Open Access

Dynamics for Nonlinear Difference Equation

Advances in Difference Equations20092009:235691

DOI: 10.1155/2009/235691

Received: 19 April 2009

Accepted: 9 October 2009

Published: 13 October 2009

Abstract

We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.

1. Introduction

Consider the following higher order difference equation:

(1.1)

where , the parameters and are nonnegative real numbers and the initial conditions and are nonnegative real numbers such that

(1.2)

It is easy to see that if one of the parameters is zero, then the equation is linear. If , then (1.1) can be reduced to a linear one by the change of variables . So in the sequel we always assume that the parameters and are positive real numbers.

The change of variables reduces (1.1) into the following equation:

(1.3)

where .

Note that is always an equilibrium point of (1.3). When , (1.3) also possesses the unique positive equilibrium .

The linearized equation of (1.3) about the equilibrium point is

(1.4)

so, the characteristic equation of (1.3) about the equilibrium point is either, for ,

(1.5)

or, for

(1.6)

The linearized equation of (1.3) about the positive equilibrium point has the form

(1.7)

with the characteristic equation either, for ,

(1.8)

or, for

(1.9)

When , , (1.1) has been investigated in [14]. When , , (1.1) reduces to the following form:

(1.10)

El-Owaidy et al. [3] investigated the global asymptotical stability of zero equilibrium, the periodic character and the existence of unbounded solutions of (1.10).

On the other hand, when , (1.1) is just the discrete delay logistic model investigated in [4, ]. Therefore, it is both theoretically and practically meaningful to study (1.1).

Our aim in this paper is to extend and generalize the work in [3]. That is, we will investigate the global behavior of (1.1), including the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence and asymptotic behavior of nonoscillatory solutions of the equation. Our results extend and generalize the corresponding ones of [3].

For the sake of convenience, we now present some definitions and known facts, which will be useful in the sequel.

Consider the difference equation

(1.11)

where is a positive integer, and the function has continuous partial derivatives.

A point is called an equilibrium of (1.11) if

(1.12)

That is, for is a solution of (1.11), or equivalently, is a fixed point of .

The linearized equation of (1.11) associated with the equilibrium point is

(1.13)

We need the following lemma.

Lemma 1.1 (see [46]).
  1. (i)

    If all the roots of the polynomial equation

     
(1.14)
lie in the open unit disk , then the equilibrium of (1.11) is locally asymptotically stable.
  1. (ii)

    If at least one root of (1.11) has absolute value greater than one, then the equilibrium of (1.11) is unstable.

     

For the related investigations for nonlinear difference equations, see also [711] and the references cited therein.

2. Global Asymptotic Stability of Zero Equilibrium

In this section, we investigate global asymptotic stability of zero equilibrium of (1.3). We first have the following results.

Lemma 2.1.

The following statements are true.

(a)If , then the equilibrium point of (1.3) is locally asymptotically stable.

(b)If , then the equilibrium point of (1.3) is unstable. Moreover, for , (1.3) has a -dimension local stable manifold and a -dimension local unstable manifold.

(c)If , is odd and is even, then the positive equilibrium point of (1.3) is unstable.

Proof.
  1. (a)

    When , it is clear from (1.5) and (1.6) that every characteristic root satisfies or , and so, by Lemma 1.1(i), is locally asymptotically stable.

     
  2. (b)

    When , if , then it is clear from (1.5) that every characteristic root satisfies , and so, by Lemma 1.1(ii), is unstable. If , then (1.6) has characteristic roots satisfying , which corresponds to a -dimension local stable manifold of (1.3), and characteristic roots satisfying , which corresponds to a -dimension local unstable manifold of (1.3).

     
  3. (c)

    If is odd and is even, then, regardless of or , correspondingly, the characteristic equation (1.8) or (1.9) always has one characteristic root lying the interval . It follows from Lemma 1.1(ii) that is unstable.

     

Remark 2.2.

Lemma 2.1(a) includes and improves [3, Theorem  3.1(i)]. Lemma 2.1(b) and (c) include and generalize [3, Theorem 3.1(ii) and (iii)], respectively.

Now we state the main results in this section.

Theorem 2.3.

Assume that , then the equilibrium point of (1.3) is globally asymptotically stable.

Proof.

We know from Lemma 2.1 that the equilibrium point of (1.3) is locally asymptotically stable. It suffices to show that for any nonnegative solution of (1.3).

Since

(2.1)
converges for any . Let , then
(2.2)
Thereout, one has
(2.3)
that is,
(2.4)

which implies . The proof is over.

Remark 2.4.

Theorem 2.3 includes [3, Theorem  3.3].

3. Existence of Eventual Period Two Solution

In this section, one studies the eventual nonnegative prime period two solutions of (1.3). A solution of (1.3) is said to be eventual prime periodic two solution if there exists an such that for and holds for all .

Theorem 3.1.
  1. (a)

    Assume is odd and is even, then (1.3) possesses eventual prime period two solutions if and only if .

     
  2. (b)

    Assume is odd and is odd, then (1.3) possesses eventual prime period two solutions if and only if .

     
  3. (c)

    Assume is even and is even. Then the necessary condition for (1.3) to possess eventual prime period two solutions is and .

     
  4. (d)

    Assume is even and is odd. Then, (1.3) has no eventual prime period two solutions.

     
Proof.
  1. (a)

    If (1.3) has the eventual nonnegative prime period two solution then, we eventually have and . Hence,

     
(3.1)

If , then we can derive from (3.1) that if or vice versa, which contradicts the assumption that is the eventual prime period two solution of (1.3). So, . Accordingly, and , which indicate that when or that and do not exist when , which are also impossible. Therefore, .

Conversely, if , then choose the initial conditions such as and , or such as and . We can see by induction that is the prime period two solution of (1.3).
  1. (b)

    Let be the eventual prime period two solution of (1.3), then, it holds eventually that and . Hence,

     
(3.2)
If , then . This is impossible. So . Moreover, and or and , that is, is the prime period two solution of (1.3).
  1. (c)

    Assume that (1.3) has the eventual nonnegative prime period two solution then eventually

     
(3.3)
Obviously, implies or vice versa. This is impossible. So . It is easy to see from (3.3) that and satisfy the equation
(3.4)

that is, and are two distinct positive roots of . From (3.4) we can see that does not have two distinct positive roots at all when , alternatively, (1.3) does not have the nonnegative prime period two solution at all when . Therefore, we assume in the following.

Let in (3.4), then the equation has at least two distinct positive roots.

By simple calculation, one has

(3.5)

If , we can see for all . This means that is strictly increasing in the interval and hence the equation, cannot have two distinct positive roots. So, next we consider , which implies . Denote . We need to discuss several cases, respectively, as follows.

Case 1.

It holds that . Then for all , hence, is convex. Again, . So it is impossible for to have two distinct positive roots.

Case 2.

It holds that and . Then, for , and so ; for , and so . At this time, one always has . Then cannot have two distinct positive roots.

Case 3.

It holds that , and . Then, for , and so and hence , that is, has no solutions for ; for , , that is, is convex for . Noticing , it is also impossible for to have two distinct positive roots for .

Case 4.

It holds that , and . This is only case where could have two distinct positive roots, which implies and .
  1. (d)

    Let , φ, ψ, φ, ψ, be the eventual nonnegative prime period two solution of (1.3), then, it is eventually true that

     
(3.6)

It is easy to see that and . So, we have

(3.7)

that is, and are two distinct positive roots of . Obviously, when , the has no positive roots.

Now let and set . Then the function, has at least two distinct positive roots. However, for any , which indicates that is strictly increasing in the interval . This implies that the function does not have two distinct positive roots at all in the interval . In turn, (1.3) does not have the prime period two solution when .

4. Existence of Oscillatory Solution

For the oscillatory solution of (1.3), we have the following results.

Theorem 4.1.

Assume , is odd and is even. Then, there exist solutions of (1.3) which oscillate about with semicycles of length one.

Proof.

We only prove the case where (the proof of the case where is similar and will be omitted). Choose the initial values of (1.3) such that
(4.1)
or
(4.2)

We will only prove the case where (4.1) holds. The case where (4.2) holds is similar and will be omitted. According to (1.3), one can see that

(4.3)

So, the proof follows by induction.

5. Existence of Unbounded Solution

With respect to the unbounded solutions of (1.3), the following results are derived.

Theorem 5.1.

Assume , is odd, and is even, then (1.3) possesses unbounded solutions.

Proof.

We only prove the case where (the proof of the case where is similar and will be omitted). Choose the initial values of (1.3) such that
(5.1)
In the following, assume . From the proof of Theorem 4.1, one can see that when is odd and that for even. Together with
(5.2)
It is derived that
(5.3)
So, is decreasing for odd whereas is increasing for even. Let
(5.4)

then one has

(1) for odd and for even,

(2) .

Now, either for some even in which case the proof is complete, or for all even . We shall prove that this latter is impossible. In fact, we prove that for all even .

Assume for some even , then one has, by (5.2), . Noticing (1), one hence further gets . However is odd, according to (1), . This is a contradiction.

Therefore, for any even . Accordingly, are unbounded subsequences of this solution of (1.3) for even . Simultaneously, for odd , we get

(5.5)

The proof is complete.

Remark 5.2.

Theorem 5.1 includes and generalizes [3, Theorem  3.5].

6. Existence and Asymptotic Behavior of Nonoscillatory Solution

In this section, we consider the existence and asymptotic behavior of nonoscillatory solution of (1.3). Because all solutions of (1.3) are nonnegative, relative to the zero equilibrium point , every solution of (1.3) is a positive semicycle, a trivial nonoscillatory solution! Thus, it suffices to consider the positive equilibrium when studying the nonoscillatory solutions of (1.3). At this time, .

Firstly, we have the following results.

Theorem 6.1.

Every nonoscillatory solution of (1.3) with respect to approaches .

Proof.

Let be any one nonoscillatory solution of (1.3) with respect to . Then, there exists an such that
(6.1)
or
(6.2)
We only prove the case where (6.1) holds. The proof for the case where (6.2) holds is similar and will be omitted. According to (6.1), for , one has
(6.3)
So, is decreasing for with upper bound . Hence, exists and is finite. Denote
(6.4)
Then . Taking limits on both sides of (1.3), we can derive
(6.5)

which shows and completes this proof.

A problem naturally arises: are there nonoscillatory solutions of (1.3)? Next, we will positively answer this question. Our result is as follows.

Theorem 6.2.

However (1.3) possesses asymptotic solutions with a single semicycle (positive semicycle or negative semicycle).

The main tool to prove this theorem is to make use of Berg' inclusion theorem [12]. Now, for the sake of convenience of statement, we first state some preliminaries. For this, refer also to [13]. Consider a general real nonlinear difference equation of order with the form

(6.6)

where , . Let and be two sequences satisfying and as . Then (maybe under certain additional conditions), for any given , there exist a solution of (6.6) and an such

(6.7)

Denote

(6.8)

which is called an asymptotic stripe. So, if , then it is implied that there exists a real sequence such that and for .

We now state the inclusion theorem [12].

Lemma 6.3.

Let be continuously differentiable when , for , and . Let the partial derivatives of satisfy
(6.9)
as uniformly in for , , as far as . Assume that there exist a sequence and constants such that
(6.10)
for as , and suppose there exists an integer , with , such that
(6.11)

Then, for sufficiently large , there exists a solution of (6.6) satisfying (6.7).

Proof of Theorem 6.2.

We only prove the case where (the proof of the case where is similar and will be omitted). Put ( is denoted into for short). Then (1.3) is transformed into
(6.12)
An approximate equation of (6.12) is
(6.13)
provided that as . The general solution of (6.13) is
(6.14)
where and are the roots of the polynomial
(6.15)
Obviously, . So, has a positive root lying in the interval (0, 1). Now, choose the solution for this . For some , define the sequences and , respectively, as follows:
(6.16)

Obviously, and as .

Now, define again the function

(6.17)
Then the partial derivatives of w.r.t. , respectively, are
(6.18)

When , . So, , , as uniformly in for , .

Moreover, from the definition of the function and (6.17) and (6.18), after some calculation, we find

(6.19)

Now choose . Noting

(6.20)
we have Again,
(6.21)
where
(6.22)
Therefore, one has
(6.23)

Up to here, all conditions of Lemma 6.3 with and are satisfied. Accordingly, we see that, for arbitrary and for sufficiently large , say , (6.12) has a solution in the stripe , where and are as defined in (6.16). Because , for . Thus, (1.3) has a solution satisfying for . Since (1.3) is an autonomous equation, still is its solution, which evidently satisfies for . Therefore, the proof is complete.

Remark 6.4.

If we take in (6.16), then . At this time, (1.3) possesses solutions which remain below its equilibrium for all , that is, (1.3) has solutions with a single negative semicycle.

Remark 6.5.

The appropriate equation (6.12) is just the linearized equation of (1.3) associated with .

Remark 6.6.

The existence and asymptotic behavior of nonoscillatory solution of special cases of (1.3) has not been found to be considered in the known literatures.

Declarations

Acknowledgments

This work of the second author is partly supported by NNSF of China (Grant: 10771094) and the Foundation for the Innovation Group of Shenzhen University (Grant: 000133). Y. Wang work is supported by School Foundation of JiangSu Polytechnic University(Grant: JS200801).

Authors’ Affiliations

(1)
College of Mathematics and Computational Science, Shenzhen University
(2)
School of Physics & Mathematics, Jiangsu Polytechnic University

References

  1. Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjecture. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002.Google Scholar
  2. Amleh AM, Kirk V, Ladas G: On the dynamics of . Mathematical Sciences Research Hot-Line 2001,5(7):1–15.MATHMathSciNetGoogle Scholar
  3. El-Owaidy HM, Ahmed AM, Youssef AM: The dynamics of the recursive sequence . Applied Mathematics Letters 2005,18(9):1013–1018. 10.1016/j.aml.2003.09.014MATHMathSciNetView ArticleGoogle Scholar
  4. Kocić VL, Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications. Volume 256. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993.Google Scholar
  5. Agarwal RP: Difference Equations and Inequalities. 1st edition. Marcel Dekker, New York, NY, USA; 1992.MATHGoogle Scholar
  6. Agarwal RP: Difference Equations and Inequalities. 2nd edition. Marcel Dekker, New York, NY, USA; 2000.MATHGoogle Scholar
  7. Su Y-H, Li W-T: Global attractivity of a higher order nonlinear difference equation. Journal of Difference Equations and Applications 2005,11(10):947–958. 10.1080/10236190500273333MATHMathSciNetView ArticleGoogle Scholar
  8. Li X: The rule of trajectory structure and global asymptotic stability for a nonlinear difference equation. Indian Journal of Pure and Applied Mathematics 2007,38(6):1–9.MATHGoogle Scholar
  9. Li X, Agarwal RP: The rule of trajectory structure and global asymptotic stability for a fourth-order rational difference equation. Journal of the Korean Mathematical Society 2007,44(4):787–797. 10.4134/JKMS.2007.44.4.787MATHMathSciNetView ArticleGoogle Scholar
  10. Li X, Zhu D, Jin Y: Some properties of a kind of Lyness equations. Chinese Journal of Contemporary Mathematics 2004,24(2):147–155.MATHGoogle Scholar
  11. Li X, Zhu D: Qualitative analysis of Bobwhite Quail population model. Acta Mathematica Scientia 2003,23(1):46–52.MATHMathSciNetGoogle Scholar
  12. Berg L: Inclusion theorems for non-linear difference equations with applications. Journal of Difference Equations and Applications 2004,10(4):399–408. 10.1080/10236190310001625280MATHMathSciNetView ArticleGoogle Scholar
  13. Li X: Existence of solutions with a single semicycle for a general second-order rational difference equation. Journal of Mathematical Analysis and Applications 2007,334(1):528–533. 10.1016/j.jmaa.2006.12.072MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Dongmei Chen et al. 2009

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