Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form

The goal of this paper is to study the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the system of two difference equations of exponential form: where , and are positive constants and the initial values are positive real values. Also, we determine the rate of convergence of a solution that converges to the equilibrium of this system.


Introduction
In [1], the authors studied the boundedness, the asymptotic behavior, the periodicity, and the stability of the positive solutions of the difference equation: where , ,  are positive constants and the initial values  −1 ,  0 are positive numbers.Motivated by the above paper we will extend the above difference equation to a system of difference equations; our goal will be to investigate the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the following system of exponential form: where , , , , ℎ are positive constants and the initial values  0 ,  0 are positive real values.
Difference equations and systems of difference equations of exponential form can be found in [2][3][4][5][6].Moreover, as difference equations have many applications in applied sciences, there are many papers and books that can be found concerning the theory and applications of difference equations; see [7][8][9] and the references cited therein.(2) In the first lemma we study the boundedness and persistence of the positive solutions of (2).Lemma 1.Every positive solution of (2) is bounded and persists.

Global Behavior of Solutions of System
Proof.Let (  ,   ) be an arbitrary solution of (2).From (2) we can see that Therefore, from (3) and ( 4) the proof of the lemma is complete.
In order to prove the main result of this section, we recall the next theorem without its proof.See [10,11].
be a continuous functions such that the following hold: (a) (, ) is decreasing in both variables and (, ) is decreasing in both variables for each (, ) ∈ R; then  1 =  1 and  2 =  2 .Then the following system of difference equations, has a unique equilibrium (, ) and every solution (  ,   ) of the system (7) with ( 0 ,  0 ) ∈ R converges to the unique equilibrium (, ).In addition, the equilibrium (, ) is globally asymptotically stable.
Now we state the main theorem of this section.
Theorem 3. Consider system (2).Suppose that the following relation holds true: Then system (2) has a unique positive equilibrium (, ) and every positive solution of (2) tends to the unique positive equilibrium (, ) as  → ∞.In addition, the equilibrium (, ) is globally asymptotically stable.
Proof.We consider the functions where It is easy to see that (, V), (, V) are decreasing in both variables for each (, V) ∈  × .In addition, from ( 9) and (10) we have (, V) ∈ , (, V) ∈  as (, V) ∈  ×  and so  :  ×  → ,  :  ×  → .Now let  1 ,  1 ,  2 ,  2 be positive real numbers such that Moreover arguing as in the proof of Theorem 2, it suffices to assume that From ( 11) we get which imply that Moreover, we get Then by adding the two relations ( 14) we obtain Therefore from (16) we have Then using ( 8), (12), and (17) gives us  1 =  1 and  2 =  2 .Hence from Theorem 2 system (2) has a unique positive equilibrium (, ) and every positive solution of (2) tends to the unique positive equilibrium (, ) as  → ∞.In addition, the equilibrium (, ) is globally asymptotically stable.This completes the proof of the theorem.

Rate of Convergence
In this section we give the rate of convergence of a solution that converges to the equilibrium  = (, ) of the system (2) for all values of parameters.The rate of convergence of solutions that converge to an equilibrium has been obtained for some two-dimensional systems in [12,13].
The following results give the rate of convergence of solutions of a system of difference equations: where x  is a -dimensional vector,  ∈ C × is a constant matrix, and  : Z + → C × is a matrix function satisfying where ‖ ⋅ ‖ denotes any matrix norm which is associated with the vector norm; ‖ ⋅ ‖ also denotes the Euclidean norm in R 2 given by ‖x‖ =     (, ) Theorem 4 (see [14]).Assume that condition (19) holds.If x  is a solution of system (18), then either x  = 0 for all large  or exists and is equal to the modulus of one of the eigenvalues of matrix .
Theorem 5 (see [14]).Assume that condition (19) holds.If x  is a solution of system (18), then either x  = 0 for all large  or exists and is equal to the modulus of one of the eigenvalues of matrix .
The equilibrium point of the system (2) satisfies the following system of equations: If  +  < , we can easily see that the system (23) has an unique equilibrium  = (, ).
(25) By using the system (23), value of the Jacobian matrix of  at the equilibrium point  = (, ) = (, ) is Our goal in this section is to determine the rate of convergence of every solution of the system (2) in the regions where the parameters , , , , ℎ ∈ (0, ∞), ( +  < ) and initial conditions  0 and  0 are arbitrary, nonnegative numbers.Theorem 6.The error vector e  = ( where |  (  ())| is equal to the modulus of one of the eigenvalues of the Jacobian matrix evaluated at the equilibrium   ().
Proof.First, we will find a system satisfied by the error terms.
The error terms are given as By calculating similarly, we get From ( 28) and (29) we have Set Then system (30) can be represented as where where   → 0,   → 0,   → 0, and   → 0 as  → ∞.Now, we have system of the form (18): (37) Thus, the limiting system of error terms can be written as