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Zbl 0958.45011
Agarwal, Ravi P.; O'Regan, Donal
Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease. (English)
[J] Nonlinear Anal., Theory Methods Appl. 40, No.1-8, A, 21-35 (2000). ISSN 0362-546X

The object of the paper is the following nonlinear integral equation $$x(t)=\int^t_{t-\tau} k(t,s)f\bigl(s,x(s) \bigr)ds\quad (t\in \bbfR),\tag 1$$ 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 {\it R. W. Leggett} and {\it 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.
[J.Banaś (Rzeszów)]

MSC 2000:: *45M15 Periodic solutions of integral equations
45G10 Nonsingular nonlinear integral equations
92C60 Medical epidemiology

Keywords: periodic solutions; nonlinear integral equation; spread of infectious diseases; fixed point theorems

Citations: Zbl 0448.47044

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