Dynamical Behavior of a System of Second-Order Nonlinear Difference Equations

cited. This paper is concerned with local stability, oscillatory character of positive solutions to the system of the two nonlinear difference equations 𝑥 𝑛+1 = 𝐴 + 𝑥 𝑝𝑛−1 /𝑦 𝑝𝑛 and 𝑦 𝑛+1 = 𝐴 + 𝑦 𝑝𝑛−1 /𝑥 𝑝𝑛 , 𝑛 = 0,1, .. . , where 𝐴 ∈ (0,∞) , 𝑝 ∈ [1, ∞) , 𝑥 𝑖 ∈ (0,∞) , and 𝑦 𝑖 ∈ (0, ∞) , 𝑖 = −1, 0 .


Introduction
Difference equation or discrete dynamical system is a diverse field which impacts almost every branch of pure and applied mathematics. Every dynamical system +1 = ( ) determines a difference equation and vice versa. Recently, there has been great interest in studying difference equation systems. One of the reasons for this is a necessity for some techniques which can be used in investigating equations arising in mathematical models describing real life situations in population biology, economics, probability theory, genetics, psychology, and so forth.
The theory of difference equations occupies a central position in applicable analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points.
Papaschinopoulos and Schinas [6] studied the system of two nonlinear difference equation: where , are positive integers.
Firstly we recall some basic definitions that we need in the sequel.
( , ) is bounded and persists if there exist positive constants , such that A solution ( , ) of (5) is said to be nonoscillatory about (0, 0) if both and are either eventually positive or eventually negative. Otherwise, it is said to be oscillatory about (0, 0).
A solution ( , ) of (5) is said to be nonoscillatory about equilibrium ( , ) if both − and − are either eventually positive or eventually negative. Otherwise, it is said to be oscillatory about equilibrium ( , ).

Main Results
In this section, we will prove the following results concerning system (5). (ii) The equilibrium point ( + 1, + 1) of system (5) is locally asymptotically stable if > 2 − 1.
(iii) From the proof of (ii), it is true.
International Journal of Differential Equations 3
Remark 5. If 0 ≤ ≤ 2 − 1, then system (5) has a unique positive equilibrium ( + 1, + 1). If > 2 − 1, then system (5) has multipositive equilibrium; however system (5) always has an equilibrium ( + 1, + 1). In this paper, we only investigate the dynamical behavior of the solution to this system associated with the equilibrium ( + 1, + 1). It is of further interest to study the behavior of system (5) about other equilibriums in the future.
International Journal of Differential Equations

Numerical Results
For confirming the results of this section, we consider numerical examples, which represent different types of solutions to (5).