Dynamical Properties for a Class of Fourth-Order Nonlinear Difference Equations
© Dongsheng Li et al. 2008
Received: 6 May 2007
Accepted: 18 September 2007
Published: 28 November 2007
We consider the dynamical properties for a kind of fourth-order rational difference equations. The key is for us to find that the successive lengths of positive and negative semicycles for nontrivial solutions of this equation periodically occur with same prime period 5. Although the period is same, the order for the successive lengths of positive and negative semicycles is completely different. The rule is or or . By the use of the rule, the positive equilibrium point of this equation is proved to be globally asymptotically stable.
1. Introduction and Preliminaries
Rational difference equation, as a kind of typical nonlinear difference equations, is always a subject studied in recent years. Especially, some prototypes for the development of the basic theory of the global behavior of nonlinear difference equations of order greater than one come from the results of rational difference equations. For the systematical investigations of this aspect, refer to the monographs [1–3], the papers [4–9], and the references cited therein.
Mainly, by analyzing the rule for the length of semicycle to occur successively, we clearly describe out the rule for the trajectory structure of its solutions. With the help of several key lemmas, we further derive the global asymptotic stability of positive equilibrium of (1.1). To the best of our knowledge, (1.1) has not been investigated so far; therefore, it is theoretically meaningful to study its qualitative properties.
It is essential in this note for us to obtain the general rule for the trajectory structure of solutions of (1.1) as follows.
The rule for the trajectory structure of any solution of (1.1) is as follows:
the solution is either eventually trivial,
The positive equilibrium point of (1.1) is a global attractor of all its solutions.
It follows from the results stated in the sequel that Theorem 1.1 is true.
2. Nontrivial Solution
3. Several Key Lemmas
The results of Lemmas 3.1, 3.2, and 3.3 can be easily obtained from (1.1), and so we omit their proofs here.
By virtue of (3.12), (3.13), (3.17), and (3.20), we see that (c) is true.
The proofs of (d) and (e) are similar to those of (c). The proof for this lemma is complete.
This shows that Lemma 3.5 is true.
4. Oscillation and Nonoscillation
Suppose that there exist eventually negative nonoscillatory solutions of (1.1). Then, there exists a positive integer such that for . Thereout, for , This contradicts Lemma 3.4(a). So, there are not eventually negative nonoscillatory solutions of (1.1), as desired.
5. Rule of Cycle Length
By Lemma 3.4(a), one can see that the length of a negative semicycle is at most 4, and that of a positive semicycle is at most 3. On the basis of the strictly oscillatory character of the solution, we see, for some integer , that one of the following sixteen cases must occur:
If case (1) occurs, of course, it will be a nonoscillatory solution of (1.1).
Moreover, the rule for the cases (2.3), (3.5), (3.13), (3.15), and (3.17) is the same as that of case (2.1). And cases (3.1), (3.3), and (3.15) are completely similar to case (3) except possibly for the first semicycle. And cases (3.16), (3.18), (3.19), and (3.20) are like case (8) with a possible exception for the first semicycle.
Up to now, the proof of Theorem 5.1 is complete.
6. Global Asymptotic Stability
The positive equilibrium point of (1.1) is locally asymptotically stable.
and so it is clear from [3, Remark 1.3.7] that the positive equilibrium point of (1.1) is locally asymptotically stable. The proof is complete.
The positive equilibrium point of (1.1) is globally asymptotically stable.
We can divide the solutions into two types:
Consider now to be nonoscillatory about the positive equilibrium point of (1.1). By virtue of Lemma 3.4(b), it follows that the solution is monotonic and bounded. So, exists and is finite. Taking limits on both sides of (1.1), one can easily see that (6.2) holds.
Now, let be strictly oscillatory about the positive equilibrium point of (1.1). By virtue of Theorem 5.1, one understands that the rule for the lengths of positive and negative semicycles occurring successively is
Similarly, using (b), one can see that is increasing with upper bound 1. So, its limit exists and is finite too. Furthermore, the limits of are equal to that of , and one can assume the limits of it to be . It is easy to see that It suffices to show that
When case (ii) or (iii) occurs, using the method similar to that proving case (i), we can prove that (6.2) is also true. Thus, the proof of Theorem is complete.
This work was partly supported by NNSF of China (Grant no. 10771094), and the Foundations for "New Century Engineering of 121 Talents in Hunan Province" and "Chief Professor of Mathematical Discipline in Hunan Province."
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