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. 1. 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 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 are positive constants and the initial values are positive real values. Difference equations and systems of difference equations of exponential form can be found in [2–6]. 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 [7–9] and the references cited therein. 2. Global Behavior of Solutions of System (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. Proof. Let be an arbitrary solution of (2). From (2) we can see that In addition, from (2) and (3) we get 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]. Theorem 2. Let and be a continuous functions such that the following hold:(a) is decreasing in both variables and is decreasing in both variables for each ;(b)if is a solution of then and . Then the following system of difference equations, has a unique equilibrium and every solution of the system (7) with 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 ,