# Analysis of a nonlinear integral equation modelling infection diseases

## Abstract

To describe the spread of virus diseases with contact rate that varies seasonally, the following delay integral equation has been proposed by K.L. Cooke and J.L. Kaplan

$x\left( t\right) =\int_{t-\tau}^{t}f\left( s,x\left( s\right) \right) ds$

This model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to describe and improve recent results on this equation, obtained by the authors in the last decade. Our analysis is concerned with the existence, uniqueness, approximation and continuous dependence on data of the positive solutions of the initial value problem, and of the periodic solutions. We use topological methods (fixed point theorems, continuation principle) and monotone iterative techniques.

## Authors

Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Eduard Kirr
Faculty of Mathematics and Informatics University “Babes-Bolyai”, Cluj, Romania

## Keywords

nonlinear integral equation; positive solutions; periodic solutions; fixed point; continuation principle; monotone iterations; continuous dependence;population dynamics.

## Paper coordinates

R. Precup, E. Kirr, Analysis of a nonlinear integral equation modelling infection diseases, In: “Proceedings of the International Conference on Analysis and Numerical Computation, Timişoara, May 19-21, 1997”, Şt. Balint ed., West Univ. of Timişoara, 1997, 178-195.

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##### Journal

“Proceedings of the International Conference on Analysis and Numerical Computation, Timişoara

##### Publisher Name

West Univ. of Timişoara