`Discrete Dynamics in Nature and SocietyVolumeΒ 2008, Article IDΒ 860152, 12 pageshttp://dx.doi.org/10.1155/2008/860152`
Research Article

On the Global Asymptotic Stability of a Second-Order System of Difference Equations

Department of Mathematics, Faculty of Education, University of Selcuk, 42099 Meram Yeni Yol, Konya, Turkey

Received 4 June 2008; Accepted 8 September 2008

Copyright Β© 2008 Ibrahim Yalcinkaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A sufficient condition is obtained for the global asymptotic stability of the following system of difference equations where the parameter and the initial values (for .

1. Introduction

Recently, there has been an increasing interest in the study of qualitative analyses of rational difference equations and systems of difference equations. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economy, physics, and so forth. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the global behaviors of their solutions (see [1β15] and the references cited therein).

In [9, 10] Papaschinopoluos and Schinas studied the behavior of the positive solutions of the system of two Lyness difference equationswhere are positive constants and initial values are positive.

IriΔanin and SteviΔ [6] studied, among others, the following system:where

In [2], Amleh et al. proved that all positive solutions of the difference equationswhere initial values are positive, converge to 1 as

Moreover, Xianyi and Deming [8] proved that the unique positive equilibrium of the difference equationwhere and are positive, is globally asymptotically stable.

In [1], we extended the results obtained in [8] to the following difference equation:where is nonnegative integer, and are positive and are globally asymptotically stable.

Also in [14], we extended the results obtained in [8] to the following system of difference equations:where and the initial values (for ) are globally asymptotically stable.

In this paper, we consider the following system of difference equations:where and the initial values (for ). Our main aim is to investigate the global asymptotic behavior of its solutions.

It is clear that the change of variablesreduces the system (1.7) to the systemwhere the initial values for (.

We need the following definitions and theorem.

Let be some interval of real numbers and letbe continuously differentiable functions. Then, for all initial values , the system of difference equationshas a unique solution Definition 1.1. A point is called an equilibrium point of the system (1.11) if

It is easy to see that the system (1.9) has the unique positive equilibrium [7].Definition 1.2. Let be an equilibrium point of the system (1.11).
(a) An equilibrium point is said to be stable if for any there is such that for every initial points and for which , the iterates of and satisfy for all An equilibrium point is said to be unstable if it is not stable (the Euclidean norm in given by ) is denoted by .
(b) An equilibrium point is said to be asymptotically stable if there exists such that as for all and that satisfy [7].
Definition 1.3. Let be an equilibrium point of a map where and are continuously differentiable functions at The Jacobian matrix of at is the matrix
The linear map given byis called the linearization of the map at [7].
Theorem 1.4 (linearized stability theorem [7]). Let be a continuously differentiable function defined on an open set in , and let in be an equilibrium point of the map
(a)If all the eigenvalues of the Jacobian matrix have modulus less than one, then the equilibrium point is asymptotically stable.(b)If at least one of the eigenvalues of the Jacobian matrix has modulus greater than one, then the equilibrium point is unstable.(c)An equilibrium point of the map is locally asymptotically stable if and only if every solution of the characteristic equationlies inside the unit circle, that is, if and only if
Definition 1.5. Let be positive equilibrium point of the system (1.11).
A βstringβ of consecutive terms (resp., is said to be a positive semicycle if (resp., , (resp., and (resp.,
A βstringβ of consecutive terms (resp., is said to be a negative semicycle if (resp., , (resp., and (resp.,
A βstringβ of consecutive terms is said to be a positive (resp., negative) semicycle if are positive (resp., negative) semicycles. Finally, a βstringβ of consecutive terms is said to be a semicycle positive (resp., negative) with respect to and negative (resp., positive) with respect to if is a positive (resp., negative) semicycle and is a negative (resp., positive) semicycle [9].

We now make new definitions. These definitions can be used for different subsequences of (resp., ).Definition 1.6. Let be positive equilibrium point of the system (1.11).
A βstringβ of consecutive terms ( resp., is said to be a positive sub-semicycle associated with (resp., ) if (resp., ), (resp., ), and (resp., ).
A βstringβ of consecutive terms ( resp., is said to be a negative sub-semicycle associated with (resp., if (resp., , (resp., and (resp.,
A βstringβ of consecutive terms is said to be a positive (resp., negative) sub-semicycle if are positive (resp., negative) sub-semicycles. Finally, a βstringβ of consecutive terms is said to be a sub-semicycle positive (resp., negative) with respect to and negative (resp., positive) with respect to if is a positive (resp., negative) sub-semicycle and is a negative (resp., positive) sub-semicycle.
Definition 1.7. Let be positive equilibrium point of the system (1.11).
A βstringβ of consecutive terms (resp., , is said to be a positive sub-semicycle associated with (resp., if (resp., , (resp., and (resp.,
A βstringβ of consecutive terms (resp., , is said to be a negative sub-semicycle associated with (resp., if (resp., , (resp., and (resp.,
A βstringβ of consecutive terms is said to be a positive (resp., negative) sub-semicycle if are positive (resp., negative) sub-semicycles. Finally, a βstringβ of consecutive terms is said to be a sub-semicycle positive (resp., negative) with respect to and negative (resp., positive) with respect to , if is a positive (resp., negative) sub-semicycle and is a negative (resp., positive) sub-semicycle.

2. Some Auxiliary Results

In this section, we give the following lemmas which show us the behavior of semicycles of positive solutions of system (1.9). The proof of Lemma 2.1 is clear from (1.9). So, it will be omitted.Lemma 2.1. Assume that is a solution of the system (1.9) and consider the following cases:
(Case a)
(Case b)
(Case c)
(Case d)
If one of the above cases occurs, then every positive solution of system (1.9) is equal to .
Lemma 2.2. Assume that is a positive solution of the system (1.9) which is not eventually equal to . Then the following statements are true:
(i)(ii)
Proof. In view of system (1.9), we obtain for from which the inequalities in (i) and (ii) follow.Lemma 2.3. Assume that is a solution of system (1.9) and suppose that the case,
(Case 1) ( for holds.
Then, and are positive sub-semicycles of system (1.9) with an infinite number of terms and they monotonically tend to the positive equilibrium .
Proof. If (for then by Lemma 2.2(ii), it follows thatthat is, these positive sub-semicycles have an infinite number of terms. Furthermore, according to Lemma 2.2(i), we know that and are strictly decreasing for all So, the limits exist and are finite. From (1.9), we can write taking limits on both sides of (2.4), we have and thus Similarly, one can see that Therefore, the proof is complete.Lemma 2.4. Assume that is a solution of system (1.9), and consider the following cases:
(Case 2) and
(Case 3) and
(Case 4) and
(Case 5) and
(Case 6) and
(Case 7) and
(Case 8) and
(Case 9) and
(Case 10)
If one of the above cases occurs, then the following hold.
(i)Every positive sub-semicycle associated with and (resp., and of system (1.9) consists of one term.(ii)Every negative sub-semicycle associated with and (resp., and of system (1.9) consists of two terms.(iii)Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.(iv)Every negative sub-semicycle of length two is followed by a positive sub-semicycle of length one.
Proof. If Case 2 occurs, then in view of inequality (ii) of Lemma 2.2 we have: andwhich imply that every positive sub-semicycle associated with and of system (1.9) of length one is followed by a negative sub-semicycle of length two, which in turn is followed by a positive sub-semicycle of length one.
Similarly, if Case 2 occurs, then in view of inequality (ii) of Lemma 2.2 we havewhich imply that every positive sub-semicycle associated with and of system (1.9) of length two is followed by a negative sub-semicycle of length four, which in turn is followed by a positive sub-semicycle of length two.
Proofs of the other cases are similar, so they will be omitted. Therefore, the proof is complete.

We omit the proofs of the following two results since they can easily be obtained in a way similar to the proof of Lemma 2.4.Lemma 2.5. Assume that is a solution of system (1.9) and consider the following cases:
(Case 11) and
(Case 12) and
(Case 13) and
If one of the above cases occurs, then the following hold.
(i) is a positive sub-semicycle of system (1.9) with an infinite number of terms (monotonically tend to the positive equilibrium .(ii)Every positive sub-semicycle associated with and of system (1.9) consists of one term. (iii) Every negative sub-semicycle associated with and of system (1.9) consists of two terms.(iv)Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.(v)Every negative sub-semicycle of length two is followed by a positive sub-semicycle of length one.
Lemma 2.6. Assume that is a solution of system (1.9) and consider the following cases:
(Case 14) and
(Case 15) and
(Case 16) and
If one of the above cases occurs, then the following hold.
(i) is a positive sub-semicycle of system (1.9) with an infinite number of terms (monotonically tend to the positive equilibrium .(ii)Every positive sub-semicycle associated with and of system (1.9) consists of one term.(iii)Every negative sub-semicycle associated with and of system (1.9) consists of two terms.(iv)Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.(v)Every negative sub-semicycle of length two is followed by a positive sub-semicycle of length one.

3. Main Result

Theorem 3.1. The positive equilibrium point of the system (1.9) is globally asymptotically stable.Proof. We must show that the positive equilibrium point of the system (1.9) is both locally asymptotically stable and as (or equivalently and as The characteristic equation of the system (1.9) about the positive equilibrium point isand so it is clear from Theorem 1.4 that positive equilibrium point of the system (1.9) is locally asymptotically stable. It remains to verify that every positive solution of the system (1.9) converges to as Namely, we want to prove
If the solution of (1.9) is nonoscillatory about the positive equilibrium point of the system (1.9), then according to Lemmas 2.1 and 2.3, respectively, we know that the solution is either eventually equal to or an eventually positive one which has an infinite number of terms and monotonically tends the positive equilibrium point of the system (1.9) and so (3.2) holds. Therefore, it suffices to prove that (3.2) holds for strictly oscillatory solutions. Now, let be strictly oscillatory about the positive equilibrium point of the system (1.9). By virtue of Lemmas 2.2(ii) and 2.4, one can see that every positive sub-semicycle associated with (resp. of this solution has one term, and every negative sub-semicycle associated with (resp., except perhaps for the first has exactly two terms. Every positive sub-semicycle of length one is followed by a negative sub-semicycle of length two.
We consider the sub-semicycles associated with and
For the convenience of statement, without loss of generality, we use the following notation. We denote by and the terms of a positive sub-semicycle of length one, followed by and which are the terms of a negative sub-semicycle of length two. Afterwards, there are the positive sub-semicycles and in turn followed by the negative sub-semicycles, and so on.
Therefore, we have the following sequences consisting of positive and negative sub-semicycles (for ):
We have the following assertions:
(i) and (ii) and (iii) and
In fact, inequality (i) immediately follows from Lemma 2.2(i). From the observations that one can see that (ii) is valid.
As for (iii), it is obtained from for
Combining the above inequalities, we derive
From (3.6), one can see that and are increasing with upper bound 1. So the limits exist and are finite. Accordingly, in view of (3.6), we obtain
Now, we consider the sub-semicycles associated with and
Similarly, for the convenience of statement, without loss of generality, we use the following notation. We denote by and the terms of a positive sub-semicycle of length one, followed by and which are the terms of a negative sub-semicycle of length two. Afterwards, there are the positive sub-semicycles and in turn followed by the negative sub-semicycles, and so on.
Therefore, we have the following sequences consisting of positive and negative sub-semicycles (for ):
We have the following assertions:
(i) and (ii) and (iii) and
Combining the above inequalities, we derive
From (3.10), one can see that and are increasing with upper bound 1. So the limits exist and are finite. Accordingly, in view of (3.10), we obtain
It suffices to verify that
To this end, note that
Take the limits on both sides of the above equality and obtain which imply that Similarly, one can see that
Moreover, by virtue of Lemmas 2.2(ii) and 2.5 (resp., 2.6), one can see that (3.2) holds. Therefore, the proof is complete.

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