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

f(f k−1 (x)) and f(x) = x for all x ∈ I 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 beenmade to nonmonotonic cases [8–11], self-mappings on circles [12, 13], set-valued functions [14, 15], and highdimensional 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 C stability for iterative roots on a closed interval with exact one fixed point at an endpoint. This result is substantially a local C 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 C 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 selfmapping on the interval are locally C stable but globally


Introduction
Let  := [0, 1] and let   (, ),  ≥ 0, be the set of all   selfmappings 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   () := ( −1 ()) and  0 () =  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][6][7][8].
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  0 stability for iterative roots on a closed interval with exact one fixed point at an endpoint.This result is substantially a local  0 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  0 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 selfmapping on the interval are locally  1 stable but globally  1 unstable.
In this paper we generally consider   ( ≥ 1) stability of iterative roots.It is clear that the global  1 instability given in [25] implies the general global   ( ≥ 2) instability because  1 approximation is the most fundamental requirement for   approximation.However, the above result of local  1 stability does not guarantee the local   ( ≥ 2) stability.In this paper we concentrate on the local   ( ≥ 2) stability for iterative roots of orientation-preserving self-mapping on .Clearly, the given mapping is a strictly increasing function.The local   ( ≥ 2) 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.

Preliminaries
In order to state our results clearly, we first pay attention to the existence of   ( ≥ 2) iterative roots of increasing   selfmappings on a compact interval including exactly one fixed point which is hyperbolic.In some sense, this is a local case.
In the following, we are going to prove inequality (10).It is clear that (10) for all  ≥  0 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   0 given in (2.4) of [25].

The Main Result
Our aim of this section is to prove the following stability result.
Theorem 2. Given integers ,  ≥ 2, let  ∈ H +1 − () with some  ∈ (0, 1) and let (  ) be a sequence of functions in 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.
Proof.From (6) we can see that for all  ∈ [0, ] ∩ Z 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  = 0, 1, . . ., .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 lim by ( 8) because the composition operator is continuous by Example 4.4.5 in [27].Moreover, since 0 is the unique stable fixed point of  in  and by (8), there is an integer  ∈ N such that   (),    () ∈   for all  ∈ N and  ∈ .Then, according to Schröder's equation, we can obtain the formulae where φ := |   and φ :=   |   .Then by Lemma 1 and from ( 24), (26), and the uniform continuity of as  → +∞ for all  = 1, . . ., .Hence, ( 21) is proved and the proof is completed.
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  ∈ N can be presented by respectively.In order to prove the convergence of (  ) in , note that, for sufficiently large  ∈ N such that  1/   () ∈ () for all  ∈ , we have as  → +∞ for all  = 1, . . ., , which implies that lim  → +∞ ‖  − ‖  = 0 and completes the proof.
Theorem 2 is also valid for  = 1, which is the same as Theorem 2.1 in [25] for  = 1.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 Γ with respect to , the order of iteration, in (33).Using those estimates, corresponding to Γ given in (32), we obtain the quantity which tends to +∞ as  → +∞.For this reason it is hard to prove the boundedness of  ℓ+1 ().As remarked after the proof of Lemma 1, our estimation in ( 9) and (10) enables us to give the boundedness of  ℓ+1 () and complete the proof of (23).