Journal of Applied Mathematics

Volume 2014 (2014), Article ID 743032, 8 pages

http://dx.doi.org/10.1155/2014/743032

Research Article

Local Stability for Iterative Roots of Orientation-Preserving Self-Mappings on the Interval

Yingying Zeng

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received 9 April 2014; Accepted 1 May 2014; Published 21 May 2014

Academic Editor: Yongkun Li

Copyright © 2014 Yingying Zeng. 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.

Stability of iterative roots is important in their numerical computation. It is known that under some conditions iterative roots of orientation-preserving self-mappings are both globally stable and locally stable but globally unstable. Although the global instability implies the general global () instability, the local stability does not guarantee the local () stability. In this paper we generally prove the local () stability for iterative roots. For this purpose we need a uniform estimate for the approximation to the conjugation in linearization, which is given by improving the method used for the case.

1. Introduction

Let and let , , be the set of all self-mappings defined on . An iterative root of order of a given self-mapping is a self-mapping such that

where denotes the th iterate of , defined by and for all inductively. The study of iterative roots was started long ago, at least about two hundred years ago when Babbage published his paper [1]. In recent decades, regarded as a weak version of the embedding flow problem for dynamical systems [2, 3], the problem of iterative roots attracted great attention in the field of dynamical systems [3, 4] and functional equations [5–8]. Based on the work for monotonic mappings [6, 7], advances have been made to nonmonotonic cases [8–11], self-mappings on circles [12, 13], set-valued functions [14, 15], and high-dimensional mappings [16, 17].

Because of the potential in extensive applications (e.g., to information processing [18, 19] and graph theory [20]), numerical computation [21, 22] of iterative roots became an important task, which demands approximation to iterative roots and considers stability of iterative roots. In 2007 Xu and Zhang [23] proved stability for iterative roots on a closed interval with exact one fixed point at an endpoint. This result is substantially a local stability because the stability is totally decided by the behaviors of the iterative root in a sufficiently small neighborhood of the fixed point. In [24] results of global stability are given, where the global sense means the stability on a closed interval bounded by two fixed points. Recently, it was proved in [25] that iterative roots of every orientation-preserving self-mapping on the interval are locally stable but globally unstable.

In this paper we generally consider () stability of iterative roots. It is clear that the global instability given in [25] implies the general global () instability because approximation is the most fundamental requirement for approximation. However, the above result of local stability does not guarantee the local () stability. In this paper we concentrate on the local () stability for iterative roots of orientation-preserving self-mapping on . Clearly, the given mapping is a strictly increasing function. The local () stability is proved by approximation to the conjugation in linearization. In order to give an estimate to the approximation uniformly with respect to the order of iteration, we improve the method used in [25] to obtain lower growth rate for given functions under iteration.

2. Preliminaries

In order to state our results clearly, we first pay attention to the existence of () iterative roots of increasing self-mappings on a compact interval including exactly one fixed point which is hyperbolic. In some sense, this is a local case. For each and integer , let

(cf. Figures 1 and 2) together with the norm

In what follows we only discuss the first class because can be converted to by considering .

Figure 1: .

Figure 2: .

Given integers , a function belonging to the class has a unique th order iterative root defined on , which is strictly increasing and is given by the formula

where is the principal solution of Schröder’s equation

The principle solution is given by , satisfying differentiable in with and and strictly increasing by Theorem 6.1 in [6]. Moreover, the proofs of Theorem 3.5.1 in [7] and Theorem 4.5 in [6] show that

Note that the aim of this paper is to consider the local () stability for iterative roots. Then we recall the formula for higher order derivatives of composition ([26, page 3]). Namely, for integer ,

where , and for all . Here is a positive universal constant, which is independent of and . Then we have the following lemma.

Lemma 1. Let with some and and let be a sequence of functions in satisfying condition

Then, for a given number , there exist an , an , and an such that

for all , , and and for all .

Proof. Let and choose a sufficiently small such that and . Then by the mean value theorem,

for all . In particular, for all and . It follows that

where .

Then we give the proof of the first inequality given in (9) by induction on greater than . From (7), write

When , the second term is absent. Applying (12) and by induction, we have

where . Further, assume that the first inequality in (9) holds for all . Let . Noting that and

we get

inductively, where

Put . Then the first inequality given in (9) is proved.

It follows from (8) that there is an such that if , then and . Thus, by the same procedure as before, we can prove the second inequality given in (9).

In the following, we are going to prove inequality (10). It is clear that (10) holds when . Further, assume that (10) holds for . It follows that

for all and for all . Thus, we can obtain (10) by induction. This completes the proof.

In Lemma 1 we gave a better estimate for and and their derivatives than that given in [25, Lemma 2.1]. In Lemma 1 the growth rate on is much lower in the sense that the constant given in (9) tends to 0 as faster than the constant  given in (2.4) of [25].

3. The Main Result

Our aim of this section is to prove the following stability result.

Theorem 2. Given integers , let with some and let be a sequence of functions in . If

then

where and are unique th order iterative roots of and , respectively, defined on .

In order to prove Theorem 2 we need the following lemma on stability of Schröder’s equation.

Lemma 3. Given an integer , let with some and let be a sequence of functions in satisfying condition (19). Then

where and are the principal solutions of Schröder’s equations and , respectively.

Proof. From (6) we can see that

for all and . In what follows we intend to discuss our results in a sufficiently small interval first and extend them to the whole interval , where is given in Lemma 1.

In order to prove the convergence of the sequence in , we claim that there exists a constant , which is independent of , such that

where . If (23) is true, then we get

implying the stability in . Next, we extend the result (24) from to the whole interval . As indicated in [25], we have

by (8) because the composition operator is continuous by Example 4.4.5 in [27]. Moreover, since is the unique stable fixed point of in and by (8), there is an integer such that for all and . Then, according to Schröder’s equation, we can obtain the formulae

where and . Then by Lemma 1 and from (24), (26), and the uniform continuity of , we get and

as for all . Hence, (21) is proved and the proof is completed.

In the following, we will prove the claimed (23) by induction on . Clearly, (2.11) in [25] and what is indicated above the proof of Theorem 2.1 in [25] show that (23) holds for and , respectively. Then we suppose that (23) is also satisfied for all , where , and we will prove (23) for . Our strategy is to prove that, for given , there exists such that

and is a bounded sequence whose upper bound satisfies (23). Clearly, for we can find constant satisfying (28). Then, assume that there exists such that (28) holds for any . Applying Lemma 1 and the inductive hypothesis, we have

where . By Lemma 1 and condition (19), we obtain that ,

where and is given in Lemma 1, for all , , , and . It follows that

because , where

Let

It is easy to check that

for all . By putting , we can prove (23) for and (23) is proved by induction.

Having this preparation, we can give a proof to the main result of this paper.

Proof of Theorem 2. By (4) the iterative roots and for each can be presented by

respectively. In order to prove the convergence of in , note that, for sufficiently large such that for all , we have

as for all integers by (21) and the fact that the inverse operator is continuous and is uniformly continuous. Hence, and

as for all , which implies that and completes the proof.

Theorem 2 is also valid for , which is the same as Theorem 2.1 in [25] for . However, it is hard to use the estimates, for example, (2.4) and (2.5) in [25], to generalize the result to the general parallel. In fact, we cannot use those estimates to give a uniform constant with respect to , the order of iteration, in (33). Using those estimates, corresponding to given in (32), we obtain the quantity

which tends to as . For this reason it is hard to prove the boundedness of . As remarked after the proof of Lemma 1, our estimation in (9) and (10) enables us to give the boundedness of and complete the proof of (23).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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