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Dynamics of a max-type system of difference equations

Dynamics of a max-type system of difference equations In this paper, we study the following max-type system of difference equations $$\begin{aligned} x_{n} = \max \left\{ \frac{1}{y_{n-m}} , \frac{\alpha _n}{x_{n-r}}\right\} ,\quad y_{n} = \max \left\{ \frac{1}{x_{n-m}} , \frac{\beta _n}{y_{n-r}}\right\} , \quad n=0,1,\ldots , \end{aligned}$$ x n = max 1 y n - m , α n x n - r , y n = max 1 x n - m , β n y n - r , n = 0 , 1 , … , where $$\alpha _n,\beta _n\in (0, 1)$$ α n , β n ∈ ( 0 , 1 ) are two sequences with $$\sup \{\max \{\alpha _n,\beta _n\}:n=0,1,\ldots \} <1$$ sup { max { α n , β n } : n = 0 , 1 , … } < 1 , $$r,m\in \{1,2,\ldots \}$$ r , m ∈ { 1 , 2 , … } with $$r\ne m$$ r ≠ m and the initial values $$x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+\infty )$$ x - d , y - d , x - d + 1 , y - d + 1 , … , x - 1 , y - 1 ∈ ( 0 , + ∞ ) with $$d=\max \{r,m\}$$ d = max { r , m } . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Dynamics of a max-type system of difference equations

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-016-0124-x
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study the following max-type system of difference equations $$\begin{aligned} x_{n} = \max \left\{ \frac{1}{y_{n-m}} , \frac{\alpha _n}{x_{n-r}}\right\} ,\quad y_{n} = \max \left\{ \frac{1}{x_{n-m}} , \frac{\beta _n}{y_{n-r}}\right\} , \quad n=0,1,\ldots , \end{aligned}$$ x n = max 1 y n - m , α n x n - r , y n = max 1 x n - m , β n y n - r , n = 0 , 1 , … , where $$\alpha _n,\beta _n\in (0, 1)$$ α n , β n ∈ ( 0 , 1 ) are two sequences with $$\sup \{\max \{\alpha _n,\beta _n\}:n=0,1,\ldots \} <1$$ sup { max { α n , β n } : n = 0 , 1 , … } < 1 , $$r,m\in \{1,2,\ldots \}$$ r , m ∈ { 1 , 2 , … } with $$r\ne m$$ r ≠ m and the initial values $$x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+\infty )$$ x - d , y - d , x - d + 1 , y - d + 1 , … , x - 1 , y - 1 ∈ ( 0 , + ∞ ) with $$d=\max \{r,m\}$$ d = max { r , m } .

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Jan 27, 2016

References