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

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 𝐶⁰ stable and locally 𝐶¹ stable but globally 𝐶¹ unstable. Although the global 𝐶¹ instability implies the general global 𝐶^𝑟 (𝑟≥2) instability, the local 𝐶¹ stability does not guarantee the local 𝐶^𝑟 (𝑟≥2) stability. In this paper we generally prove the local 𝐶^𝑟 (𝑟≥2) stability for iterative roots. For this purpose we need a uniform estimate for the approximation to the conjugation in 𝐶^𝑟 linearization, which is given by improving the method used for the 𝐶¹ case.

1. Introduction

Let 𝐼∶=[0,1] and let 𝐶^𝑟(𝐼,𝐼), 𝑟≥0, be the set of all 𝐶^𝑟 self-mappings defined on 𝐼. An iterative root of order 𝑘 of a given self-mapping 𝐹∶𝐼→𝐼 is a self-mapping 𝑓∶𝐼→𝐼 such that 𝑓^𝑘(𝑥)=𝐹(𝑥),∀𝑥∈𝐼, (1) where 𝑓^𝑘 denotes the 𝑘th iterate of 𝑓, defined by 𝑓^𝑘(𝑥)∶=𝑓(𝑓^𝑘−1(𝑥)) and 𝑓⁰(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 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 𝐶⁰ stability for iterative roots on a closed interval with exact one fixed point at an endpoint. This result is substantially a local 𝐶⁰ 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 𝐶⁰ 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 𝐶¹ stable but globally 𝐶¹ unstable.

In this paper we generally consider 𝐶^𝑟 (𝑟≥1) stability of iterative roots. It is clear that the global 𝐶¹ instability given in [25] implies the general global 𝐶^𝑟 (𝑟≥2) instability because 𝐶¹ approximation is the most fundamental requirement for 𝐶^𝑟 approximation. However, the above result of local 𝐶¹ stability does not guarantee the local 𝐶^𝑟 (𝑟≥2) stability. In this paper we concentrate on the local 𝐶^𝑟 (𝑟≥2) stability for iterative roots of orientation-preserving self-mapping on 𝐼. Clearly, the given mapping is a strictly increasing function. The local 𝐶^𝑟 (𝑟≥2) stability is proved by approximation to the conjugation in 𝐶^𝑟 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 𝐶^𝑟 (𝑟≥2) iterative roots of increasing 𝐶^𝑟 self-mappings on a compact interval including exactly one fixed point which is hyperbolic. In some sense, this is a local case. For each 𝜆∈(0,1) and integer 𝑟≥0, let ℋ^𝑟₋(𝜆)∶=ℎ∈𝐶^𝑟(𝐼,𝐼)∶ℎ(0)=0,ℎ^′(0)=𝜆,ℎ^′(𝑥)ℎ(𝑥)(,ℋ>0,<𝑥,∀𝑥∈0,1]_𝑟+(𝜆)∶=ℎ∈𝐶^𝑟(𝐼,𝐼)∶ℎ(1)=1,ℎ^′(1)=𝜆,ℎ^′(𝑥)ℎ(𝑥)[>0,>𝑥,∀𝑥∈0,1) (2) (cf. Figures 1 and 2) together with the norm ‖ℎ‖_𝑟∶=sup_𝑥∈𝐼|ℎ(𝑥)|+⋯+sup_𝑥∈𝐼ℎ^(𝑟)(𝑥). (3) In what follows we only discuss the first class because ℋ^𝑟₊(𝜆) can be converted to ℋ^𝑟₋(𝜆) by considering 𝐺(𝑥)=1−𝐹(1−𝑥).

Figure 1 

ℎ∈ℋ𝑟−(𝜆).

Figure 2 

ℎ∈ℋ𝑟+(𝜆).

Given integers 𝑘,𝑟≥2, a function 𝐹 belonging to the class ⋃_𝜆∈(0,1)ℋ^𝑟₋(𝜆) has a unique 𝑘th order 𝐶^𝑟 iterative root 𝑓 defined on 𝐼, which is strictly increasing and is given by the formula 𝑓(𝑥)∶=𝜑⁻¹𝜆^1/𝑘𝜑(𝑥), (4) where 𝜑∶𝐼→ℝ is the principal solution of Schröder’s equation 𝜑(𝐹(𝑥))=𝜆𝜑(𝑥). (5) The principle solution is given by 𝜑(𝑥)=lim_𝑛→+∞𝜆^−𝑛𝐹^𝑛(𝑥), satisfying 𝐶^𝑟 differentiable in 𝐼 with 𝜑^′(0)=1 and 𝜑(0)=0 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 𝜑^(𝑟)(𝑥)=lim_𝑛→+∞𝜆^−𝑛𝐹^𝑛(𝑟)(𝑥). (6)

Note that the aim of this paper is to consider the local 𝐶^𝑟 (𝑟≥2) stability for iterative roots. Then we recall the formula for higher order derivatives of composition ([26, page 3]). Namely, for integer 𝑚≥1, (𝐺∘𝐻)^(𝑚)(𝑥)=_{𝜔∈Ω(𝑚)}𝑐_𝜔𝐺^(𝑝)(𝐻(𝑥))^𝑝_𝑞=1𝐻^(𝑗𝑞)(𝑥), (7) where Ω(𝑚)∶={(𝑝;𝑗₁,…,𝑗_𝑝)∶1≤𝑝≤𝑚, 𝑗₁+⋯+𝑗_𝑝=𝑚 and 𝑗_𝑞≥1 for all 𝑞=1,…,𝑝}. Here 𝑐_𝜔 is a positive universal constant, which is independent of 𝐺 and 𝐻. Then we have the following lemma.

Lemma 1. Let 𝐹∈ℋ^𝑟₋(𝜆) with some 𝑟≥2 and 𝜆∈(0,1) and let (𝐹_𝑖) be a sequence of functions in ℋ^𝑟₋(𝜆) satisfying condition lim_𝑖→+∞𝐹_𝑖−𝐹_𝑟=0. (8) Then, for a given number 𝜇∈(𝜆,1), there exist an 𝐿>1, an 𝜀>0, and an 𝑁₀∈ℕ such that 𝐹^𝑛(𝑚)(𝑥)≤𝐿𝜆^𝑛,𝐹^𝑛_𝑖^(𝑚)(𝑥)≤𝐿𝜆^𝑛,𝐹^𝑛_𝑖(𝑥)−𝐹^𝑛(𝑥)≤1+𝐿𝜆𝜇𝜇−𝜆^𝑛−1−𝐿𝜆𝜇−𝜆^𝑛𝐹_𝑖−𝐹₁ (9)(10) for all 𝑖≥𝑁₀, 𝑛∈ℕ, and 0≤𝑚≤𝑟 and for all 𝑥∈𝐼_𝜀∶=[0,𝜀].

Proof. Let 𝐾∶=2‖𝐹‖_𝑟 and choose a sufficiently small 𝜀>0 such that 𝜆+𝐾𝜀<1 and sup_{𝑥∈𝐼𝜀}|𝐹^′(𝑥)|<𝜇. Then by the mean value theorem, |𝐹(𝑥)|=|𝐹(𝑥)−𝐹(0)|=𝐹^′𝜉_𝑥𝑥|≤𝐹_′|(0)|𝑥|+𝐹^′𝜉_𝑥−𝐹^′(0𝑥||𝑥||𝑥|)|≤𝜆+𝐾² (11) for all 𝑥∈𝐼. In particular, |𝐹^𝑛−1(𝑥)|≤(𝜆+𝐾𝜀)^𝑛−1 for all 𝑛≥1 and 𝑥∈𝐼_𝜀. It follows that 𝐹^𝑛(𝑥)≤𝜆𝐹^𝑛−1(𝑥)+𝐾𝐹^𝑛−1(𝑥)²≤𝜆+𝐾(𝜆+𝐾𝜀)^𝑛−1𝐹^𝑛−1(𝑥≤_{𝑛−1𝑠=0})𝜆+𝐾(𝜆+𝐾𝜀)^𝑠|𝑥|=𝜆^𝑛𝑛−1_𝑠=01+𝐾𝜆⁻¹(𝜆+𝐾𝜀)^𝑠|𝑥|≤𝐿₀𝜆^𝑛,∀𝑥∈𝐼_𝜀, (12) where 𝐿₀∶=∏^+∞_𝑠=0{1+𝐾𝜆⁻¹(𝜆+𝜂𝜀)^𝑠}∈(1,+∞).
Then we give the proof of the first inequality given in (9) by induction on 𝑚 greater than 1. From (7), write 𝐹^𝑛(𝑚)(𝑥)=𝐹^′𝐹^𝑛−1(𝑥)𝐹^{𝑛−1(𝑚)}(𝑥)+_{𝑝≥2𝜔∈Ω(𝑚)}𝑐_𝜔𝐹^(𝑝)𝐹^𝑛−1(𝑥)^𝑝_𝑞=1𝐹^{𝑛−1(𝑗𝑞)}(𝑥). (13) When 𝑚=1, the second term is absent. Applying (12) and by induction, we have 𝐹^𝑛′(𝑥)=𝐹^′𝐹^𝑛−1(𝑥)𝐹^𝑛−1′(𝑥≤𝐹_′)(0)+𝐹^′𝐹^𝑛−1(𝑥)−𝐹^′(0)𝐹^𝑛−1′(𝑥≤)𝜆+𝐾𝐿₀𝜆^𝑛−1𝐹^𝑛−1′(𝑥≤_{𝑛−1𝑠=0})𝜆+𝐾𝐿₀𝜆^𝑠|𝑥|=𝜆^𝑛𝑛−1_𝑠=01+𝐾𝐿₀𝜆^𝑠−1|𝑥|≤𝐿₁𝜆^𝑛,∀𝑥∈𝐼_𝜀, (14) where 𝐿₁∶=∏^+∞_𝑠=0{1+𝐾𝐿₀𝜆^𝑠−1}∈(1,+∞). Further, assume that the first inequality in (9) holds for all 𝑚≤ℓ<𝑟. Let Θ∶=∑_{𝑝≥2,𝜔∈Ω(ℓ+1)}𝑐_𝜔𝐾∏^ℓ_𝑞=1𝐿_𝑞. Noting that 𝜆∈(0,1) and 𝐹^′𝐹^𝑛−1(𝑥)≤𝐹^′(0)+𝐹^′𝐹^𝑛−1(𝑥)−𝐹^′(0)≤𝜆+𝐾𝐿₀𝜆^𝑛−1,∀𝑥∈𝐼_𝜀, (15) we get 𝐹^𝑛(ℓ+1)(𝑥=𝐹_′)𝐹^𝑛−1(𝑥)𝐹^{𝑛−1(ℓ+1)}(𝑥)+_{()𝑝≥2𝜔∈Ωℓ+1}𝑐_𝜔𝐹^(𝑝)𝐹^𝑛−1(𝑥)^𝑝_𝑞=1𝐹^{𝑛−1(𝑗𝑞)}(𝑥)≤𝜆+𝐾𝐿₀𝜆^𝑛−1𝐹^{𝑛−1(ℓ+1)}(𝑥)+Θ𝜆^2(𝑛−1)≤𝜆^𝑛𝑛−2_𝑠=−11+𝐾𝐿₀𝜆^𝑠+Θ^𝑛−1_𝑡=0𝜆^{2𝑡𝑛−1}_𝑠=𝑡+1𝜆+𝐾𝐿₀𝜆^𝑠=𝜆^𝑛𝑛−2_𝑠=−11+𝐾𝐿₀𝜆^𝑠+Θ𝜆^𝑛𝑛−1_𝑡=0𝜆^{𝑡−1𝑛−2}_𝑠=𝑡1+𝐾𝐿₀𝜆^𝑠≤𝐿_ℓ+1𝜆^𝑛 (16) inductively, where 𝐿_ℓ+1∶=^+∞_𝑠=−11+𝐿₀𝐾𝜆^𝑠+Θ^+∞_𝑡=0𝜆^𝑡−1+∞_𝑠=01+𝐾𝐿₀𝜆^𝑠∈(1,+∞). (17) Put 𝐿∶=max{𝐿₀,𝐿₁,…,𝐿_𝑟}. Then the first inequality given in (9) is proved.
It follows from (8) that there is an 𝑁₀∈ℕ such that if 𝑖≥𝑁₀, then ‖𝐹_𝑖‖_𝑟≤2‖𝐹‖_𝑟=𝐾 and sup_{𝑥∈𝐼𝜀}|𝐹^′_𝑖(𝑥)|<𝜇. 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 𝑛=1. Further, assume that (10) holds for 𝑛=ℓ∈ℕ. It follows that 𝐹_𝑖^ℓ+1(𝑥)−𝐹^ℓ+1(𝑥=𝐹_𝑖)𝐹^ℓ_𝑖(𝑥)−𝐹𝐹^ℓ(𝑥)≤𝐹_𝑖𝐹^ℓ_𝑖(𝑥)−𝐹_𝑖𝐹^ℓ(𝑥)+𝐹_𝑖𝐹^ℓ(𝑥)−𝐹𝐹^ℓ(𝑥)≤sup_{𝜉∈𝐼𝜀}𝐹^′_𝑖(𝜉)⋅𝐹^ℓ_𝑖(𝑥)−𝐹^ℓ(𝑥)+sup_{𝜉∈𝐼𝜀}𝐹_𝑖−𝐹^′(𝜉)⋅𝐹^ℓ(𝑥𝐹)≤𝜇^ℓ_𝑖(𝑥)−𝐹^ℓ(𝑥)+𝐿𝜆^ℓ𝐹_𝑖−𝐹₁≤𝜇1+𝐿𝜆𝜇𝜇−𝜆^ℓ−1−𝐿𝜆𝜇−𝜆^ℓ×𝐹_𝑖−𝐹₁+𝐿𝜆^ℓ𝐹_𝑖−𝐹₁≤1+𝐿𝜆𝜇𝜇−𝜆^ℓ−𝐿𝜆𝜇−𝜆^ℓ+1𝐹_𝑖−𝐹₁ (18) for all 𝑖≥𝑁₀ and for all 𝑥∈𝐼_𝜀. Thus, we can obtain (10) by induction. This completes the proof.

In Lemma 1 we gave a better estimate for 𝐹^𝑛 and 𝐹^𝑛_𝑖 and their derivatives than that given in [25, Lemma 2.1]. In Lemma 1 the growth rate on 𝑛 is much lower in the sense that the constant 𝐿𝜆^𝑛 given in (9) tends to 0 as 𝑛→+∞ faster than the constant 𝜇^𝑛₀  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 𝑘,𝑟≥2, let 𝐹∈ℋ₋^𝑟+1(𝜆) with some 𝜆∈(0,1) and let (𝐹_𝑖) be a sequence of functions in ℋ₋^𝑟+1(𝜆). If lim_𝑖→+∞𝐹_𝑖−𝐹_𝑟+1𝐹=0,_(𝑚)(0)=𝐹_𝑖^(𝑚)(0),∀𝑚=2,…,𝑟, (19) then lim_𝑖→+∞𝑓_𝑖−𝑓_𝑟=0, (20) where 𝑓 and 𝑓_𝑖 are unique 𝑘th order 𝐶^𝑟 iterative roots of 𝐹 and 𝐹_𝑖, respectively, defined on 𝐼.

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

Lemma 3. Given an integer 𝑟≥2, let 𝐹∈ℋ₋^𝑟+1(𝜆) with some 𝜆∈(0,1) and let (𝐹_𝑖) be a sequence of functions in ℋ₋^𝑟+1(𝜆) satisfying condition (19). Then lim_𝑖→+∞𝜑_𝑖−𝜑_𝑟=0, (21) where 𝜑∶𝐼→ℝ and 𝜑_𝑖∶𝐼→ℝ are the principal solutions of Schröder’s equations 𝜑(𝐹(𝑥))=𝜆𝜑(𝑥) and 𝜑_𝑖(𝐹_𝑖(𝑥))=𝜆𝜑_𝑖(𝑥), respectively.

Proof. From (6) we can see that 𝜑^(𝑚)(𝑥)=lim_𝑛→+∞𝜆^−𝑛𝐹^𝑛(𝑚)(𝑥),𝜑_𝑖^(𝑚)(𝑥)=lim_𝑛→+∞𝜆^−𝑛𝐹^𝑛_𝑖^(𝑚)(𝑥) (22) for all 𝑚∈[0,𝑟]∩ℤ and 𝑥∈𝐼. In what follows we intend to discuss our results in a sufficiently small interval 𝐼_𝜀 first and extend them to the whole interval 𝐼, where 𝜀 is given in Lemma 1.
In order to prove the convergence of the sequence (𝜑_𝑖) in 𝐼_𝜀, we claim that there exists a constant 𝑀_𝑚, which is independent of 𝑛, such that 𝜆^−𝑛𝐹^𝑛_𝑖^(𝑚)(𝑥)−𝐹^𝑛(𝑚)(𝑥)≤𝑀_𝑚𝐹_𝑖−𝐹_𝑟+1,∀𝑥∈𝐼_𝜀, (23) where 𝑚=0,1,…,𝑟. If (23) is true, then we get 𝜑_𝑖^(𝑚)(𝑥)−𝜑^(𝑚)(𝑥)=lim_𝑛→+∞𝜆^−𝑛𝐹^𝑛_𝑖^(𝑚)(𝑥)−𝐹^𝑛(𝑚)(𝑥)≤𝑀_𝑚𝐹_𝑖−𝐹_𝑟+1,∀𝑥∈𝐼_𝜀, (24) implying the stability in 𝐼_𝜀. Next, we extend the result (24) from 𝐼_𝜀 to the whole interval 𝐼. As indicated in [25], we have lim_𝑖→+∞𝐹^𝑛_𝑖−𝐹^𝑛_𝑟+1=0,∀𝑛∈ℕ, (25) 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 𝐹 in 𝐼 and by (8), there is an integer 𝑁∈ℕ such that 𝐹^𝑁(𝑥),𝐹^𝑁_𝑖(𝑥)∈𝐼_𝜀 for all 𝑖∈ℕ and 𝑥∈𝐼. Then, according to Schröder’s equation, we can obtain the formulae 𝜑(𝑥)=𝜆^−𝑁𝜑𝐹^𝑁(𝑥),𝜑_𝑖(𝑥)=𝜆^−𝑁𝜑_𝑖𝐹^𝑁_𝑖(𝑥),∀𝑥∈𝐼, (26) where 𝜑∶=𝜑|_𝐼𝜀 and 𝜑_𝑖∶=𝜑_𝑖|_𝐼𝜀. Then by Lemma 1 and from (24), (26), and the uniform continuity of 𝜑^(𝑚), we get lim_𝑖→+∞‖𝜑_𝑖−𝜑‖₀=0 and 𝜑_𝑖^(𝑚)(𝑥)−𝜑^(𝑚)(𝑥)=𝜆^−𝑁_{(𝑚)𝜔∈Ω}𝑐_𝜔𝜑_𝑖^(𝑝)𝐹^𝑁_𝑖(𝑥)^𝑝_𝑞=1𝐹^𝑁_𝑖^(𝑗𝑞)(𝑥)−_{(𝑚)𝜔∈Ω}𝑐_𝜔𝜑^(𝑝)𝐹^𝑁(𝑥)^𝑝_𝑞=1𝐹^{𝑁(𝑗𝑞)}(𝑥)≤𝜆^−𝑁_{(𝑚)𝜔∈Ω}𝑐_𝜔𝜑_𝑖^(𝑝)𝐹^𝑁_𝑖(𝑥)^𝑝_𝑞=1𝐹^𝑁_𝑖^(𝑗𝑞)(𝑥)−𝜑^(𝑝)𝐹^𝑁(𝑥)^𝑝_𝑞=1𝐹^{𝑁(𝑗𝑞)}(𝑥)≤𝜆^−𝑁_{(𝑚)𝜔∈Ω}𝑐_𝜔𝜑_𝑖^(𝑝)𝐹^𝑁_𝑖(𝑥)−𝜑^(𝑝)𝐹^𝑁(𝑥)×^𝑝_𝑞=1𝐹^𝑁_𝑖^(𝑗𝑞)(𝑥)+𝜑^(𝑝)𝐹^𝑁(𝑥)×^𝑝_𝑞=1𝐹^𝑁_𝑖^(𝑗𝑞)(𝑥)−^𝑝_𝑞=1𝐹^{𝑁(𝑗𝑞)}(𝑥)≤𝜆^−𝑁_{(𝑚)𝜔∈Ω}𝑐_𝜔𝐿^𝑝𝜆^𝑝𝑁×𝜑_𝑖^(𝑝)𝐹^𝑁_𝑖(𝑥)−𝜑^(𝑝)𝐹^𝑁_𝑖(𝑥)+𝜑^(𝑝)𝐹^𝑁_𝑖(𝑥)−𝜑^(𝑝)𝐹^𝑁(𝑥)+𝐿^𝑝−1𝜆^{(𝑝−1)𝑁}𝜑^(𝑝)𝐹^𝑁(𝑥)×𝐹^𝑁_𝑖−𝐹^𝑁_𝑟+1≤𝜆^−𝑁_{(𝑚)𝜔∈Ω}𝑐_𝜔𝐿^𝑝𝜆^𝑝𝑁×𝑀_𝑝𝐹_𝑖−𝐹_𝑟+1+𝜑^(𝑝)𝐹^𝑁_𝑖(𝑥)−𝜑^(𝑝)𝐹^𝑁(𝑥)+𝐿^𝑝−1𝜆^{(𝑝−1)𝑁}𝜑^(𝑝)𝐹^𝑁(𝑥)×𝐹^𝑁_𝑖−𝐹^𝑁_𝑟+1⟶0 (27) as 𝑖→+∞ for all 𝑚=1,…,𝑟. Hence, (21) is proved and the proof is completed.
In the following, we will prove the claimed (23) by induction on 𝑚. Clearly, (2.11) in [25] and what is indicated above the proof of Theorem 2.1 in [25] show that (23) holds for 𝑚=1 and 𝑚=0, respectively. Then we suppose that (23) is also satisfied for all 𝑚∈[1,ℓ], where 2≤ℓ<𝑟, and we will prove (23) for 𝑚=ℓ+1. Our strategy is to prove that, for given 𝑠∈ℕ, there exists 𝑀_ℓ+1(𝑠) such that 𝜆^−𝑠𝐹^𝑠_𝑖^(ℓ+1)(𝑥)−𝐹^𝑠(ℓ+1)(𝑥)≤𝑀_ℓ+1(𝑠)𝐹_𝑖−𝐹_𝑟+1 (28) and (𝑀_ℓ+1(𝑠)) is a bounded sequence whose upper bound satisfies (23). Clearly, for 𝑠=0 we can find constant 𝑀_ℓ+1(0) satisfying (28). Then, assume that there exists 𝑀_ℓ+1(𝑠) such that (28) holds for any 𝑠=0,…,𝑛−1. Applying Lemma 1 and the inductive hypothesis, we have 𝜆^−𝑛𝐹^𝑛_𝑖^(ℓ+1)(𝑥)−𝐹^𝑛(ℓ+1)(𝑥)=𝜆^−𝑛𝐹^′_𝑖𝐹_𝑖^𝑛−1(𝑥)𝐹_𝑖^{𝑛−1(ℓ+1)}(𝑥)+_{𝑝≥2𝜔∈Ω(ℓ+1)}𝑐_𝜔𝐹_𝑖^(𝑝)𝐹_𝑖^𝑛−1(𝑥)^𝑝_𝑞=1𝐹_𝑖^{𝑛−1(𝑗𝑞)}(𝑥)−𝐹^′𝐹^𝑛−1(𝑥)𝐹^{𝑛−1(ℓ+1)}(𝑥)−_{𝑝≥2𝜔∈Ω(ℓ+1)}𝑐_𝜔𝐹^(𝑝)𝐹^𝑛−1(𝑥)^𝑝_𝑞=1𝐹^{𝑛−1(𝑗𝑞)}(𝑥)≤𝜆^−𝑛𝐹^′_𝑖𝐹_𝑖^𝑛−1(𝑥)𝐹_𝑖^{𝑛−1(ℓ+1)}(𝑥)−𝐹^′𝐹^𝑛−1(𝑥)𝐹^{𝑛−1(ℓ+1)}(𝑥+_{()𝑝≥2𝜔∈Ωℓ+1})𝑐_𝜔𝐹_𝑖^(𝑝)𝐹_𝑖^𝑛−1(𝑥)^𝑝_𝑞=1𝐹_𝑖^{𝑛−1(𝑗𝑞)}(𝑥)−𝐹^(𝑝)𝐹^𝑛−1(𝑥)^𝑝_𝑞=1𝐹^{𝑛−1(𝑗𝑞)}(𝑥)≤𝜆⁻¹𝑀_ℓ+1(𝑛−1)𝐹^′_𝑖𝐹_𝑖^𝑛−1(𝑥)𝐹_𝑖−𝐹_𝑟+1+𝐿𝜆⁻¹𝐴_1,𝑛−1(𝑥)+_{𝑝≥2𝜔∈Ω(ℓ+1)}𝑐_𝜔𝐿^𝑝𝜆^{(𝑝−1)𝑛−𝑝}𝐴_{𝑝,𝑛−1}(𝑥)+^𝑝_𝑞=1𝑀_𝑗𝑞𝐿^𝑝−1𝜆^{(𝑝−1)𝑛−𝑝}×𝐹^(𝑝)𝐹^𝑛−1(𝑥)𝐹_𝑖−𝐹_𝑟+1, (29) where 𝐴_𝑝,𝑛(𝑥)∶=|𝐹_𝑖^(𝑝)(𝐹^𝑛_𝑖(𝑥))−𝐹^(𝑝)(𝐹^𝑛(𝑥))|. By Lemma 1 and condition (19), we obtain that |𝐹^′_𝑖(𝐹_𝑖^𝑛−1(𝑥))|≤|𝐹^′_𝑖(0)|+|𝐹^′_𝑖(𝐹_𝑖^𝑛−1(𝑥))−𝐹^′_𝑖(0)|≤𝜆+𝐿𝜂𝜆^𝑛−1, 𝐹^(𝑝)𝐹^𝑛−1(𝑥)≤𝐹^(𝑝)(0)+𝐹^(𝑝)𝐹^𝑛−1(𝑥)−𝐹^(𝑝)(0)≤𝜂+𝐿𝜂𝜆^𝑛−1,𝐴_𝑝,𝑛(𝑥)≤𝐹_𝑖^(𝑝)𝐹^𝑛_𝑖(𝑥)−𝐹^(𝑝)𝐹^𝑛_𝑖(𝑥)−𝐹_𝑖^(𝑝)(0)+𝐹^(𝑝)(0+𝐹_(𝑝))𝐹^𝑛_𝑖(𝑥)−𝐹^(𝑝)𝐹^𝑛(𝑥)≤𝐿𝜆^𝑛𝐹_𝑖−𝐹_𝑟+1+𝜂𝐹^𝑛_𝑖(𝑥)−𝐹^𝑛(𝑥)≤𝐿𝜆^𝑛𝐹_𝑖−𝐹_𝑟+1+𝜂1+𝐿𝜆𝜇𝜇−𝜆^𝑛−1−𝐿𝜆𝜇−𝜆^𝑛×𝐹_𝑖−𝐹₁=1+𝐿𝜆𝜇−𝜆𝜂𝜇^𝑛−1+𝜂1−𝜇−𝜆𝐿𝜆^𝑛𝐹_𝑖−𝐹_𝑟+1, (30) where 𝜂∶=2‖𝐹‖_𝑟+1 and 𝜇 is given in Lemma 1, for all 𝑥∈𝐼_𝜀, 0<𝑝≤𝑟, 𝑖≥𝑁₀, and 𝑛∈ℕ. It follows that 𝜆^−𝑛𝐹^𝑛_𝑖^(ℓ+1)(𝑥)−𝐹^𝑛(ℓ+1)(𝑥≤)1+𝐿𝜂𝜆^𝑛−2𝑀_ℓ+1(𝑛−1)+1+𝐿𝜆𝜇−𝜆𝐿𝜂𝜇^𝑛−2𝜆⁻¹+𝜂1−𝐿𝜇−𝜆²𝜆^𝑛−2𝐹_𝑖−𝐹_𝑟+1+_{𝑝≥2𝜔∈Ω(ℓ+1)}𝑐_𝜔1+𝐿𝜆𝐿𝜇−𝜆^𝑝𝜂𝜇^𝑛−2𝜆^{(𝑝−1)𝑛−𝑝}+𝜂1−𝐿𝜇−𝜆^𝑝+1𝜆^{𝑝𝑛−𝑝−1}+1+𝐿𝜆^𝑛−1𝐿^𝑝−1𝜂𝜆^{𝑝(𝑝−1)𝑛−𝑝}_𝑞=1𝑀_𝑗𝑞×𝐹_𝑖−𝐹_𝑟+1≤1+𝐿𝜂𝜆^𝑛−2𝑀_ℓ+1(𝑛−1)+1+𝐿𝜆𝜇−𝜆𝐿𝜂𝜇^𝑛−2𝜆⁻¹+𝜂1−𝐿𝜇−𝜆²𝜇^𝑛−2+_{𝑝≥2𝜔∈Ω(ℓ+1)}𝑐_𝜔1+𝐿𝜆𝐿𝜇−𝜆^𝑝𝜂𝜇^2𝑛−4+𝜂1−𝐿𝜇−𝜆^𝑝+1𝜇^2𝑛−3+(1+𝐿)𝐿^𝑝−1𝜂𝜇^𝑝𝑛−2_𝑞=1𝑀_𝑗𝑞×𝐹_𝑖−𝐹_𝑟+1≤1+𝐿𝜂𝜇^𝑛−2𝑀_ℓ+1(𝑛−1)+Γ𝜇^𝑛𝐹_𝑖−𝐹_𝑟+1 (31) because 𝜆<𝜇∈(0,1), where Γ∶=𝜇⁴1+𝐿𝜆𝜇−𝜆𝐿𝜂𝜆⁻¹+𝜂1−𝐿𝜇−𝜆²+_{()𝑝≥2𝜔∈Ωℓ+1}𝑐_𝜔1+𝐿𝜆𝐿𝜇−𝜆^𝑝𝜂+𝜂1−𝐿𝜇−𝜆^𝑝+1+(1+𝐿)𝐿^𝑝−1𝜂^𝑝_𝑞=1𝑀_𝑗𝑞. (32) Let 𝑀_ℓ+1(𝑛)∶=1+𝐿𝜂𝜇^𝑛−2𝑀_ℓ+1(𝑛−1)+Γ𝜇^𝑛=^𝑛−2_𝑗=11+𝐿𝜂𝜇^𝑗𝑀_ℓ+1(2)+Γ^𝑛_𝑗=3𝜇^𝑗𝑛−2_{𝑡=𝑗−1}1+𝐿𝜂𝜇^𝑡. (33) It is easy to check that 𝑀_ℓ+1(𝑛)≤^+∞_𝑗=11+𝐿𝜂𝜇^𝑗𝑀_ℓ+1(2)+Γ^+∞_𝑗=3𝜇^𝑗+∞_𝑡=01+𝐿𝜂𝜇^𝑡 (34) for all 𝑛∈ℕ. By putting 𝑀_ℓ+1∶=∏^+∞_𝑗=1(1+𝐿𝜂𝜇^𝑗)𝑀_ℓ+1(2)+Γ∑^+∞_𝑗=3𝜇^𝑗∏^+∞_𝑡=0(1+𝐿𝜂𝜇^𝑡), we can prove (23) for 𝑚=ℓ+1 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 𝐶^𝑟 iterative roots 𝑓 and 𝑓_𝑖 for each 𝑖∈ℕ can be presented by 𝑓(𝑥)=𝜑⁻¹𝜆^1/𝑘𝜑(𝑥),𝑓_𝑖(𝑥)=𝜑_𝑖⁻¹𝜆^1/𝑘𝜑_𝑖(𝑥),∀𝑥∈𝐼, (35) respectively. In order to prove the convergence of (𝑓_𝑖) in 𝐼, note that, for sufficiently large 𝑖∈ℕ such that 𝜆^1/𝑘𝜑_𝑖(𝑥)∈𝜑(𝐼) for all 𝑥∈𝐼, we have 𝜑_𝑖^−1(𝑝)𝜆^1/𝑘𝜑_𝑖(𝑥)^𝑝_𝑞=1𝜑^(𝑗𝑞)_𝑖(𝑥)−𝜑^−1(𝑝)𝜆^1/𝑘𝜑(𝑥)^𝑝_𝑞=1𝜑^(𝑗𝑞)(𝑥)≤𝜑_𝑖^−1(𝑝)𝜆^1/𝑘𝜑_𝑖(𝑥)−𝜑^−1(𝑝)𝜆^1/𝑘𝜑(𝑥)×^𝑝_𝑞=1𝜑^(𝑗𝑞)_𝑖(𝑥)+𝜑^−1(𝑝)𝜆^1/𝑘𝜑(𝑥)×^𝑝_𝑞=1𝜑^(𝑗𝑞)_𝑖(𝑥)−^𝑝_𝑞=1𝜑^(𝑗𝑞)(𝑥)≤2^𝑝𝜑_𝑟^𝑝𝜑_𝑖⁻¹−𝜑⁻¹_𝑟+𝜑^−1(𝑝)𝜆^1/𝑘𝜑_𝑖(𝑥)−𝜑^−1(𝑝)𝜆^1/𝑘𝜑(𝑥)+𝑝2^𝑝−1𝜑_𝑟^𝑝−1𝜑⁻¹_𝑟𝜑_𝑖−𝜑_𝑟⟶0 (36) as 𝑖→+∞ for all integers 𝑝∈[1,𝑟] by (21) and the fact that the inverse operator is continuous and (𝜑⁻¹)^(𝑝) is uniformly continuous. Hence, lim_𝑖→+∞‖𝑓_𝑖−𝑓‖₀=0 and 𝑓_𝑖^(𝑚)(𝑥)−𝑓^(𝑚)(𝑥=_{(𝑚)𝜔∈Ω})𝑐_𝜔𝜆^𝑝/𝑘𝜑_𝑖^−1(𝑝)𝜆^1/𝑘𝜑_𝑖(𝑥)^𝑝_𝑞=1𝜑^(𝑗𝑞)_𝑖(𝑥)−_{𝜔∈Ω(𝑚)}𝑐_𝜔𝜆^𝑝/𝑘𝜑^−1(𝑝)𝜆^1/𝑘𝜑(𝑥)^𝑝_𝑞=1𝜑^(𝑗𝑞)(𝑥)≤_{(𝑚)𝜔∈Ω}𝑐_𝜔𝜆^𝑝/𝑘𝜑_𝑖^−1(𝑝)𝜆^1/𝑘𝜑_𝑖(𝑥)^𝑝_𝑞=1𝜑^(𝑗𝑞)_𝑖(𝑥)−𝜑^−1(𝑝)𝜆^1/𝑘𝜑(𝑥)^𝑝_𝑞=1𝜑^(𝑗𝑞)(𝑥)⟶0 (37) as 𝑖→+∞ for all 𝑚=1,…,𝑟, which implies that lim_𝑖→+∞‖𝑓_𝑖−𝑓‖_𝑟=0 and completes the proof.

Theorem 2 is also valid for 𝑟=1, which is the same as Theorem 2.1 in [25] for 𝑟=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 𝑟 parallel. In fact, we cannot use those estimates to give a uniform constant Γ with respect to 𝑛, the order of iteration, in (33). Using those estimates, corresponding to Γ given in (32), we obtain the quantity Γ^∗(𝑛)∶=𝜇⁴₀𝐿²+(𝑛−1)𝐿𝜂𝜆⁻¹+_{()𝑝≥2𝜔∈Ωℓ+1}𝑐_𝜔𝐿^𝑝+1+(𝑛−1)𝐿^𝑝𝜂+(1+𝐿)𝐿^𝑝−1𝜂^𝑝_𝑞=1𝑀_𝑗𝑞, (38) which tends to +∞ as 𝑛→+∞. For this reason it is hard to prove the boundedness of 𝑀_ℓ+1(𝑛). As remarked after the proof of Lemma 1, our estimation in (9) and (10) enables us to give the boundedness of 𝑀_ℓ+1(𝑛) 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.