On a System of Second-Order Nonlinear Difference Equations

This paper is concerned with dynamics of the solution to the system of two second-order nonlinear difference equations 1 1 1 n n n n x x A x y + − − = + , 1 1 1 n n n n y y A x y + − − = + ,  n = 0,1, , where ( ) 0, A∈ ∞ , ( ) 0, i x− ∈ ∞ , ( ) 0, i y− ∈ ∞ , i = 0, 1. Moreover, the rate of convergence of a solution that converges to the equilibrium of the system is discussed. Finally, some numerical examples are considered to show the results obtained.


Introduction
Difference equations or discrete dynamical systems are diverse field which impacts almost every branch of pure and applied mathematics.Every dynamical system ( ) , , , determines a difference equation and vise versa.Recently, there has been great interest in studying the system of difference equations.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, economic, probability theory, genetics psychology, etc.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.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.Recently there has been published quite a lot of works concerning the behavior of positive solutions of systems of difference equations [1]- [8].These results are not only valuable in their own right, but they can provide insight into their differential counterparts.
Papaschinopoulos et al. [1] investigated the global behavior for a system of the following two nonlinear difference equations.
where A is a positive real number; p and q are positive integers, and 0 0 , , , , , Clark and Kulenovic [2] [3] investigated the system of rational difference equations.
, , , 0, a b c d ∈ ∞ and the initial conditions 0 x and 0 y are arbitrary nonnegative numbers.Yang [4] studied the system of high-order difference equations.are positive real numbers.Ibrahim [7] has obtained the positive solution of the difference equation system in the modeling competitive populations., .
Din et al. [8] studied the global behavior of positive solution to the fourth-order rational difference equations where the parameters , , , , , α β γ α β γ and the initial conditions , , 0,1, 2, = are positive real numbers.Although difference equations are sometimes very simple in their forms, they are extremely difficult to understand thoroughly the behavior of their solutions.In book [9], Kocic and Ladas have studied global behavior of nonlinear difference equations of higher order.Similar nonlinear systems of difference equations were investigated (see [10]- [19]).
Our aim in this paper is to investigate the solutions, stability character and asymptotic behavior of the system of difference equations where

H. M. Bao
Clearly, if 0 A > , system (1) has always a positive equilibrium point ( ) x y be a positive solution of (1), then the following statements holds: 1) , , Proof.Assertion 1) is obviously true.Now it only need to prove assertion 2).From (1) and in view of 1), we have, for Let , k k u v be the solution of following system, respectively . We prove by induction that , , 3.

Stability
Theorem 2. Assume that 2 3 A > , then the unique positive equilibrium point ( ) is locally asymptotically stable.
Proof.We can obtain easily the linearized system of (1) about the positive equilibrium ( ) Let 1 2 3 4 , , , λ λ λ λ denote the eigenvalues of matrix B, let ( ) be a diagonal matrix, where ( ) It is well known that B has the same eigenvalues as 1 DBD − , we have that This implies that the equilibrium ( ) , c c of ( 1) is locally asymptotically stable.Theorem 3. Assume that 1 A > .Then every positive solution of (1) converges to ( ) x y be an arbitrary positive solution of (1).Let { } { }

Rate of Convergence
In this section we will determine the rate of convergence of a solution that converges to the equilibrium point ( ) , c c of the system (1).The following result gives the rate of convergence of solution of a system of difference equations ( ) where n X is a four dimensional vector, : where ⋅ denotes any matrix norm which is associated with the vector norm.Theorem 5. [20] Assume that condition (12) hold, if X n is a solution of (11), then either 0 n X = for all large n or exists and is equal to the moduls of one the eigenvalues of the matrix A.
Assume that lim , lim , we will find a system of limiting equations for the system (1).The error terms are given as x x e y y = − = − , therefore it follows that e C e D e where ( ) ( ) Hence, the limiting system of error terms at ( ) 0, 0 can be written as where ( ) ( ) ( ) Using Theorem 5, we have the following result.x y be a positive solution of the system (1).Then, the error vector n E of every solution of (1) satisfies both of the following asymptotic relations ( ) ( ) where ( ) 0, 0 F J λ is equal to the moduls of one the eigenvalues of the Jacobian matrix evaluted at the equilibrium ( ) 0, 0 .

Numerical Examples
In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider an interesting numerical example in this section.
Example 5.1.Consider the system (1) with initial conditions , Moreover, choosing the parameters 1.7 A = . Then system (1) can be written as

Conclusions and Future Work
In this paper, the dynamical behavior of second-order discrete system is studied.It can be concluded that: 1) The positive equilibrium point 2) The equilibrium rate of convergence is discussed.Some numerical examples are provided to support theoretical results.It is our future work to study the oscillation behavior of system (1).

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the uniqueness of the positive equilibrium ( ) Then the positive equilibrium ( ) , c c of (1) is globally asymptotically stable for all positive solutions.