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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 235808, 14 pages
http://dx.doi.org/10.1155/2010/235808
Research Article

Quantitative Bounds for Positive Solutions of a Stević Difference Equation

College of Computer Science, Chongqing University, Chongqing 400044, China

Received 8 November 2009; Revised 5 March 2010; Accepted 7 April 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Wanping Liu and Xiaofan Yang. 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 behavior of positive solutions to the following particular case of a difference equation by Stević , , where ,, , and presents theoretically computable explicit lower and upper bounds for the positive solutions to this equation. Besides, a concrete example is given to show the computing approaches which are effective for small parameters. Some analogous results are also established for the corresponding Stević max-type difference equation.

1. Introduction

The study regarding the behavior of positive solutions to the difference equation where and was put forward by Stević at many conferences (see, e.g., [13]). For numerous papers in this area and some closely related results, see [139] and the references cited therein.

In [4, 24], the authors proved some conditions for the global asymptotic stability of the positive equilibrium to the difference equation given by with

Motivated by these papers, the authors of [8] studied the quantitative bounds for the recursive equation (1.2) where , and and quantitative bounds of the form were provided. Exponential convergence was shown to persist for all solutions. The authors also took as an example, and eventually obtained the concrete bounds as follows: In [20], Stević investigated positive solutions of the following difference equation: where , and gave a complete picture concerning the boundedness character of the positive solutions to (1.4) as well as of positive solutions of the following counterpart in the class of max-type difference equations: where are positive real numbers.

Motivated by the above work and works in [6, 9, 10, 12, 17, 21, 22], our aim in this paper is to discuss the quantitative bounds of the solutions to the following higher-order difference equation: where , and the initial values are positive. Following the methods and ideas from [8], we obtain theoretically computable explicit bounds of the form which are independent of the positive initial values

Our results extend those ones in [8], in which the case was considered, and also in some way improve those in [20], in which the case was considered.

On the other hand, inspired by the study in [19] we also investigate the quantitative bounds for the positive solutions to the following max-type recursive equation: where and some similar results are established.

We want to point out that the boundedness characters of (1.1) and (1.8) for the case and , including our particular case, have been recently solved by Stević and presented at several conferences (see also [25]).

2. Auxiliary Results

In this section, we will present several preliminary lemmas needed to prove the main results in Section 3.

The following lemma can be easily proved.

Lemma 2.1. Equation (1.6) has a unique positive equilibrium point

Now, let us define a first-order difference equation given by where are identical to those of (1.6), , and the initial value .

If , then (2.1) reduces to the sequence defined in [8].

Lemma 2.2. Equation (2.1) has a unique positive equilibrium if and or .

Proof. Suppose that is an equilibrium point of (2.1), then we have Let , then it suffices to show that has only one positive fixed point. The derivative of is (i)If , then obviously for .(ii)If and , then follows from .(iii)If and , we have Hence
Through above analysis, if and or , then is monotonically increasing on . Hence the uniqueness of positive equilibrium of (2.1) follows from , and .

Lemma 2.3. If and or , then the unique equilibrium point of (1.6) has the form , where is the unique positive equilibrium of (2.1).

Proof. Defining , simply we have that has a unique positive zero denoted by , that is, If , then , and thus If and , then Hence From above analysis, we conclude that and are the unique equilibriums of (2.1) and (1.6), respectively.

3. Quantitative Bounds of Solutions to (1.6)

In this section, through analyzing the boundedness of (1.6) we mainly present two explicit bounds for the positive solutions to (1.6).

Let the positive sequence be a solution to (1.6), then for we define It follows from (3.1) and (1.6) that Combining (3.1) and (1.6), we can simply obtain that By (3.2) and (3.3), the identity holds for all

Note that for , and hence it follows from (3.2) that Let us define two sequences and recursively in the following way: for all , and the initial values satisfy Apparently for , and the problem of bounding (1.6) reduces to consideration of the recursive dependent sequences

Lemma 3.1. The sequences and are nondecreasing and nonincreasing, respectively.

Proof. It follows from (3.7) that for we have and Hence assume that and for . By induction, we have that Through similar calculations, we have , and by induction the lemma is proved.

Theorem 3.2. For (2.1) with , let and for . Then the inequality holds for all .

Proof. From (3.7) and the definitions of we have that for . Thus, assume that and for . Then Similar calculations lead to , and inductively the theorem can be proved.

Theorem 3.3. If the solution to (2.1) with converges to the unique equilibrium under the conditions in Lemma 2.2 and there exist two sequences and which are lower and upper bounds for (2.1) such that , and , then the solutions to (1.6) have explicit bounds of the following form: for all

Proof. The proof follows directly from Lemma 2.2 and Theorem 3.2, and thus is omitted.

Note that Theorem 3.3 and Lemma 2.3 imply the following corollary.

Corollary 3.4. If the solution to (2.1) with converges to the unique equilibrium under the conditions in Lemma 2.2, then the unique equilibrium of (1.6) is a global attractor.

By Theorem 3.3, it suffices to determine the explicit bounds for (2.1). In the following, a simple case would be taken. For example,

if the parameters are fixed, then by (2.1) we get

Denote for ( the unique equilibrium of (2.1)), then we have that, being for , where the function is defined by for

Example 3.5 (). Then reduces to the following form: Obviously, is monotonically increasing for and . By simplifying , we have Having the function defined via , we get the derivative of as follows: for all Thus for and Therefore whenever . In addition, for and both and are monotonically increasing. Hence we have and whenever
Now set , for , and . Note that and . Thus suppose that and , for . Then by induction we have Therefore since the fact that for even and for odd, we obtain that, for , Employing Theorem 3.3, we get the bounds

4. Quantitative Bounds for Solutions to (1.8)

In this section, the upper and lower bounds of solutions to (1.8) are given, and first we present a lemma concerning the equilibrium points of (1.8).

Lemma 4.1. If , then (1.8) has a unique positive equilibrium ; and if , then (1.8) has a unique positive equilibrium .

The proof is simple and thus omitted.

Suppose that is a positive solution to (1.8), and by the transformation we have that It follows from (4.1) and (1.8) that Employing (4.2) and (4.3), we obtain that, for , For two nonnegative sequences and , let for .

Then according to (4.4) and the sequences we have that .

Note that for , and thus from (4.2) we get for .

Lemma 4.2. Let for , then the sequences and are nondecreasing and nonincreasing, respectively.

Proof. Assume that and for . Then we have that Analogous argument gives that , and the lemma can be proved inductively.

Now we will take in the other first-order recursive equation where equal those of (1.8), , and the initial value .

Through Lemma 4.2, simpler bounds for are given below.

Theorem 4.3. The inequality holds for , where and .

Proof. Let for , and by the definitions of and we have that and for .
Assume that and for , then Similar computations lead to the inequality , and the theorem follows by induction.

Theorem 4.3 and (4.8) imply the following result.

Theorem 4.4. Suppose that there exist two positive sequences and which are bounds for a positive solution to (4.8) with such that Then the solutions to (1.8) have explicit upper and lower bounds of the following form: for all

5. Conclusion

In this paper, we investigate a particular case of a higher-order difference equation by Stević which is a natural extension of that one in [8], and mainly present improved results which give computable approaches for quantitative bounds of solutions to (1.6). However, the methods are only effective for small parameters, because complex polynomials will arise in the process of computing for large parameters .

On the basis of Corollary 3.4 and Theorem 4.4, we suggest to study the behaviors, particularly the convergence and stability, of positive solutions to the following two recursive equations: where , and

Acknowledgment

The authors are grateful to the referees for their huge number of valuable suggestions, which considerably improved the presentation of the consequences in the paper.

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