Dynamical Behavior of a System of Third-Order Rational Difference Equation

It is known that difference equation appears naturally as discrete analogous and as numerical solutions of differential equation and delay differential equation having many applications in economics, biology, computer science, control engineering, and so forth. The study of discrete dynamical systems described by difference equations has now been paid great attention by many mathematical researchers. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed in many papers. In 1998, DeVault et al. [1] proved every positive solution of the difference equation:


Introduction
It is known that difference equation appears naturally as discrete analogous and as numerical solutions of differential equation and delay differential equation having many applications in economics, biology, computer science, control engineering, and so forth.The study of discrete dynamical systems described by difference equations has now been paid great attention by many mathematical researchers.Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions can be discussed in many papers.
In 1998, DeVault et al. [1] proved every positive solution of the difference equation: where  ∈ (0, ∞) oscillates about the positive equilibrium  = 1 +  of (1).Moreover every positive solution of (1) is bounded away from zero and infinity.Also the positive equilibrium of (1) is globally asymptotically stable.
In 2003, Abu-Saris and DeVault [2] studied the following recursive difference equation: where  ∈ (1, +∞),  − ,  −+1 , . . .,  0 are positive real numbers.For similar results the reader can refer to [3][4][5][6][7][8][9].Difference equations or discrete dynamical systems are a diverse field which impact almost every branch of pure and applied mathematics.We refer to [10,11] for basic theory of difference equations and rational difference equations.It is very interesting to investigate the qualitative behavior of the discrete dynamical systems of nonlinear difference equations.Recently there has been a lot of work concerning the global asymptotic stability, the periodicity, and the boundedness of nonlinear difference equations.Moreover similar results in [12][13][14][15][16][17] have been derived for systems of two nonlinear difference equations.
Papaschinopoulos and Schinas [12] investigated the global behavior for a system of the following two nonlinear difference equations: where  is a positive real number, ,  are positive integers, and  − , . . .,  0 ,  − , . . .,  0 are positive real numbers.
(ii) If  > 1, then, for  ≥ 4, we have the following: Proof.Assertion (i) is obviously true.We now prove assertion (ii).From ( 6) and in view of (i), we have, for all  ≥ 4, that Let V  ,   be the solution of following system, respectively, such that We prove by induction that Suppose that ( 13) is true for  =  ≥ 4. Then from (10) we get the following: Therefore ( 13) is true.From (11) we have the following: Then from (10), (13), and ( 14) the proof of the relation (9) follows immediately.
Proof.Let {  ,   } be an arbitrary positive solution of (6).Let From Theorem 1, we have The previous and ( 6) imply that which can derive that If Λ 1 >  1 and Λ 2 >  2 , it follows from Lemma 3 and the condition . Then a contradiction occurs.Thus we have either We assume Λ 1 =  1 (the discussion for the case Λ 2 =  2 is similar).Then lim  → ∞   exists.Using (27) and in view of Then lim  → ∞   exists.From the uniqueness of the positive equilibrium (, ) of ( 6), we conclude that lim  → ∞   = , lim  → ∞   = .
Combining Theorems 2 and 4, we obtain the following theorem.

Numerical Example
In order to illustrate the results of the previous sections and to support our theoretical discussions, we consider several interesting numerical examples in this section.These examples represent different types of qualitative behavior of solutions to nonlinear difference equations and system of nonlinear difference equations.