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Existence of positive periodic solutions for difference equations with feedback control. (English) Zbl 1085.39009

The authors’ purpose in this work is, by using Mawhin’s continuation theorem of coincidence degree theory, to study the existence of positive periodic solutions of the following nonautonomous difference model with feedback control: \[ \begin{cases} N(n+1)=N(n)\exp\left[r(n)\left(1-\frac{N(n-m)}{k(n)}-c(n)\mu(n)\right)\right],\\ \Delta\mu(n)=-a(n)\mu(n)+b(n)N(n-m), \end{cases} \] where \(a:\mathbb Z\to(0,1)\), \(c,k,r,b:\mathbb Z\to\mathbb R^+\) are all \(\omega\)-periodic functions and \(m\) is a positive integer, \(\mathbb Z,\mathbb R^+\) denote the sets of all integers and all positive real numbers, respectively, \(\Delta\) is the first-order forward difference operator \(\Delta\mu(n)=\mu(n+1)-\mu(n)\).

MSC:

39A11 Stability of difference equations (MSC2000)
93B52 Feedback control
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