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Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease. (English) Zbl 0958.45011

The object of the paper is the following nonlinear integral equation

x(t)= t-τ t k(t,s)fs , x ( s )ds(t),(1)

where τ>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 R. W. Leggett and 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.

MSC:
45M15Periodic solutions of integral equations
45G10Nonsingular nonlinear integral equations
92C60Medical epidemiology