Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease
type: Item
instance of: scholarly article
title: Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease
zbMATH Open document ID: 0958.45011
DOI: 10.1016/S0362-546X(00)85002-6
author: Ravi P. Agarwal
author: Donal O'Regan
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2001-04-10T00:00:00Z
review text: The object of the paper is the following nonlinear integral equation \[ x(t)=\int^t_{t-\tau} k(t,s)f\bigl(s,x(s) \bigr)ds\quad (t\in \mathbb{R}),\tag{1} \] where \(\tau>0\) is a fixed constant and \(f(t,x)\) is a real function being periodic in \(t\). The equation (1) is a generalization of an integral equation modelling the spread of infectious diseases. Using three fixed point theorems (nonlinear alternative of Leray-Schauder type, the Krasnoselskii fixed point theorem in a cone and a fixed point theorem of \textit{R. W. Leggett} and \textit{L. R. Williams} [J. Math. Anal. Appl. 76, 91-97 (1980; Zbl 0448.47044)]), the authors established a few interesting existence results for the equation (1). The assumptions of those theorems are rather complicated and too long to be presented here.
Mathematics Subject Classification ID: 45M15
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 92C60
zbMATH DE Number: 1472205
zbMATH Keywords: periodic solutions
zbMATH Keywords: nonlinear integral equation
zbMATH Keywords: spread of infectious diseases
zbMATH Keywords: fixed point theorems
reviewed by: Józef Banaś
MaRDI profile type: Publication
cites work: A periodicity threshold theorem for epidemics and population growth
cites work: Q3213489
cites work: Q3994493
cites work: A fixed point theorem with application to an infectious disease model
description: scientific article; zbMATH DE number 1472205