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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 $$x(t)=\int^t_{t-\tau} k(t,s)f\bigl(s,x(s) \bigr)ds\quad (t\in \bbfR),\tag 1$$ 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.

MSC:
 45M15 Periodic solutions of integral equations 45G10 Nonsingular nonlinear integral equations 92C60 Medical epidemiology
Full Text:
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. [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) · Zbl 0273.45001 [4] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · 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.