Long-term behavior of a cyclic max-type system of difference equations

We study the long-term behavior of positive solutions of the cyclic system of difference equations $$ x^{(i)}_{n+1}=\max\Big\{\alpha,\frac{(x^{(i+1)}_n)^p}{(x^{(i+2)}_{n-1})^q}\Big\}, \quad i=1,\ldots,k,\; n\in\mathbb{N}_0, $$ where $k\in\mathbb{N}$, $\min\{\alpha, p, q\}>0$ and where we regard t...

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Bibliographic Details
Published in:Electronic journal of differential equations 2015-09, Vol.2015 (234), p.1-12
Main Authors: Tatjana Stevic, Bratislav Iricanin
Format: Article
Language:English
Subjects:
boundedness character
cyclic system
global attractivity
Max-type system of difference equations
positive solutions
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Summary:We study the long-term behavior of positive solutions of the cyclic system of difference equations $$ x^{(i)}_{n+1}=\max\Big\{\alpha,\frac{(x^{(i+1)}_n)^p}{(x^{(i+2)}_{n-1})^q}\Big\}, \quad i=1,\ldots,k,\; n\in\mathbb{N}_0, $$ where $k\in\mathbb{N}$, $\min\{\alpha, p, q\}>0$ and where we regard that $x^{(i_1)}_n=x^{(i_2)}_n$ when $i_1\equiv i_2$ (mod $k$). We determine the set of parameters $\alpha$, p and q in $(0,\infty)^3$ for which all such solutions are bounded. In the other cases we show that the system has unbounded solutions. For the case p=q we give some sufficient conditions which guaranty the convergence of all positive solutions. The main results in this paper generalize and complement some recent ones.
ISSN:1072-6691