A new existence theory for positive periodic solutions to functional differential equations
type: Item
instance of: scholarly article
title: A new existence theory for positive periodic solutions to functional differential equations
zbMATH Open document ID: 1073.34082
DOI: 10.1016/S0898-1221(04)90120-4
author: Daqing Jiang
author: Xiaojie Xu
author: A-ying Wan
publication date: 2005-03-08T00:00:00Z
review text: The aim of the paper is to obtain the existence of positive periodic solutions for the nonlinear functional-differential equation \[ y'(t)=-a(t)y(t)+g(t,y(t-\tau(t))), \tag{1} \] where \(a\in C(\mathbb{R},(0,\infty))\), \(\tau\in C(\mathbb{R},\mathbb{R})\), \(g\in C(\mathbb{R}\times[0,\infty),[0,\infty))\) and \(a,\tau, g(t,\cdot)\) are all \(\omega\)-periodic functions, \(\omega>0\) is a constant. By using a fixed-point theorem in a cone, the authors establish the following result: equation (1) has at least one positive \(\omega\)-periodic solution provided one of the following conditions holds: \[ \liminf_{u\downarrow 0}\min_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}>1\quad \text{and} \quad \limsup_{u\uparrow\infty}\max_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}<1;\tag{i} \] \[ \limsup_{u\downarrow 0}\max_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}<1\quad \text{and} \quad \liminf_{u\uparrow\infty}\min_{0\leq t\leq\omega}\frac{g(t,u)}{a(t)u}=\infty.\tag{ii} \]
reviewed by: Jurang Yan
Mathematics Subject Classification ID: 34K13
zbMATH DE Number: 2142385
zbMATH Keywords: functional-differential equation
zbMATH Keywords: periodic solution
zbMATH Keywords: fixed-point theorem
MaRDI profile type: Publication
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full work available at URL: s0898-1221(04)90120-4
OpenAlex ID: W2092426687
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2142385