Electron. J. Diff. Equ., Vol. 2015 (2015), No. 234, pp. 1-12.

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

Tatjana Stevic, Bratislav Iricanin

Abstract:
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.

Submitted June 6 2015. Published September 11, 2015.
Math Subject Classifications: 39A10, 39A20.
Key Words: Max-type system of difference equations; cyclic system; positive solutions; boundedness character; global attractivity.

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  Tatjana Stevic
Faculty of Electrical Engineering
Belgrade University, Bulevar Kralja Aleksandra 73
11000 Beograd, Serbia
email: tanjas019@gmail.com
Bratislav Iricanin
Faculty of Electrical Engineering
Belgrade University, Bulevar Kralja Aleksandra 73
11000 Beograd, Serbia
email: iricanin@etf.rs

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