Table of Contents
Journal of Difference Equations
Volume 2014, Article ID 275312, 11 pages
http://dx.doi.org/10.1155/2014/275312
Research Article

Global Asymptotic Stability for Linear Fractional Difference Equation

Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA

Received 17 March 2014; Accepted 11 May 2014; Published 1 July 2014

Academic Editor: Zong-Xuan Chen

Copyright © 2014 A. Brett et al. 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

Consider the difference equation ,  , where all parameters ,  , and the initial conditions ,   are nonnegative real numbers. We investigate the asymptotic behavior of the solutions of the considered equation. We give easy-to-check conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

1. Introduction

Consider the difference equation where , the parameters ,  , and the initial conditions , are nonnegative real numbers. The important special cases of (1) are the well-known Riccati equation the second order linear fractional difference equation and the third order linear fractional difference equation that we get from (1) for . The global behavior and the exact solutions of (2) even for real parameters have been found in [1]. The global behavior of solutions of (3), in many subcases when one or more parameters are zero, was established in [1]. There are still some conjectures left whose answers will complete the global picture of the asymptotic behavior for the solutions of (3). As far as the third order linear fractional difference equation is concerned, there are a large number of sporadic results that are systemized in a book [2]. The characterization of the global asymptotic behavior of the solutions of (1) for seems to be much harder than for the second order equation (3). Consequently an attempt at giving the characterization of the global asymptotic behavior for the solutions of (1) seems to be a formidable task at this time. However using some known global attractivity results we can describe the global asymptotic behavior for the solutions of (1) in some subspaces of the parametric space and the space of initial conditions. See [26] for a complete description of the behavior of some special cases of (1), in particular for the cases known as periodic trichotomies. See [7] where the difference in global behavior between the second and third order linear fractional difference equation is emphasized. The results on the global periodicity, that is, the results which describe all special cases of (1) where all solutions are periodic of the same period, were obtained in [8, 9]. Most results in [26, 10, 11] are based on known global attractivity or global asymptotic stability results obtained in [1, 2, 1217].

This paper is an attempt at establishing some global stability results for the equilibrium solution(s) of (1). Our results give effective conditions for global asymptotic stability of the equilibrium solution(s) of (1) expressed in terms of the inequalities on the coefficients. It is worth mentioning that the long standing conjecture for (3) is that local asymptotic stability implies global asymptotic stability of the equilibrium [2, 4]. In the case of the third order equation (1) with , the standing conjecture is that local asymptotic stability and boundedness of all solutions imply global asymptotic stability of the equilibrium. If the second conjecture is proved to hold, it will still be very difficult to verify the conditions for local asymptotic stability of the equilibrium as these conditions are very difficult to check for linear fractional equations of order higher than 2. See [2] for many special cases of third order linear fractional equation with very complicated conditions for local asymptotic stability. Thus the presented results are of importance even if the abovementioned conjecture is proved to be true.

The following general global results will be applied to (1); see [18]. Consider the difference equation where . Sometimes it is more advantageous to investigate (4) by embedding (4) into a higher iteration of the form where (see [16, 18, 19]) and then linearizing (4) or (5) by rewriting them (see [18]) into a nonautonomous linear equation of the form where and the functions are in general functions of both and the state variables ,  . See [18, 20] for examples of such linearizations.

Theorem 1. Let . Suppose that (4) has the linearization (6) where the functions are such that Then

As we have observed in [18], condition (7) is actually a contraction condition in the Banach contraction principle.

In addition, we will need the following stability result which is a consequence of our results in [18].

Theorem 2. Suppose that (4) can be linearized into the form where is an equilibrium of (4) and the functions . If , then the equilibrium of (4) is stable.

Proof. Observe that Assume that . Take . Then (9) implies and so by induction for .

2. Preliminaries

First observe that when (1) becomes the linear nonhomogeneous equation where for all and whose equilibrium satisfies .

We now establish our first result.

Theorem 3. Let and .(1)If and , then the zero equilibrium of (1) is globally asymptotically stable.(2)If and , then the zero equilibrium of (1) is stable.(3)If and , then whenever, for some .(4)If and , then the unique positive equilibrium of (1) is globally asymptotically stable.

Proof. When and , (1) becomes (1)In this case and the result follows from Theorems 1 and 2.(2)In this case and the result follows from Theorem 2.(3)Since ,   and  , then the result follows from Theorem 2 in [18].
When and , (1) becomes and has a unique positive equilibrium provided . Then
Let where ,  . Then, for ,   satisfies (4)Since , then, by Theorem 1, and so . Thus is a global attractor. By applying Theorem 2 to (15) we get that the equilibrium is stable and so the positive equilibrium of (1) is globally asymptotically stable.

For the remainder of this paper we will assume that for at least one .

Now we investigate the stability of the zero equilibrium of (1). Note that (1) has a zero equilibrium if and only if and .

Theorem 4. Let and . Assume that .(1)If , then the zero equilibrium of (1) is globally asymptotically stable.(2)If , then the zero equilibrium of (1) is stable.

Proof. When (1) becomes which can be written in the linearized form (6) where : Define ,   for .(1)The proof follows from Theorems 1 and 2 as .(2)The proof follows from Theorem 2 as .

3. Positive Equilibrium

In this section we investigate the stability of the unique positive equilibrium of (1) by using Theorems 1 and 2.

Note that, for , the function has the following properties:(a)if and , then is increasing in on the interval ;(b)if and , then is decreasing in on the interval .

The following result gives some other cases when is monotonic.

Remark 5. Consider the function given by (19) on where is a unique positive fixed point of this function. Assume that for . Then for set Then for Thus we have that, for ,   on the interval and on the interval .

In the case when is monotonic in all its arguments one can try to use global attractivity and global asymptotic stability results established in [1, 2, 15, 16].

In order to apply Theorem 1 to (1) we first need to linearize (1) into the form (6) which can be done as follows: Now applying the equilibrium equation we get that for Let for and . Then satisfies where for

The conditions ,   and , which are equivalent to ,  , and , can be reformulated in a more explicit way.

Proposition 6. Let for some and let be the positive equilibrium of (1). Then for (a) if and only if ,(b) if and only if ,(c) if and only if .

Proof. Consider the following.
Case  1  . In this case (1) has the positive equilibrium provided . Now case (a) becomes if and only if which proves (a). The proofs of parts (b) and (c) are similar.
Case  2  . Then (1) has the positive equilibrium Assume that . Then implies which yields and so Now assume that . Then and so Otherwise, suppose that Since then which is a contradiction.Therefore, and so .Similarly we can show that if and only if from which the result follows.Assume that . Suppose that Then either or and so either or which are both contradictions. Thus Similarly we can show that implies .

We can now obtain easy-to-check conditions which show when the positive equilibrium of (1) is globally asymptotically stable. We will then apply these conditions to various cases of (1).

Theorem 7. Let . Assume that one of the following holds:(1);(2)there exist such that for every solution of (1) for all and , where .
Then the positive equilibrium of (1) is globally asymptotically stable on the interval .

Proof. As we have seen (1) can be written in the form of the linearized equation (24), where the coefficients are given as (25).(1)Observe that for Then by Theorem 1, and so . Thus is a global attractor on the interval .From (23) we have that , for . Then which by Theorem 2 implies that the equilibrium is stable.(2)Assume that there exist such that for every solution of (1) for all . Then for and, so by Theorem 1, is a global attractor on the interval . Since, by assumption, is an attracting interval, then is a global attractor on the interval .By Theorem 2 applied to (23), is stable. Consequently, is globally asymptotically stable on the interval .

Many cases of (1) have some combination of ,  , and . In view of this we will adopt the following notations where , , and : Then and . Also ,  , and .

Before we apply Theorem 7 to various cases of (1) we establish the following useful lemma.

Lemma 8. Let and . Then

Proof. Observe that in Proposition 6 for and are positive real numbers. Thus by Proposition 6 with and we have that(a)if and only if if and only if Cases (b) and (c) follow similarly.

Theorem 9. Let . Then the positive equilibrium of (1) is globally asymptotically stable on the interval provided one of the following holds:(1)  for all ;(2)  for and ;(3)  for and ;(4)for some ,  ,   and , where .

Proof. The positive equilibrium of (1) satisfies (1)Let for all . Then and so . Then (1) becomes, for , (2)Let for . Then for we have . Thus and the result follows from Theorem 7.(3)Let for . Then for we have . By Lemma 8 with and the result follows from Theorem 7.(4)For some ,   and . Then Observe that . Since , then and so the result follows from Theorem 7.

There are many cases of (1) when we can establish a lower bound for all the solutions of (1).

Remark 10. The results on boundedness of all solutions of (1) are well known; see [2, 19]. For instance, if for every such that we have , then the uniform lower bound for all solutions of (1), for , is On the other hand, if for every such that we have , then the uniform lower bound for all solutions of (1), for , is where See Example 19.

The results of Theorem 9 can be extended for those cases of (1) which have a lower bound for every solution of (1); see Remark 10.

Theorem 11. Let and let be such that, for ,   for every solution of (1).(1)If and either  for and ;  for and ; orfor some   ,  , andwhere ,then the positive equilibrium of (1) is globally asymptotically stable on the interval .(2)If and either for all ; for and ; orfor some   ,  , and where ,then the positive equilibrium of (1) is globally asymptotically stable on the interval .

Proof. (1) Assume that .(a)Let for . Since , then and the result follows from Theorem 7.(b)Let for . By Lemma 8 with we getand the result follows from Theorem 7. (c)Assume that for some    and . Then and the result follows from Theorem 7.
(2) Assume that .(a)Let for all . Then and (1) becomes . (b)Let for . Then and the result follows from Theorem 7. (c)Assume that for some    and . Then and the result follows from Theorem 7.

By using Theorem 2 and similar methods as in the proof of Theorems 7, 9, and 11 we can obtain the conditions for the stability of the positive equilibrium.

Theorem 12. Let . Assume that one of the following holds:(1);(2)there exist such that for every solution of (1) for all and , where .
Then the positive equilibrium of (1) is stable on the interval .

Proof. (1) Observe that for  Thus the result follows from Theorem 2.
(2) The result follows similarly as in Theorem 7 part (2).

Theorem 13. Let . Then the positive equilibrium of (1) is stable on the interval provided one of the following holds:(1)  for and ;(2)  for and ;(3)for some  , and where .

Theorem 14. Let and let be such that for every solution of (1) for .(1)If and either for and   orfor some   ,  , and where ,then the positive equilibrium of (1) is stable on the interval .(2)If and either for and orfor some   , and where ,then the positive equilibrium of (1) is stable on the interval .

When , the following results show that the positive equilibrium of (1) may be globally asymptotically stable on a subspace of the initial conditions. First we will need the following lemma.

Lemma 15. Let , and for some   ,  . Suppose that where the nonnegative functions . Assume that, for this ,  . Then .

Proof. Let . Then (69) has the generalized identity
Choose . First, suppose that for . Then by (69) Thus .
Second, suppose that for some . Then . Let . Then By assumption for this and so Thus .

Theorem 16. Let and . Assume that . Then the positive equilibrium of (1) is globally asymptotically stable on the interval .

Proof. Clearly . Since for , then and so for . Then by Lemma 15 with for and we get that . Hence . Thus for . Now applying Lemma 15 again with for and we get that and so . Thus for . Hence by induction we get that for . Therefore applying Theorem 1 to (24) we have that and by applying Theorem 2 to (24) is stable.

We now apply Theorem 16 to various cases of (1).

Theorem 17. Let and . Then the positive equilibrium of (1) is globally asymptotically stable on the interval provided one of the following holds:(1)  for all ;(2)  for , and ;(3)  for   and either and   or and ;(4)for some    , and .

Proof. (1) If , then Theorem 9 part (1) applies. If , (1) becomes
(2)  In this case we have for and since , then . Thus and the result follows from Theorem 16.
(3) In this case we have for .(a)Assume that and . By Lemma 8 with . Thus and so the result follows from Theorem 16.(b)Assume that and . Then Thus and so the result follows from Theorem 16.
(4) In this case from the equilibrium equation we get that
Suppose that and . Then and so the result follows from Theorem 16.

We illustrate our results with some examples.

Example 18. Equation with nonnegative initial conditions and positive coefficients was considered in [4], and it was proved that the condition implies global asymptotic stability. By Theorem 7 the condition implies global asymptotic stability of the unique positive equilibrium . If , then the sufficient condition for global asymptotic stability of the unique positive equilibrium of (84) becomes

Example 19. Equation with nonnegative initial conditions and positive coefficients was considered in [4], and it was proved that the condition implies global asymptotic stability. Theorem 7 implies that, for the condition where the positive equilibrium is globally asymptotically stable. If then the sufficient condition for global asymptotic stability of the unique positive equilibrium of (87) becomes

Example 20. Equation