Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease
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.
scientific article; zbMATH DE number 1472205