Copyright © 2009 Fangkuan Sun 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

This paper studies the dynamic behavior of the positive solutions to the difference equation xn=A+xn−kp/xn−1r, n=1,2,…, where A,p, and r are positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation for p∈(0,1).

1. Introduction

In recent years, there has been intense interest in the dynamic behavior of the positive solutions to a class of difference equations of the form (1.1) where and are positive real numbers. Now, let us make a brief review on the advances in this class of difference equations.

In 1999, Amleh et al. [1] studied the second-order rational difference equation (1.2)

Later, Berenhaut and Stević [2], Stević [3], and El-Owaidy et al. [4] extended this work to the following more general second-order difference equation: (1.3)

On the other hand, DeVault et al. [5] investigated the following higher-order version of (1.2): (1.4)

By combining (1.3) and (1.4), Berenhaut and Stević [6] examined a larger class of difference equations, which are of the form (1.5)

Very recently, Berenhaut et al. [7] studied the following generalization of (1.5): (1.6)

For some related work, the interested reader is referred to [1, 3, 8–19].

Inspired by the previous work and by the work owing to Stević [15], this paper studies the behavior of the recursive equation (1.7) We establish some interesting results regarding the stability and oscillation character of this equation for .

2. Stability Character

In this section we investigate the stability character of the positive solutions to (1.7).

A point is an equilibrium point of (1.7) if and only if it is a root for the function (2.1) that is, (2.2)

Lemma 2.1. Let , then (1.7) has a unique equilibrium point .

Proof

Case 1. . Then . Case 2. . Then defined by (2.1) is decreasing on and increasing on . Since and , then has a unique zero . Case 3. . Since is increasing on and , then has a unique zero .

Lemma 2.2. Let . Assume that is the equilibrium point of (1.7). If , then is locally asymptotically stable.

Proof. By the Linearized Stability Theorem [11], is locally asymptotically stable if and only if . A simple calculations shows that (2.3) where is defined by (2.1). Then since , we have and . The proof is complete.

Lemma 2.3. If , then every positive solution to (1.7) is bounded.

Proof. Note that each can be written in the form for some and . From (1.7) and since for every , we have that (2.4) for every and . Let be the solution to the difference equation (2.5)

From (2.4) and by induction we see that . Hence it is enough to prove that the sequences are bounded.

Since the function is increasing and concave for , it follows that there is a unique fixed point of the equation and that the function satisfies (2.6)

Using this fact it is easy to see that if , the sequence is nondecreasing and bounded from above by , and if , it is nonincreasing and bounded from below by . Hence for every , each of the sequences is bounded. The claimed result follows.

Lemma 2.4 (see [18]). Let be distinct nonnegative integers. Consider the difference equation (2.7) Suppose satisfies the following conditions. is a continuous function that is nondecreasing in the first argument and is nonincreasing in the second argument. The system (2.8) has a unique solution . Then is the global attractor of all solutions to (2.7).

Theorem 2.5. Let , then the unique equilibrium to (1.7) is globally asymptotically stable.

Proof. By Lemma 2.3, there must exist positive constants and such that . Let , it is easy to verify that holds. In addition, if (2.9) then (2.10) Assume that , then or .

In case , we have , which contradicts with (2.10).

In case , we have , again a contradiction.

Thus . By Lemma 2.4, the required result follows.

Theorem 2.6. Let and . Then every positive solution to (1.7) converges to the unique equilibrium .

Proof. By Lemma 2.3, every positive solution to (1.7) is bounded, which implies that there are finite and . Assume that . Taking the and in (1.7), it follows that (2.11) From this and , it follows that (2.12) yielding (2.13)

Define function . Since (2.14) we deduce that is increasing, and thus (2.13) cannot hold. Therefore we have , which implies the result.

Theorem 2.7. Let , and . Then every positive solution to (1.7) converges to the unique equilibrium .

Proof. From (2.11) we have (2.15) Consequently, we obtain . Suppose that , we get (2.16) where , leading to (2.17)

This implies that , which is a contradiction. Hence, .

3. Oscillation Character

In this section we investigate the oscillation character of the positive solutions to (1.7).

Theorem 3.1. Let be a positive solution to (1.7). Then either consists of a single semicycle or oscillates about the equilibrium with semicycles having at most terms.

Proof. Suppose that has at least two semicycles. Then there exists such that either or . Assume that . (The argument for the case is similar and is omitted). Now suppose that the positive semicycle beginning with the term has terms. Then and so (3.1) This completes the proof.

Theorem 3.2. Suppose that is even and let be a solution to (1.7), which has consecutive semicycles of length one, then every semicycle after this point is of length one.

Proof. There exists such that either (3.2) or (3.3) We prove the former case. The proof for the latter is similar and is omitted. Now, we have (3.4)

The result then follows by induction.

Lemma 3.3. Let . Then (1.7) has no nontrivial periodic solutions of (not necessarily prime) period .

Proof. Suppose that is a positive solution to (1.7) satisfying for all , then implies that for all . The proof is complete.

Theorem 3.4. Assume that . Let be a positive solution to (1.7), which consists of a single semicycle, then converges to the equilibrium .

Proof. Suppose (the case for is similar and is omitted) for all , then (3.5) implying that (3.6) and so (3.7) From here it is clear that for there exists such that (3.8) But then is a periodic solution of (not necessarily prime) period . By Lemma 3.3 the result holds.

Acknowledgments

The author is grateful to the anonymous referees for their valuable comments and suggestions. This work was supported by the Natural Science Foundation of China (no. 10771227) and the Project for New Century Excellent Talents of Educational Ministry of China (no. NCET-05-0759).

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