Abstract

We study some qualitative properties of the solutions of a system of difference equations, which describes an economic model. The study of the local stability of the equilibrium points is carried out. We give some important results of the invariant and the boundedness of the solutions to the considered system. The global convergence of the solutions is presented and investigated.

1. 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, [1–4]. 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: where and (0, ) with the initial conditions and (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 ,(ii) if ; ; and imply and .
Then, system (2) has a unique positive equilibrium point , and every solution of system (2) converges to .
The equilibrium points of system (1) are the solutions of the following system:
or
El-Metwally and Elsadany [19] have shown that system (1) has the equilibrium points , and where and satisfy and , respectively, and proved that (i)the equilibrium point of system (1) is locally asymptotically stable if and unstable if ;(ii)the equilibrium points and of system (1) are unstable;(iii)the equilibrium point of system (1) is stable if and unstable if .

2. 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).

Theorem 1. Assume that is a solution of system (1). Then,(i) implies that for all ,(ii) implies that for all ,(iii)if , then for all provided that .

Proof. It follows from (1) that So Cases (i) and (ii) are immediately proved. Now set Then, Therefore, is the absolute minimum of . That is, Note that (5) implies and, hence, has the same sign of for all . The proof is so complete.

Remark 2. Theorem 1 reduces system (1) into the following single difference equation:

Theorem 3. Assume that . Then, every solution of system (1) with and satisfies that and for all .

Proof. Let be a continuous function defined by Then, system (1) can be rewritten in the form Now assume that is a solution of system (1) with positive initial values. Then, it suffices to show that is positive for all . 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 . Therefore, for all. Since and are arbitrary positive numbers, we can conclude that for all.

Theorem 4. Assume that is a solution of system (1) with for some . Assume also that one of the following statements is true. (i). (ii). (iii).

Then, for all .

Proof. Let be such that . It follows from system (1) that Set for . Then, it follows from (16) that . Also, we obtain that Then, . If (i) holds, then , and hence is increasing on . Therefore, . If (ii) holds, then (17) yields .
Now suppose that (iii) holds. In this case, it follows from (18) that , where for all . It is not difficult to see that is the absolute maximum of on where . According to (iii) and since . That is in all cases we obtain that whenever gives . So it is easy to prove by induction that for all . The proof of is similar and so will be omitted. This completes the proof.

Theorem 5. For every solution of system (1), the following statements hold. (i), .(ii), .

Proof. We obtain, for , 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.

Corollary 6. Assume that is a positive solution of system (1) with for . Then, for all .

Corollary 7. Every positive solution of system (1) is bounded. Moreover, In Theorem 4, we gave conditions under which every positive solution of system (1) to be in provided that . In the next result we show that every positive solution of system (1) eventually lies in .

Theorem 8. Assume that is a positive solution of system (1), and assume that one of the following conditions is true.(i). (ii) and , where .

Then, there exists such that for all .

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 , there exists such that . Set . Since when 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 , and then On the other hand, the equation has the positive roots Observe that if and only if which holds by (22). Therefore, . Consequently, for all which yields by (23) that . Using the increasing property of on and inequality (24), we see that . Since , it follows that This completes the proof.

Theorem 9. Assume that is a positive solution of system (1). If either or where , then there exists such that for all .

Proof. Assume that , and the function are defined as in the previous proof. Then, where . Thus, Hence, attains its maximum value at ; 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.

3. 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 of system (1).

Theorem 10. Assume that . Then, (0,0) is a global attractor of all positive solutions of system (1).

Proof. Let be a solution of system (1). It follows from system (1) that Then, there exist and such that and . Since the only possible values of in the present case are and . This completes the proof.

In the following theorems, we investigate the global attractivity of the positive equilibrium point of system (1) where is given by .

Theorem 11. Assume that . Then the unique positive equilibrium point of system (1) is a global attractor of all positive solutions of system (1).

Proof. Let 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 . Then, it follows by Theorem 1 that for all . Since , then , where . Thus, . Therefore, we obtain from system (1) that Then, the sequence 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 is also convergent to the only positive equilibrium point :. Thus, converges to .
Case 2. Assume that . 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 . Then, the unique positive equilibrium point of system (1) is a global attractor of all positive solutions of system (1).

Proof. Let be a solution of system (1). It follows from system (1) that Thus, we see from Corollary 7 that Then, the sequence 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 is also convergent to the unique positive equilibrium point :. Therefore converges to , and then the proof is so complete.