differential equations
type: Item
instance of: scholarly article
title: Multiple positive solutions of second-order differential equations
zbMATH Open document ID: 1077.34025
author: Xiaojing Yang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-06-27T00:00:00Z
review text: Consider the periodic boundary value problem \[ x'' + f (t,x,x')=0, \quad x(0)= x(T),\quad x'(0)=x'(T),\tag{P} \] where \(f: \mathbb R\times \mathbb R\times \mathbb R \rightarrow \mathbb R\) is continuously differentiable, \(f(t,0,0)>0,\) \(T\)-periodic in \(t\) and satisfies some Nagumo condition. The author introduces the concept of strict lower and upper solutions to problem (P) and assumes that there exist \(n-1\) strict lower solutions \(\alpha_k (t)\) and \(n\) strict upper solutions \(\beta_k(t)\) satisfying \[ 0<\beta_1 (t) < \alpha_1 (t) <\beta_2 (t) < \dots <\alpha_{n-1} (t) < \beta_n (t). \] Then (P) has at least \(2n-1\) positive periodic solutions. The proof is based on the coincidence degree theory.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 2183187
zbMATH Keywords: positive periodic solutions
zbMATH Keywords: upper and lowersolutions
zbMATH Keywords: coincidence degree
MaRDI profile type: Publication
full work available at URL: j.na.2005.03.013
OpenAlex ID: W2050334243
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DOI: 10.1016/J.NA.2005.03.013
description: scientific article; zbMATH DE number 2183187