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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 \[ x(t)=\int^t_{t-\tau} k(t,s)f\bigl(s,x(s) \bigr)ds\quad (t\in \mathbb{R}),\tag{1} \] 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 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:

45M15 Periodic solutions of integral equations
45G10 Other nonlinear integral equations
92C60 Medical epidemiology

Citations:

Zbl 0448.47044
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References:

[1] R.P. Agarwal, D. O’Regan, A fixed point theorem of Leggett Williams type with applications to single and multivalued maps, to appear.; R.P. Agarwal, D. O’Regan, A fixed point theorem of Leggett Williams type with applications to single and multivalued maps, to appear.
[2] Cooke, K. L.; Kaplan, J. L., A periodicity threshold theorem for epidemics and population growth, Mat. Biosci., 31, 87-104 (1976) · Zbl 0341.92012
[3] Corduneanu, C., Integral Equations and Stability of Feedback Systems (1973), Academic Press: Academic Press New York · Zbl 0268.34070
[4] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045
[5] Leggett, R. W.; Williams, L. R., A fixed point theorem with application to an infectious disease model, J. Math. Anal. Appl., 76, 91-97 (1980) · Zbl 0448.47044
[6] M. Meehan, D. O’Regan, Multiple nonnegative solutions to nonlinear integral equations on compact and semiinfinite intervals, to appear.; M. Meehan, D. O’Regan, Multiple nonnegative solutions to nonlinear integral equations on compact and semiinfinite intervals, to appear.
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