On a System of Difference Equations of an Economic Model

The increasing study of realistic mathematical models is a reflection of their use in helping to understand the dynamic processes involved in areas such as population dynamics, biology, epidemiology, ecology, and economy. More realistic models should include some of the past states of these systems; that is, ideally, a real system should be modeled by difference equations with time delays. Most of these models are described by nonlinear delay difference equations; see, for example, [1–4]. The subject of the qualitative study of the nonlinear delay population models is very extensive, and the current research work tends to center around the relevant global dynamics of the considered systems of difference equations such as oscillation, boundedness of solutions, persistence, global stability of positive steady sates, permanence, and global existence of periodic solutions. See [5–18] and the references therein. In this paper, we intend to cover some of these global aspects of the qualitative behavior of a system of a discrete model in the economy area, where we deal with the studying of some qualitative properties of solutions of the following system of difference equations:


Introduction
The increasing study of realistic mathematical models is a reflection of their use in helping to understand the dynamic processes involved in areas such as population dynamics, biology, epidemiology, ecology, and economy.More realistic models should include some of the past states of these systems; that is, ideally, a real system should be modeled by difference equations with time delays.Most of these models are described by nonlinear delay difference equations; see, for example, [1][2][3][4].The subject of the qualitative study of the nonlinear delay population models is very extensive, and the current research work tends to center around the relevant global dynamics of the considered systems of difference equations such as oscillation, boundedness of solutions, persistence, global stability of positive steady sates, permanence, and global existence of periodic solutions.See [5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.
In this paper, we intend to cover some of these global aspects of the qualitative behavior of a system of a discrete model in the economy area, where we deal with the studying of some qualitative properties of solutions of the following system of difference equations:  +1 = (1 − )   +   (1 −   )  −(  +  ) ,  +1 = (1 − )   +   (1 −   )  −(  +  ) ,  = 0, 1, . . ., where  and  ∈ (0, ∞) with the initial conditions  0 and  0 ∈ (0, ∞).We study the boundedness and the invariant of the solutions of system (1) and also investigate global convergence for the solutions of system (1).System (1) is an important type of economic models which describes a discrete-time map generated by bounded rationally duopoly game with exponential demand function.See [19].
The following theorem was presented in [6], and it will be useful in the investigation of the global stability of system (1).
Theorem A. Consider the following system of difference equations: Suppose that (i) (; ) is nondecreasing in  and is nonincreasing in , and (; ) is nonincreasing in  and is nondecreasing in , Then, system (2) has a unique positive equilibrium point (, ), and every solution of system (2) converges to (, ).

Boundedness and Invariant
In this section, we concern ourselves with the boundedness character of solutions of system (1).Under appropriate conditions, we give some bounded results related to system (1).
Proof.It follows from (1) that So Cases (i) and (ii) are immediately proved.Now set Then, Therefore, (2) is the absolute minimum of ().That is, Note that (5) implies and, hence, ( +1 −  +1 ) has the same sign of (  −   ) for all  > 0. The proof is so complete.
Proof.Let (, ) be a continuous function defined by Then, system (1) can be rewritten in the form Now assume that {(  ,   )} ∞ =0 is a solution of system (1) with positive initial values.Then, it suffices to show that (, ) is positive for all  > 0,  > 0. Observe that Therefore,  has no positive critical points.Let  and  be arbitrary positive numbers and consider the domain Then, Using elementary differential calculus, we obtain that the absolute minimum of each one of the above functions is 1−− / 2 .Therefore, (, ) ≥ 1 −  − / 2 > 0 for all (, ) ∈ .
Proof.We obtain, for  0 ≥ 0, from (1) that Then, it follows by Theorems 3 and 4 that Case (i) is true.The proof of Case (ii) is similar and so will be omitted.
The following corollaries are coming immediately from Theorem 5.
Then, there exists  0 ≥ 0 such that (  ,   ) ∈ (0, 1] 2 for all  ≥  0 . Proof.The proof of the theorem, when (i) holds, is followed by Corollary 7. Now consider that (ii) is true.Then, it follows from Corollary 7 that for every constant  > 0, there exists  0 ≥ 0 such that   ≤ / +  = ,  ≥  0 .Set  =  − .Since  →  −] when  → 0 and the inequalities in (ii) hold, depending on the continuity in ] of the left hand side of each inequality in (ii), one can choose  so small such that Now, we obtain from (1) that where () = (1 − ) +  − ( −  2 ),  ≤ , and then On the other hand, the equation has the positive roots Observe that This completes the proof.
Theorem 9. Assume that {(  ,   )} ∞ =0 is a positive solution of system (1).If either or where ] = /, then there exists  0 ≥ 0 such that (  ,   ) ∈ (0, 1] 2 for all  ≥  0 . Proof.Assume that , , and the function (  ) are defined as in the previous proof.Then, where () = (1 −  + ) −  2  2 ,  ≤ .Thus, Hence, () attains its maximum value at  = (1 −  + )/2 2 ; that is, Also, Similar to the proof of Theorem 8, we can choose  so small such that our assumptions imply that Therefore, we have either or which is our desired conclusion for   .Similarly, one can accomplish the same conclusion for   .The proof is so complete.

Global Stability Analysis
In this section, we are interested in establishing conditions under which the equilibrium points of system (1) are to be the attractors of the solutions of system (1).
In the following theorem, we investigate the global attractivity of the equilibrium point (0, 0) of system (1).
In the following theorems, we investigate the global attractivity of the positive equilibrium point (; ) of system (1) where  is given by  = (1 − ) −2 .Theorem 11.Assume that  +  −2 < 1.Then the unique positive equilibrium point (; ) of system (1) is a global attractor of all positive solutions of system (1).
Proof.Let {(  ,   )} ∞ =0 be a solution of system (1), and let   ≤  (the case   ≥  is similar, and it will be left to the reader).Now there are two cases to consider.
Case 1. Assume that  0 ≥  0 .Then, it follows by Theorem 1 that   <   for all  ≥ 1.Since   ≤ , then ℎ(  ) ≤ 0, where ℎ(  ) =  − (1 −   ) −2  .Thus,  ≤ (1 −   ) −2  .Therefore, we obtain from system (1) that Then, the sequence {  } ∞ =0 is increasing, and since it was shown that it is bounded above, then it converges to the only positive equilibrium point , and it follows by the comparison test of convergence for sequence that {  } ∞ =0 is also convergent to the only positive equilibrium point  = :.Thus, {(  ,   )} ∞ =0 converges to (; ).Case 2. Assume that  0 <  0 .Then, it follows from system (1) and Theorem 1 that The rest of the proof is similar to Case 1, and it will be left to the reader.Theorem 12. Assume that ( − ) ≥  2  3 .Then, the unique positive equilibrium point (; ) of system (1) is a global attractor of all positive solutions of system (1).
Proof.Let {(  ,   )} ∞ =0 be a solution of system (1).It follows from system (1) that (42) Thus, we see from Corollary 7 that Then, the sequence {  } ∞ =0 is increasing, and since it is bounded, then it converges to the only positive equilibrium point .Similarly, it is easy to show that the sequence {  } ∞ =0 is also convergent to the unique positive equilibrium point  = .Therefore {(  ,   )} ∞ =0 converges to (, ), and then the proof is so complete.(II)  +  < 1.Then the unique positive equilibrium point (, ) of system (1) is a global attractor of all positive solutions of system (1).