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Multiple positive periodic solutions for a nonlinear first order functional difference equation. (English) Zbl 1204.39013
The authors investigate the existence of positive periodic solutions for the following first order functional difference equation $x(n+1)-x(n)=-a(n)x(n)+f(n,x(h_1(n)),\dots,x(h_m(n))),\quad n=0,1,2,\dots,$
where $$a(n),h_i(n)$$ are periodic sequences with a common period $$T\geq 1$$, and $$f(n,x_1,\dots,x_m)$$ is a real-valued function which is $$T$$-periodic on $$n$$ and continuous on $$x_i\geq 0$$.
By using the Leggett-Williams fixed point theorem, the authors establish three theorems, which ensure that the above functional difference equation has at least three non-negative periodic solutions. The main results of this paper provide a partial answer to a problem proposed by Y. N. Raffoul [Electron. J. Differ. Equ. 2002, Paper No. 55, 8p. (2002; Zbl 1007.39005)]. In addition, the authors apply their results to a hematopoiesis model in population dynamics.

##### MSC:
 39A23 Periodic solutions of difference equations 39A10 Additive difference equations 39A12 Discrete version of topics in analysis 34K13 Periodic solutions to functional-differential equations 92D25 Population dynamics (general)
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