The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum's Sense

Abstract

We show that the composition operator 𝐻, associated with ℎ∶[𝑎,𝑏]→ℝ, maps the spaces Lip[𝑎,𝑏] on to the space 𝜅BV_𝜙[𝑎,𝑏] of functions of bounded variation in Schramm-Korenblum's sense if and only if ℎ is locally Lipschitz. Also, verify that if the composition operator generated by ℎ∶[𝑎,𝑏]×ℝ→ℝ maps this space into itself and is uniformly bounded, then regularization of ℎ is affine in the second variable.

1. Introduction

The composition operator problem (or COP, for short) refers to determining the conditions on a function ℎ∶ℝ→ℝ, such that the composition operator, associated with the function ℎ, maps a space 𝕏 of functions 𝑢∶[𝑎,𝑏]→ℝ into itself [1, 2]. There are several spaces where the COP has been resolved. For example, in 1961, Babaev [3] showed that the composition operator 𝐻, associated with the function ℎ∶ℝ→ℝ, maps the space Lip[𝑎,𝑏] of the Lipschitz functions into itself if and only if ℎ is locally Lipschitz; in 1967, Mukhtarov [4] obtained the same result for the space Lip_𝛼[𝑎,𝑏] of the Hölder functions of order 𝛼(0<𝛼<1).

The first work on the COP in the space of functions of bounded variation BV[𝑎,𝑏] was made by Josephy in 1981 [5]. In 1986, Ciemnoczołowski and Orlicz [6] got the same result for the space of the functions of bounded 𝜑-variation in Wiener’s sense. In 1974, Chaika and Waterman [7] reached a similar result for the space of functions of bounded harmonic variation HBV[𝑎,𝑏]. In the years 1991 and 1995 Merentes showed a similar result for the spaces of absolutely continuous functions AC[𝑎,𝑏] and the space of function of bounded 𝜑-variation in Riesz’s sense RV_𝜑[𝑎,𝑏] (see [8, 9]), and in 1998 Merentes and Rivas achieved the same result when the composition operator maps the space RV_𝑝[𝑎,𝑏] of the functions of bounded 𝑝-variation in Riesz’s sense (1<𝑝<∞) into the space BV[𝑎,𝑏] [10]. In 2003, Pierce and Waterman solved the COP for the spaces 𝜙BV[𝑎,𝑏] and ΛBV[𝑎,𝑏] [11]. More recently, in 2011, Appell et al. [1] conclude the same results verifying when the composition operator maps Lip[𝑎,𝑏] into BV[𝑎,𝑏]. Finally, Appell and Merentes verify the same result for the space of functions of bounded 𝜅-variation [12].

There exist spaces 𝕏 of real functions defined on an interval [𝑎,𝑏], such that 𝐻 maps 𝕏 into itself and ℎ is not locally Lipschitz. For example, in the case the space of continuous functions 𝐶[𝑎,𝑏] it follows from the Tietze-Urysohn theorem that the composition operator 𝐻 acts from 𝐶[𝑎,𝑏] into itself and the function ℎ must be continuous; that is, ℎ does not need to be Lipschitz. A similar result was obtained in the space of regulated functions [13].

A first objective of this work is to demonstrate that the composition operator, associated with the function ℎ, maps the space Lip[𝑎,𝑏] of the Lipschitz functions into the space 𝜅BV_𝜙[𝑎,𝑏] of functions of bounded variation in Schramm-Korenblum’s sense or into the space 𝜅BV[𝑎,𝑏] of functions of bounded variation in Korenblum’s sense if and only if ℎ is locally Lipschitz. We also extend this result to function spaces 𝕏,𝕐, such that Lip[𝑎,𝑏]⊂𝕏⊂𝕐, where 𝕐⊂𝜅BV[𝑎,𝑏] or 𝕐⊂BV_𝜙[𝑎,𝑏].

In a seminal article of 1982, Matkowski [14] showed that if the composition operator 𝐻, associated with the function ℎ∶[𝑎,𝑏]×ℝ→ℝ, maps the space Lip[𝑎,𝑏] of the Lipschitzian functions into itself and is a globally Lipschitzian map, then the function ℎ has the form ℎ(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),𝑡∈[𝑎,𝑏],𝑥∈ℝ, (1) for some 𝛼,𝛽∈Lip[𝑎,𝑏].

There are a variety of spaces besides Lip[𝑎,𝑏] that verify this result [15]. The spaces of Banach (𝕏,‖⋅‖) that fulfill this property are said to satisfy the Matkowski property [1].

In 1984, Matkowski and Miś [16] considered the same hypotheses on the operator 𝐻 for the space BV[𝑎,𝑏] of the function of bounded variation and concluded that (1) is true for the regularization ℎ⁻ of the function ℎ with respect of the first variable; that is, ℎ⁻(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),𝑡∈[𝑎,𝑏],𝑥∈ℝ, (2) where 𝛼,𝛽∈BV⁻[𝑎,𝑏]. Spaces that satisfy this conditions said to be verified Weak Matkowski Property [1].

A second objective of this paper is to show that if function ℎ(𝑡,⋅) is continuous in the second variable, for each 𝑡∈[𝑎,𝑏], and the composition operator 𝐻, associated with the function ℎ, is uniformly bounded, then ℎ satisfies (2).

2. Preliminaries

Let [𝑎,𝑏] be a closed interval of the real line ℝ(𝑎,𝑏∈ℝ,𝑎<𝑏). From now on, for a function 𝑢∶[𝑎,𝑏]→ℝ denote by 𝐿^𝑏_𝑎(𝑢) the Lipschitz constant of 𝑢; that is, 𝐿^𝑏_𝑎(𝑢)|𝑢(𝑡)(𝑠=sup−𝑢)|||[].𝑠−𝑡∶𝑡,𝑠∈𝑎,𝑏,𝑠≠𝑡 (3)

By Lip[𝑎,𝑏]={𝑢∶[𝑎,𝑏]→ℝ∶𝐿^𝑏_𝑎(𝑢)<∞} we will denote the space of the Lipschitz functions. It is well known that the space Lip[𝑎,𝑏] is a Banach space endowed with the norm ‖𝑢‖_Lip∶=|𝑢(𝑎)|+𝐿^𝑏_𝑎(𝑢)𝑢∈Lip[𝑎,𝑏]. (4)

Ever since the notion of a function of bounded variation appeared, it has led to an incredible number of generalizations. In 1881, Jordan [17] introduced the definition of function of bounded variation for a function 𝑢∶[𝑎,𝑏]→ℝ and showed that these kinds of function can be decomposed as the difference of two monotone functions. As a consequence of this result we have that those functions satisfy the Dirichlet criterion, that is, the functions that have pointwise convergent Fourier series.

Jordan defined such functions in the following way.

Definition 1. Let 𝑢∶[𝑎,𝑏]→ℝ and 𝜋∶𝑎≤𝑡₁<⋯<𝑡_𝑛≤𝑏 be a partition of the interval [𝑎,𝑏]. Consider 𝜎(𝑢,𝜋)∶=^𝑛−1_𝑗=1𝑢𝑡_𝑗+1−𝑢𝑡_𝑗,𝑉(𝑢;[𝑎,𝑏])=sup_𝜋𝜎(𝑢,𝜋), (5) where the supremum is taken over all partitions 𝜋 of the interval [𝑎,𝑏].
If 𝑉(𝑢;[𝑎,𝑏])<∞, then 𝑢 has bounded variation on the interval [𝑎,𝑏] and this number is called the variation in Jordan’s sense on [𝑎,𝑏]. This space of function is denoted by BV[𝑎,𝑏].

The concept of bounded variation has been the subject of intensive research, and many applications, generalizations, and improvements of them can be found in the literature (see for instance [18–20]). Some generalizations have been introduced by De La Vallée Poussin, F. Riesz, N. Wiener, L. C. Young, Yu. T. Medvedev, D. Waterman, B. Korenblum and M. Schramm.

In 1975, Korenblum in [21] considered a new kind of variation, called 𝜅-variation, introducing a function 𝜅 for distorting the expression |𝑡_𝑗−𝑡_𝑗−1| in the partition itself rather than the expression |𝑓(𝑡_𝑗)−𝑓(𝑡_𝑗−1)| in the range. Subsequently, this class of functions has been studied in detail by Cyphert and Kelingos [22]. One advantage of this alternate approach is that a function of bounded 𝜅-variation may be decomposed into the difference of two simpler functions called 𝜅-decreasing functions (for the precise definition see the following).

Definition 2. A function 𝜅∶[0,1]→[0,1] is said to be a 𝜅-function or distortion function if it satisfies the following properties:(1)𝜅is continuous with 𝜅(0)=0 and 𝜅(1)=1, (2)𝜅 is concave, increasing, and (3)lim_𝑡→0⁺(𝜅(𝑡)/𝑡)=∞.

Simple examples of distortion functions are 𝜅(𝑡)=𝑡^𝛼(0<𝛼<1),𝜅(𝑡)=𝑡1−log𝑡. (6)

From Definition 2 we can see that 𝜅 is subadditive; that is, 𝜅(𝑠+𝑡)≤𝜅(𝑠)+𝜅(𝑡)(0≤𝑠,𝑡≤1), (7) and since lim_𝑡→0⁺(𝜅(𝑡)/𝑡)=∞, then without loss of generality we can assume that 𝜅(𝑡)≥𝑡(𝑡∈[0,1]). (8)

Furthermore Korenblum introduces the following concept of variation.

Definition 3. Let 𝜅∶[0,1]→[0,1] be a distortion function and 𝑢∶[𝑎,𝑏]→ℝ and 𝜋∶𝑎≤𝑡₁<⋯<𝑡_𝑛≤𝑏 a partition of the interval [𝑎,𝑏]. Consider 𝜅()∑𝑢,𝜋∶=^{𝑛−1𝑗=1}𝑢𝑡_𝑗+1−𝑢𝑡_𝑗∑^𝑛−1_𝑗=1𝜅𝑡_𝑗+1−𝑡_𝑗/(,(𝑢)([𝑏−𝑎)𝜅𝑉=𝜅𝑉𝑢;𝑎,𝑏])∶=sup_𝜋𝜅(𝑢,𝜋), (9) where the supremum is taken over all partitions 𝜋 of the interval [𝑎,𝑏], called the 𝜅-variation of 𝑢 on [𝑎,𝑏]. In the case, 𝜅𝑉(𝑢;[𝑎,𝑏])<∞ we say that 𝑢 has 𝜅-variation on [𝑎,𝑏] and we will denote by 𝜅BV[𝑎,𝑏] the space of functions of 𝜅-variation on [𝑎,𝑏].

In the case that [𝑎,𝑏]=[0,1] the equality (9) becomes 𝜅()∑𝑢,𝜋∶=^{𝑛−1𝑗=1}𝑢𝑡_𝑗+1−𝑢𝑡_𝑗∑^𝑛−1_𝑗=1𝜅𝑡_𝑗+1−𝑡_𝑗. (10)

Some properties of the functions with bounded 𝜅-variation are summarized in the following theorem.

Theorem 4 (see [22]). Let 𝜅∶[0,1]→[0,1] be a distortion function. Then(1)𝜅BV[𝑎,𝑏] is a Banach space endowed with the norm ‖𝑢‖_𝜅=|𝑢(𝑎)|+𝜅𝑉(𝑢;[𝑎,𝑏])(𝑢∈𝜅BV[𝑎,𝑏]). (11)(2)If the function 𝑢 is monotone, then 𝜅𝑉(𝑢;[𝑎,𝑏])=|𝑢(𝑏)−𝑢(𝑎)|. (3)If 𝑢∈𝜅BV[𝑎,𝑏], then 𝑢 is bounded and ‖𝑢‖_∞≤2‖𝑢‖_𝜅.(4)Lip[𝑎,𝑏]⊂BV[𝑎,𝑏]⊂𝜅BV[𝑎,𝑏]⊂𝑅[𝑎,𝑏], where 𝑅[𝑎,𝑏] denotes the space of regulated functions. (5)If 𝑢∈𝜅BV[𝑎,𝑏], then 𝑢 can be decomposed as difference of two 𝜅-decreasing functions, that is, there exist functions 𝑣∶[𝑎,𝑏]→ℝ such that sup_𝑠<𝑡((𝑣(𝑡)−𝑣(𝑠))/𝜅((𝑡−𝑠)/(𝑏−𝑎)))<∞.

It is easy to show that if lim_𝑡→0(𝑘(𝑡)/𝑡) is finite, then 𝜅BV[𝑎,𝑏]=BV[𝑎,𝑏]. In [1] it is shown that inclusions (4) of Theorem 4 are strict.

Throughout this paper a 𝜑-function 𝜑 is a continuous increasing function 𝜑∶[0,∞)→ℝ, such that 𝜑(0)=0 and lim_𝑡→∞𝜑(𝑡)=∞.

A 𝜙-sequence is a sequence of decreasing 𝜙={𝜑_𝑛}_𝑛≥1 of convex 𝜑-function that satisfies ∑_𝑛≥1𝜑_𝑛(𝑡) diverge for 𝑡>0.

We denote by 𝐹_ℕ[𝑎,𝑏] the collection of finite or numerable family of nonoverlapping interval {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1, such that ⋃_𝑛≥1[𝑎_𝑛,𝑏_𝑛]=[𝑎,𝑏].

In 1985, Schramm [23] introduced a new concept of variation as follows.

Definition 5. Let 𝜙={𝜑_𝑛}_𝑛≥1 be a 𝜙-sequence, {𝐼_𝑛=[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏], and 𝑢∶[𝑎,𝑏]→ℝ. We define 𝜎_𝜙𝑢,𝐼_𝑛∶=^∞_𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛,𝑉_𝜙(𝑢)=𝑉_𝜙(𝑢;[𝑎,𝑏])∶=sup_{𝐼𝑛∈𝐹ℕ[]𝑎,𝑏}𝜎_𝜙𝑢,𝐼_𝑛. (12) If 𝑉_𝜙(𝑢;[𝑎,𝑏])<∞, we say that 𝑢 has bounded 𝜙-variation in the interval [𝑎,𝑏] and this number denotes the 𝜙-variation of 𝑢 in Schramm’s sense in [𝑎,𝑏]. The class of functions that have bounded 𝜙-variation in the interval [𝑎,𝑏] is denoted by 𝑉_𝜙[𝑎,𝑏]. The vectorial space generated by this class is denoted by BV_𝜙[𝑎,𝑏].

The next lemma is useful for building the space generated by several classes of functions.

Lemma 6. Let 𝕏 be a vector space and 𝐴⊂𝕏 a nonempty and symmetric set. Then (1)0∈𝐴. (2)The vector space generated for 𝐴 is equal to ⟨𝐴⟩={𝑥∈𝕏∶∃𝜆>0𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜆𝑥∈𝐴}=_𝜆>0𝜆𝐴. (13)

Some properties of functions of bounded 𝜙-variation in Schramm’s sense are given in the following theorem.

Theorem 7 (see [23]). Let 𝜙={𝜑_𝑛}_𝑛≥1 be 𝜙-sequence then (1)BV_𝜙[𝑎,𝑏] is a Banach space endowed with the norm ‖𝑢‖_𝜙∶=|𝑢(𝑎)|+𝜇(𝑢)𝑢∈BV_𝜙[𝑎,𝑏], (14) where 𝜇(𝑢)∶=inf_𝜆>0{𝜆>0∶𝑉_𝜙(𝑢/𝜆)≤1}.(2) If 𝑢 is monotone, then 𝑉_𝜙(𝑢)=𝜑₁(|𝑢(𝑏)−𝑢(𝑎)|).(3)If 𝑢∈𝑉_𝜙[𝑎,𝑏], then 𝑢 is bounded and ‖𝑢‖_∞≤|𝑢(𝑎)|+2𝜑₁⁻¹(𝑉_𝜙(𝑢)).(4) 𝑉_𝜙[𝑎,𝑏] is a symmetrical and convex set. (5)BV_𝜙[𝑎,𝑏]={𝑢∶[𝑎,𝑏]→ℝ∶∃𝜆>0𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑉_𝜙(𝜆𝑢)<∞}. (6) 𝑢∈𝑉_𝜙[𝑎,𝑏], then 𝑢 has lateral limits at each point of [𝑎,𝑏].

In 1986, S. K. Kim and J. Kim [24] combined the concepts of 𝜅-variation and 𝜙-variation introduced by Korenblum and Schramm to create the concept of 𝜅𝜙-variation or variation in Schramm-Korenblum’s sense.

Definition 8. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, {𝐼_𝑛=[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏], and 𝑢∶[𝑎,𝑏]→ℝ. We define 𝜅𝜎_𝜙𝑢,𝐼_𝑛∑∶=^∞𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛/(),𝑏−𝑎𝜅𝑉_𝜙(𝑢)=𝜅𝑉_𝜙(𝑢;[𝑎,𝑏])∶=sup_{𝐼𝑛∈𝐹ℕ[𝑎,𝑏]}𝜅𝜎_𝜙𝑢,𝐼_𝑛. (15) If 𝜅𝑉_𝜙(𝑢;[𝑎,𝑏])<∞, we say that 𝑢 has bounded 𝜅𝜙-variation in the interval [𝑎,𝑏] and this number denotes the 𝜅𝜙-variation of 𝑢 in Schramm-Korenblum’s sense in [𝑎,𝑏]. The class of functions that have bounded 𝜅𝜙-variation in the interval [𝑎,𝑏] is denoted by 𝜅𝑉_𝜙[𝑎,𝑏]. The vectorial space generated by this class is denoted by 𝜅BV_𝜙[𝑎,𝑏].

A particular case of 𝜙-sequence is when all the functions 𝜑_𝑛,𝑛∈ℕ are equal to a fixed 𝜑-function 𝜑. In this situation the class 𝜅𝑉_𝜑[𝑎,𝑏] is the class of the functions that have bounded 𝜅𝜑-variation in Wiener-Korenblum’s sense. This class of functions is denoted by 𝜅𝑉_𝜑[𝑎,𝑏] and the vectorial space generated by this class of function is denoted by 𝜅BV_𝜑[𝑎,𝑏].

Some properties of functions of bounded 𝜅𝜙-variation in Schramm-Korenblum’s sense are given in the following theorem.

Theorem 9. Let 𝜅∶[0,1]→[0,1] be a distortion function and 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, then (1) 𝜅BV_𝜙[𝑎,𝑏] is a Banach space endowed with the norm ‖𝑢‖_𝜅𝜙∶=|𝑢(𝑎)|+𝜇_𝜙(𝑢)𝑢∈𝜅BV_𝜙[𝑎,𝑏], (16) where 𝜇_𝜙(𝑢)∶=inf_𝜆>0{𝜆>0∶𝜅𝑉_𝜙(𝑢/𝜆)≤1}.(2) If 𝑢 is monotone, then 𝜅𝑉_𝜙(𝑢)=𝜑₁(|𝑢(𝑏)−𝑢(𝑎)|).(3) If 𝑢∈𝜅𝑉_𝜙[𝑎,𝑏], then 𝑢 is bounded and ‖𝑢‖_∞≤|𝑢(𝑎)|+2𝜑₁⁻¹(𝜅𝑉_𝜙(𝑢)).(4) 𝜅𝑉_𝜙[𝑎,𝑏] is a symmetrical and convex set. (5) 𝜅BV_𝜙[𝑎,𝑏]={𝑢∶[𝑎,𝑏]→ℝ∶∃𝜆>0such that𝜅𝑉_𝜙(𝜆𝑢)<∞}.(6) BV[𝑎,𝑏]⊂𝑉_𝜙[𝑎,𝑏]⊂𝜅𝑉_𝜙[𝑎,𝑏], and therefore BV[𝑎,𝑏]⊂BV_𝜙[𝑎,𝑏]⊂𝜅BV_𝜙[𝑎,𝑏]. (7) 𝜅BV[𝑎,𝑏]⊂𝜅BV_𝜙[𝑎,𝑏]. (8) 𝑢∈𝜅𝑉_𝜙[𝑎,𝑏], then 𝑢 has lateral limits at each point of [𝑎,𝑏].

Proof

Part (1). See [24].

Part (2). We take {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏], then since the functions 0<𝑡→𝜑_𝑛(𝑡)/𝑡 are increasing, we obtain ∑^∞_𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛/()=𝑏−𝑎_∞𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛×^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛𝑏−𝑎⁻¹≤^∞_𝑛=1𝜑_𝑛(|𝑢(𝑏)−𝑢(𝑎)|)|𝑢(𝑏)(𝑎𝑢𝑏−𝑢)|^𝑛−𝑢𝑎_𝑛×^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛𝑏−𝑎⁻¹≤𝜑₁(|𝑢(𝑏)−𝑢(𝑎)|)|𝑢(𝑏)−𝑢(𝑎)|^∞_𝑛=1𝑢𝑏_𝑛−𝑢𝑎_𝑛=𝜑₁(|𝑢(𝑏)−𝑢(𝑎)|), (17)

from which is obtained 𝜅𝑉_𝜙(𝑢)≤𝜑₁(|𝑢(𝑏)−𝑢(𝑎)|). (18)

From the Definition of 𝑘𝑉_𝜙(𝑢) we have the reciprocal inequality.

Part (3). We consider 𝑎<𝑡≤𝑏, then 𝜑₁(|𝑢(𝑡)−𝑢(𝑎)|)𝜅((𝑡−𝑎)/(𝑏−𝑎))+𝜅((𝑏−𝑡)/(𝑏−𝑎))≤𝜅𝑉_𝜙(𝑢). (19) Then 𝜑₁(|𝑢(𝑡)−𝑢(𝑎)|)≤2𝜅𝑉_𝜙(𝑢) and from this inequality we have the required relation.

Part (4). We get Part (4) by the convexity of functions 𝜑_𝑛,𝑛∈ℕ and the definition of 𝜅𝜙-variation.

Part (5). It follows from part (4) and Lemma 6.

Part (6). Let 𝑢∈BV[𝑎,𝑏] and consider {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏]. Define 𝐴∶=𝑛∈ℕ∶𝑢𝑏_𝑛−𝑢𝑎_𝑛≤1. (20)

Then ^∞_𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛≤𝜑₁(1)_𝑛∈𝐴𝑢𝑏_𝑛−𝑢𝑎_𝑛+_𝑛∉𝐴𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛. (21)

The last sum has at most [𝑉(𝑢)] terms, where [𝑥]=max{𝑛∶𝑛≤𝑥}. Because otherwise it has at least [𝑉(𝑢)]+1 summands.

Accordingly 𝑉(𝑢)≥_𝑛∉𝐴𝑢𝑏_𝑛−𝑢𝑎_𝑛≥[𝑉(𝑢)]+1. (22)

Which is a contradiction. Therefore ^∞_𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛≤𝜑₁(1)𝑉(𝑢)+_𝑛∉𝐴𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛≤𝜑₁(1)𝑉(𝑢)+𝜑₁2‖𝑢‖_∞[𝑉(𝑢)]. (23)

This concludes that BV[𝑎,𝑏]⊂𝑉_𝜙[𝑎,𝑏].

Let us show that 𝑉_𝜙[𝑎,𝑏]⊂𝜅𝑉_𝜙[𝑎,𝑏]. In Fact let 𝑢∈𝑉_𝜙[𝑎,𝑏] and {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏], then ^∞_𝑛=1𝜑_𝑛𝑢𝑏_𝑛−𝑢𝑎_𝑛≤𝑉_𝜙(𝑢)𝜅(1)≤𝑉_𝜙(𝑢)𝜅^∞_𝑛=1𝑏_𝑛−𝑎_𝑛.𝑏−𝑎 (24)

Hence, we get that 𝜅𝑉_𝜙(𝑢)≤𝑉_𝜙(𝑢).

Part (7). Let 𝑢∈𝜅BV[𝑎,𝑏], then by part (3) 𝑢 is bounded in [𝑎,𝑏]. Let us fix 𝜆>0, such that ‖𝜆𝑢‖_∞≤1/2. Let {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏], then from the convexity of the functions 𝜑_𝑛,𝑛∈ℕ, we have ∑^∞_𝑛=1𝜑_𝑛𝜆𝑢𝑏_𝑛−𝜆𝑢𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛/()≤∑𝑏−𝑎_∞𝑛=1𝜑_𝑛(1)𝜆𝑢𝑏_𝑛−𝑢𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛/(𝑏−𝑎)≤𝜑₁(1)𝜆𝜅𝑉(𝑢). (25)

Thus Lemma 6 concludes that 𝑢∈𝜅BV_𝜙[𝑎,𝑏].

Part (8). Suppose that there is 𝑡^∗∈(𝑎,𝑏] such that lim_{𝑡↑𝑡∗}𝑢(𝑡) does not exist.

By part (3) 𝑢 is bounded then 𝛼=lim_{𝑡↑𝑡∗}inf𝑢(𝑡)<lim_{𝑡↑𝑡∗}sup𝑢(𝑡)=𝛽,𝛼,𝛽∈ℝ. (26)

For each integer 𝑛 (large enough) we can choose 𝑡_𝑛,𝑡^′_𝑛 such that 𝑡^∗−1𝑛<𝑡_𝑛<𝑡^′_𝑛<𝑡^∗,𝑢𝑡_𝑛−𝑢𝑡^′_𝑛≥𝛽−𝛼. (27)

Using the definition of 𝜅𝜙-variation, we have 𝜑₁𝑢𝑡_𝑛−𝑢𝑡^′_𝑛×𝜅𝑡_𝑛−𝑎𝑡𝑏−𝑎+𝜅^′𝑛−𝑡_𝑛𝑏−𝑎+𝜅𝑏−𝑡^′_𝑛𝑏−𝑎⁻¹≤𝜅𝑉_𝜙(𝑢). (28)

Therefore, 𝜑₁𝛽−𝛼≤𝜅𝑉_𝜙(𝑢)12+𝜅𝑛().𝑏−𝑎 (29)

By taking limit when 𝑛→∞, we obtain 𝜑₁(𝛽−𝛼)=0, which is absurd. From each it follows that lim_{𝑡↑𝑡∗}inf𝑢(𝑡)=lim_{𝑡↑𝑡∗}sup𝑢(𝑡). (30)

By a similar argument it follows that there exist lim_{𝑡↓𝑡∗}𝑢(𝑡),𝑡^∗∈[𝑎,𝑏).

Assuming that lim_𝑡→0(𝑘(𝑡)/𝑡) is finite, then V_𝜙[𝑎,𝑏]=𝜅𝑉[𝑎,𝑏]. For the last part of Theorem 9, we can give the definition of left and right regularizations of the function 𝑢∈𝜅𝑉_𝜙[𝑎,𝑏].

Definition 10. Let 𝑢∈𝜅BV_𝜙[𝑎,𝑏], then 𝑢⁻(𝑡)∶=lim_𝑠↑𝑡𝑢(𝑠)(]𝑢(𝑎)𝑢,𝑡∈𝑎,𝑏,𝑡=𝑎,₊(𝑡)∶=lim_𝑠↓𝑡𝑢(𝑠)[)𝑢(𝑏),𝑡∈𝑎,𝑏,𝑡=𝑏. (31)

The function 𝑢⁻ is called the left regularization of the function 𝑢 and the function 𝑢⁺ the right regularization of the function 𝑢.

Applying the previous definition and the last part of Theorem 9, we can define 𝜅BV⁻_𝜙[𝑎,𝑏]∶=𝑢⁻∶𝑢∈𝜅BV_𝜙[𝑎,𝑏]. (32)

Similarly, we defined 𝜅BV⁺_𝜙[𝑎,𝑏].

Recently Castillo et al. [25] introduced the concept of 𝜅𝑝-variation in Riesz-Korenblum’s sense in the following way.

Definition 11. Let 1<𝑝<∞, 𝜅∶[0,1]→[0,1] be a distortion function and 𝑢∶[𝑎,𝑏]→ℝ and 𝜋∶𝑎≤𝑡₁<⋯<𝑡_𝑛≤𝑏 a partition of the interval [𝑎,𝑏]. We define 𝜅𝑉^𝑅_𝑝(𝑢)=𝜅𝑉^𝑅_𝑝(𝑢;[a,𝑏])∶=sup_𝜋𝜅𝜎_𝑝(𝑢,𝜋), (33) where 𝜅𝜎_𝑝(𝑢,𝜋)∶=^𝑛−1_𝑗=1𝑢𝑡_𝑗+1−𝑢𝑡_𝑗^𝑝𝑡_𝑗+1−𝑡_𝑗^𝑝−1×^𝑛−1_𝑗=1𝜅𝑡_𝑗+1−𝑡_𝑗𝑏−𝑎⁻¹ (34) and the supremum is taken on the set of all partitions of [𝑎,𝑏]. If 𝜅𝑉^𝑅_𝑝(𝑢;[𝑎,𝑏])<∞, we say that 𝑢 has bounded 𝜅𝑝-variation in the interval [𝑎,𝑏]. The number 𝜅𝑉^𝑅_𝑝(𝑢;[𝑎,𝑏]) denoted the 𝜅𝑝-variation of 𝑢 in Riesz-Korenblum’s sense in [𝑎,𝑏]. The space of functions that have bounded 𝜅𝑝-variation in the interval [𝑎,𝑏] is denoted by 𝜅RV_𝑝[𝑎,𝑏].

Some properties of these functions are exposed in the following theorem.

Theorem 12 (see [25]). Let 1<𝑝<∞ and let 𝜅∶[0,1]→[0,1] be a distortion function, then (1)𝜅RV[𝑎,𝑏] is a Banach space endowed with the norm ‖𝑢‖_𝜅𝑝∶=|𝑢(𝑎)|+𝜅𝑉^𝑅_𝑝(𝑢)^1/𝑝𝑢∈𝜅RV_𝑝[𝑎,𝑏]. (35)(2)Lip[𝑎,𝑏]⊂RV_𝑝[𝑎,𝑏]⊂𝜅RV_𝑝[𝑎,𝑏]⊂𝜅BV[𝑎,𝑏], where RV_𝑝[𝑎,𝑏] denote the space of the functions 𝑢 that have bounded 𝑝-variation in Riesz’s sense [20].(3)𝜅RV_𝑝[𝑎,𝑏] is an algebra.

This concept was generalized by Castillo et al. [26] as stated in the following definition.

Definition 13. Let 𝜑 be a 𝜑-function, 𝜅∶[0,1]→[0,1] be a distortion function, and 𝑢∶[𝑎,𝑏]→ℝ and 𝜋∶𝑎≤𝑡₁<⋯<𝑡_𝑛≤𝑏 a partition of the interval [𝑎,𝑏]. We define 𝜅𝑉^𝑅_𝜑(𝑢)=𝜅𝑉^𝑅_𝜑(𝑢;[𝑎,𝑏])∶=sup_𝜋𝜅𝜎_𝜑(𝑢,𝜋), (36) where 𝜅𝜎_𝜑()∑𝑢,𝜋∶=^{𝑛−1𝑗=1}𝜑𝑢𝑡_𝑗+1−𝑢𝑡_𝑗/𝑡_𝑗+1−𝑡_𝑗𝑡_𝑗+1−𝑡_𝑗∑^𝑛−1_𝑗=1𝜅𝑡_𝑗+1−𝑡_𝑗/(𝑏−𝑎) (37) and the supremum is taken over all partitions of [𝑎,𝑏]. If 𝜅𝑉^𝑅_𝜑(𝑢;[𝑎,𝑏])<∞, we say that 𝑢 has bounded 𝜅𝜑-variation in the interval [𝑎,𝑏] and this number denotes the 𝜅𝜑-variation of 𝑢 in Riesz-Korenblum’s sense in [𝑎,𝑏]. The class of functions that have bounded 𝜅𝜑-variation in the interval [𝑎,𝑏] is denoted by 𝜅𝑉_𝜑[𝑎,𝑏]. The vectorial space generate by this class is denoted by 𝜅RV_𝜑[𝑎,𝑏].

The space of all functions that have bounded 𝜅𝜑-variation on [𝑎,𝑏] is denoted by 𝜅RV_𝜑[𝑎,𝑏]. Some properties of these functions are exposed in the following theorem.

Theorem 14 (see [26]). Let 𝜑 be a convex 𝜑-function and 𝜅∶[0,1]→[0,1] a distortion function, then (1)Lip[𝑎,𝑏]⊂𝜅𝑉^𝑅_𝜑[𝑎,𝑏]⊂𝜅BV[𝑎,𝑏]. (2)RV_𝜑[𝑎,𝑏]⊂𝜅RV_𝜑[𝑎,𝑏]⊂𝜅BV[𝑎,𝑏], where RV_𝜑[𝑎,𝑏] denote the space generated by the class of functions of bounded 𝜑-variation in Riesz’s sense [20]. (3) If lim_𝑡→∞(𝜑(𝑡)/𝑡)<∞, then 𝜅𝑉^𝑅_𝜑[𝑎,𝑏]=𝜅BV[𝑎,𝑏].(4) If 𝑢∈𝜅𝑉^𝑅_𝜑[𝑎,𝑏], then 𝑢 is bounded. (5)𝜅𝑉^𝑅_𝜑[𝑎,𝑏] is a convex and symmetric set. (6)𝜅RV_𝜑[𝑎,𝑏]={𝑢∶[𝑎,𝑏]→ℝ∶∃𝜆>0 such that 𝜅𝑉^𝑅_𝜙(𝜆𝑢)<∞}.(7)𝜅RV_𝜑[𝑎,𝑏] is a Banach space endowed with the norm ‖𝑢‖_𝜅𝜑∶=|𝑢(𝑎)|+𝜇(𝑢)𝑢∈𝜅RV_𝜑[𝑎,𝑏], (38) where 𝜇_𝜑(𝑢)∶=inf_𝜆>0{𝜆>0∶𝜅𝑉^𝑅_𝜑(𝑢/𝜆)≤1}.

3. Composition Operator between Lip[𝑎,𝑏] and 𝜅BV[𝑎,𝑏] or 𝜅BV_𝜙[𝑎,𝑏]

Given a function ℎ∶ℝ→ℝ, the composition operator 𝐻, associated to the function ℎ (case autonomous), maps each function 𝑢∶[𝑎,𝑏]→ℝ into the composition function 𝐻𝑢∶[𝑎,𝑏]→ℝ defined by 𝐻𝑢(𝑡)∶=ℎ(𝑢(𝑡)),(𝑡∈[𝑎,𝑏]). (39)

More generally, given ℎ∶[𝑎,𝑏]×ℝ→ℝ, we consider operator 𝐻 defined by 𝐻𝑢(𝑡)∶=ℎ(𝑡,𝑢(𝑡)),(𝑡∈[𝑎,𝑏]). (40)

This operator is also called superposition operator or substitution operator or Nemytskii operator. In what follows, will refer to (39) as the autonomous case and to (40) as the nonautonomous case.

A problem related with this operator is to establish necessary and sufficient conditions of function ℎ so that the operators 𝐻 map the space 𝕏 of real functions defined on [𝑎,𝑏] into itself, that is, 𝐻(𝕏)⊂𝕏, or in more general way that operator 𝐻 maps the space 𝕏 into space of functions 𝕐(𝐻(𝕏)⊂𝕐). This problem is sometimes referred to as the composition operator problem (or COP). The solution to this problem for given 𝕏 is sometimes very easy and sometimes highly nontrivial. As we mentioned in the introduction of this paper in a variety of spaces the required condition is that function ℎ is locally Lipschitz. Another interesting problem is to determine the smallest space of functions 𝕏 and the bigger space 𝕐 such that 𝐻(𝕏)⊂𝕐.

In order to obtain the main result of this section, we will use a function of the zig-zig type such as the employed by Appell et al. in [1, 15]. In this section we will show that the locally Lipschitz condition of the function ℎ is a necessary and sufficient condition such that 𝐻(Lip[𝑎,𝑏])⊂𝜅BV[𝑎,𝑏] and that in this situation 𝐻 is bounded.

The following lemma will be useful in the proof of our main theorem (Theorem 17).

Lemma 15. Let 𝑢∶[𝑎,𝑏]→ℝ, 𝑎≤𝑠<𝜂<𝑡≤𝑏, then |𝑢(𝑡)−𝑢(𝑠)|≤𝑢𝜂(𝑠𝑡−𝑠−𝑢)+𝑢(𝑡)𝜂𝜂−𝑠−𝑢.𝑡−𝜂 (41)

Proof. Let 𝑎≤𝑠<𝜂<𝑡≤𝑏. Then |𝑢(𝑡)−𝑢(𝑠)|≤𝑢𝜂(𝑠𝑡−𝑠−𝑢)+𝑢(𝑡)𝜂𝑡−𝑠−𝑢=𝑢𝜂(𝑠𝑡−𝑠−𝑢)𝜂−𝑠𝜂−𝑠+𝑢(𝑡)𝜂𝑡−𝑠−𝑢𝑡−𝜂𝑡−𝜂≤𝑢𝜂(𝑠𝑡−𝑠−𝑢)+𝑢(𝑡)𝜂𝜂−𝑠−𝑢.𝑡−𝜂 (42)

Lemma 16. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, 𝑢∈𝜅BV_𝜙[𝑎,𝑏], and 𝜆>0. Then 𝜇_𝜙(𝑢)<𝜆 if and only if 𝜅𝑉_𝜙(𝑢/𝜆)<1.

Proof. Let 𝑢∈𝜅BV_𝜙[𝑎,𝑏]. Suppose that 𝜇_𝜙(𝑢)<𝜆; then, by definition of 𝜇_𝜙(𝑢) there exists 𝑘 such that 𝜆>𝑘>𝜇_𝜙(𝑢) and 𝜅𝑉_𝜙(𝑢/𝑘)≤1. Hence, by the convexity of the functions 𝜑_𝑛, we have 𝜅𝑉_𝜙𝑢𝜆=𝜅𝑉_𝜙𝑢𝑘𝑘𝜆≤𝑘𝜆𝜅𝑉_𝜙𝑢𝑘≤𝑘𝜆≤1. (43)
Conversely, assume 𝜅𝑉_𝜙(𝑢/𝜆)<1, then 𝜆∈{𝜆>0∶𝜅𝑉_𝜙(𝑢/𝜆)≤1}; hence 𝜇_𝜙(𝑢)<𝜆.

Theorem 17. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, ℎ∶ℝ→ℝ, and 𝐻 the composition operator associated to ℎ. 𝐻 maps the space Lip[0,1] into the space 𝜅BV_𝜙[0,1] or 𝜅BV[0,1] if and only if ℎ is locally Lipschitz. Furthermore operator 𝐻 is bounded.

Proof. Let 𝑢∈Lip[0,1], 𝑟=‖𝑢‖_∞ and suppose that ℎ is locally Lipschitz, then there exist 𝑘=𝑘(𝑟), such that |ℎ(𝑡)−ℎ(𝑠)|≤𝑘(𝑟)|𝑠−𝑡|,(𝑠,𝑡∈ℝ,|𝑠|≤𝑟,|𝑡|≤𝑟). (44)
Let {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏] and 𝜆>0, such that 𝜆<1/(2𝑘(𝑟)‖𝑢‖_Lip‖𝑢‖_∞+1), then ∑^∞_𝑛=1𝜑_𝑛𝜆ℎ𝑢𝑏_𝑛−ℎ𝑢𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛/()≤∑𝑏−𝑎_∞𝑛=1𝜑_𝑛𝜆𝑘(𝑟)𝑢𝑏_𝑛−𝑢𝑎_𝑛𝜅(1)≤𝜆𝑘(𝑟)𝜑₁(1)‖𝑢‖_Lip^∞_𝑛=1𝑏_𝑛−𝑎_𝑛<∞. (45)
Then by Lemma 15 and Theorem 9, we have 𝐻(𝑢)∈𝜅BV_𝜙[𝑎,𝑏].
The proof of the only if direction will be by contradiction, that is, we assume 𝐻(Lip[0,1])⊂𝜅BV_𝜙[0,1] and ℎ is not locally Lipschitz. Since the identity function 𝐼_𝑑∶[0,1]→[0,1] belongs to Lip[0,1], then ℎ∘𝐼_𝑑∈𝜅BV_𝜙[0,1] and therefore ℎ is bounded in the interval [0,1]. Without loss of generality we may assume that ℎ∣_[0,1]∞≤14. (46)
Since ℎ is not locally Lipschitz in ℝ, there is a closed interval 𝐼 such that ℎ does not satisfy any Lipschitz condition. In order to simplify the proof we can assume that 𝐼=[0,1]. In this way for any increasing sequence of positive real numbers {𝑘_𝑛}_𝑛≥1 that converge to infinite that we will define later, we can choose sequences {𝑎_𝑛}_𝑛≥1, {𝑏_𝑛}_𝑛≥1, such that ℎ𝑏_𝑛−ℎ𝑎_𝑛>𝑘_𝑛𝑏_𝑛−𝑎_𝑛,(𝑛∈ℕ). (47)
In addition we can choose 𝑎_𝑛,𝑏_𝑛 such that 𝑎_𝑛<𝑏_𝑛,(𝑛∈ℕ). (48)
Considering subsequences if necessary, we can assume that the sequence {𝑎_𝑛}_𝑛≥1 is monotone. We can assume without loss of generality that sequence {𝑎_𝑛}_𝑛≥1 is increasing.
Since [0,1] is compact, from inequality (47) we have that there exist subsequences of {𝑎_𝑛}_𝑛≥1 and {𝑏_𝑛}_𝑛≥1 that we will denote in the same way, and that converge to 𝑎_∞∈[0,1].
Since the sequence {𝑎_𝑛}_𝑛≥1 is a Cauchy sequence, we can assume (taking subsequence if necessary) that 𝑎_𝑚−𝑎_𝑛<1𝑘_𝑛,(𝑚>𝑛). (49)
Again considering subsequences if needed using the properties of the function 𝜅 we can assume that max𝜅𝑏_𝑛−𝑎_𝑛,𝜅𝑎_𝑚−𝑎_𝑛<1𝑘_𝑛,(𝑛∈ℕ,𝑚≥𝑛). (50)
Consider the new sequence {𝑚_𝑛}_𝑛≥1 defined by 𝑚_𝑛1∶=𝑘_𝑛𝑏_𝑛−𝑎_𝑛,(𝑛∈ℕ). (51)
From inequalities (46) and (47) it follows that 𝑚_𝑛>2; therefore 𝑚_𝑛2<𝑚_𝑛≤𝑚_𝑛,(𝑛∈ℕ). (52)
Consider the sequence defined recursively {𝑡_𝑛}_𝑛≥1 by 𝑡₁∶=0,𝑡_𝑛+1∶=𝑡_𝑛+𝑎_𝑛+1−𝑎_𝑛+2𝑚_𝑛𝑏_𝑛−𝑎_𝑛,(𝑛∈ℕ). (53)
This sequence is strictly increasing and from the relations (49) and (50), we get 𝑡_𝑛⟶𝑡_∞∶=𝑡_𝑛+1−𝑡_𝑛=^∞_𝑛=1𝑎_𝑛+1−𝑎_𝑛+2^∞_𝑛=1𝑚_𝑛𝑏_𝑛−𝑎_𝑛≤3^∞_𝑛=11𝑘_𝑛. (54)
Then to ensure that 𝑡_∞∈[0,1], it is sufficient to suppose that ∑^∞_𝑛=1(1/𝑘_𝑛)≤1/3.
We define the continuous zig-zag function 𝑢∶[0,1]→ℝ, as shown in the following: 𝑢(𝑡)𝑎∶=^𝑛,𝑡=𝑡_𝑛+2𝑖𝑏_𝑛−𝑎_𝑛,𝑖=0,…,𝑚_𝑛𝑏_𝑛,𝑡=𝑡_𝑛+(2𝑖+1)𝑏_𝑛−𝑎_𝑛,𝑖=0,…,𝑚_𝑛𝑎−1_∞,𝑡_∞≤𝑡≤1affine,othercase. (55)
Put 𝑡_𝑛,𝑖∶=𝑡_𝑛+𝑖𝑏_𝑛−𝑎_𝑛,𝑛∈ℕ,𝑖=0,…,2𝑚_𝑛. (56)
We can write each interval 𝐼_𝑛=[𝑡_𝑛,𝑡_𝑛+1],𝑛∈ℕ, as the union of the family of nonoverlapping intervals 𝐼_𝑛,𝑖∶=𝑡_𝑛.𝑖,𝑡_𝑛,𝑖+1,𝑖=0,…,𝑚_𝑛𝐼−1,_{𝑛,2[𝑚𝑛]}∶=𝑡_{𝑛,2[𝑚𝑛]},𝑡_𝑛+1. (57)
And function 𝑢 is defined on 𝐼_𝑛,𝑖,𝑖=0,…,2[𝑚_𝑛] as follows: 𝑢(𝑡)=𝑡−𝑡_𝑛+2𝑖𝑏_𝑛−𝑎_𝑛+𝑎_𝑛,𝑡∈𝐼_𝑛,2𝑖,𝑢(𝑡)=𝑡−𝑡_𝑛+(2𝑖+1)𝑏_𝑛−𝑎_𝑛+𝑏_𝑛,𝑡∈𝐼_𝑛,2𝑖+1,𝑢(𝑡)=𝑡−𝑡_𝑛+1+𝑎_𝑛,𝑡∈𝐼_𝑛,2[𝑚]. (58)(59)(60)
In all these situations, the slopes of the segments of lines are equal to 1.
Hence, we have for 𝑛∈ℕ, the absolute value of the slope of the line segments in these ranges are bounded by 1, as shown below 2^−𝑛𝑏_𝑛−𝑎_𝑛𝜅⁻¹𝑏_𝑛−𝑎_𝑛≤2^−𝑛𝑘_𝑛𝑏_𝑛−𝑎_𝑛2≤1,_−(𝑛+1)𝑎_𝑛+1−𝑎_𝑛𝑡_𝑛+𝑎_𝑛+1−𝑎_𝑛≤1. (61)
We will show that 𝑢∈Lip[0,1].
Let 0≤𝑠<𝑡≤1, and then there are the following possibilities for the location of 𝑠 and 𝑡 on [0,1].
Case 1. If 𝑠,𝑡∈𝐼_𝑛,(𝑛∈ℕ) and are in the same interval 𝐼_𝑛,𝑖,𝑖=0,…,2[𝑚_𝑛].
From relations (58), (59), and (60) it follows that |𝑢(𝑡)−𝑢(𝑠)|/|𝑡−𝑠|=1.
Case 2. If 𝑠,𝑡∈𝐼_𝑛,(𝑛∈ℕ) and are in two different intervals 𝐼_𝑛,𝑖,𝑖=0,…,2[𝑚_𝑛].
There are several possibilities. (a)𝑠∈𝐼_𝑛.𝑖,𝑡∈𝐼_𝑛,𝑗,𝑖<𝑗<2[𝑚_𝑛]. (a₁)𝑗=𝑖+1. By Lemma 15 and relations (58) and (59) we have |𝑢(𝑡)−𝑢(𝑠)|||≤𝑢𝑡𝑡−𝑠_𝑛,𝑖+1−𝑢(𝑠)𝑡_𝑛,𝑖+1+𝑢(𝑡)𝑡−𝑠−𝑢_𝑛,𝑖+1𝑡−𝑡_𝑛,𝑖+1≤2. (62)(a₂)𝑗>𝑖+1. Then |𝑢(𝑡)−𝑢(𝑠)|||≤𝑏𝑡−𝑠^𝑛−𝑎_𝑛𝑡_𝑛,𝑖+2−𝑡_𝑛,,𝑖+1=1. (63)(b)𝑠∈𝐼_𝑛.𝑖,𝑡∈𝐼_𝑛,𝑗,𝑖<𝑗=2[𝑚_𝑛]. If 𝑗=𝑖+1, proceed as (𝑎₁). If 𝑗>𝑖+1, again using the Lemma 15 and relations (58), (59), and (60) we obtain |𝑢(𝑡)−𝑢(𝑠)|||≤𝑢𝑡𝑡−𝑠_{𝑛,2[𝑚𝑛]}−𝑢(𝑠)𝑡_{𝑛,2[𝑚𝑛]}+𝑢(𝑡)𝑡−𝑠−𝑢_{𝑛,2[𝑚𝑛]}𝑡_{𝑛,2[𝑚𝑛]}≤𝑏−𝑡_𝑛−𝑎_𝑛𝑡_{𝑛,2[𝑚𝑛]}−𝑡_{𝑛,2[𝑚𝑛]−1}+1≤2. (64)
Case 3. If 𝑠∈𝐼_𝑛,𝑡∈𝐼_𝑚,𝑛,𝑚∈ℕ,𝑛<𝑚.
From Lemma 15 and Case 2, we conclude that |𝑢(𝑡)−𝑢(𝑠)|≤𝑢𝑡𝑡−𝑠_𝑛+1−𝑢(𝑠)𝑡_𝑛+1+𝑢(𝑡)𝑡−𝑠−𝑢_𝑚𝑡−𝑡_𝑚≤4. (65)
Case 4. If 𝑠∈𝐼_𝑛,𝑛∈ℕ,𝑡=𝑡_∞.
Then from Lemma 15𝑢𝑡_∞−𝑢(𝑠)𝑡_∞≤𝑢𝑡−𝑠_𝑛,𝑖+1−𝑢(𝑠)𝑡_𝑛,𝑖+1+𝑎−𝑠^∞−𝑢𝑡_𝑛,𝑖+1𝑏_𝑛−𝑎_𝑛𝑎≤1+^∞−𝑎_𝑛𝑏_𝑛−𝑎_𝑛≤2. (66)
Case 5. If 𝑠<𝑡_∞<𝑡≤1.
From Lemma 15 and Case 4, |𝑢(𝑡)−𝑢(𝑠)|≤𝑢𝑡𝑡−𝑠^∞−𝑢(𝑠)𝑡_∞−𝑠≤2. (67)
Case 6. If 𝑡_∞≤𝑠<𝑡≤1.
In this circumstance 𝑢(𝑠)=𝑢(𝑡)=𝑎_∞ and the situation is trivial. Therefore we have that |𝑢(𝑡)−𝑢(𝑠)|≤|𝑡−𝑠|,(𝑠,𝑡∈[0,1]). (68)
So 𝑢 is Lipschitz in [0,1]. Moreover, for each partition of the interval [0,1] of the form 𝜋∶0=𝑡₁<𝑡₁+𝑏₁−𝑎₁<⋯<𝑡₁+2𝑚₁𝑏₁−𝑎₂<𝑡₂<𝑡₂+𝑏₂−𝑎₂<⋯<𝑡_𝑘<⋯<𝑡_𝑘+2𝑚_𝑘𝑏_𝑘−𝑎_𝑘<1 (69) and 𝑐>0, using the inequality (47), convexity of the function 𝜑_𝑛,𝑛≥1, and definition of 𝑚_𝑛,𝑛∈ℕ, we have 𝜅𝑉_𝜙(𝑐();[≥∑ℎ∘𝑢0,1])_𝑘𝑛=12𝑚_𝑛𝜑_𝑛(𝑐)ℎ𝑏_𝑛−ℎ𝑎_𝑛∑^𝑘_𝑛=12𝑚_𝑛𝜅𝑏_𝑛−𝑎_𝑛+𝜅𝑎_𝑛+1−𝑎_𝑛≥∑^𝑘_𝑛=12𝑚_𝑛𝜑_𝑛(𝑐)𝑘_𝑛𝑏_𝑛−𝑎_𝑛∑^𝑘_𝑛=12𝑚_𝑛𝜅𝑏_𝑛−𝑎_𝑛+𝜅𝑎_𝑛+1−𝑎_𝑛≥∑^𝑘_𝑛=12𝑚_𝑛𝑘_𝑛𝑏_𝑛−𝑎_𝑛𝜑_𝑛(𝑐)∑^𝑘_𝑛=11/𝑘_𝑛≥^𝑘_𝑛=1𝜑_𝑛(𝑐). (70) As the series ∑^∞_𝑛=1𝜑_𝑛(𝑐) diverge, 𝑐ℎ∘𝑢∉𝜅BV_𝜙[0,1], which is a contradiction.
Let us see that ℎ∘𝑢∉𝜅BV[𝑎,𝑏]. In fact, as in the case of 𝜅-variation we have 𝑉_𝜅(ℎ∘𝑢;[0,1])≥^𝑘_𝑛=12𝑚_𝑘𝑘_𝑛𝑏_𝑛−𝑎_𝑛≥𝑘. (71)
Therefore, ℎ∘𝑢∉𝜅BV[0,1].
To prove that the operator 𝐻 is bounded, let 𝑟>0 and 𝑢∈Lip[0,1], such that ‖𝑢‖_Lip≤𝑟, then from the definition of ‖⋅‖_Lip[0,1], we have ‖𝑢‖_∞≤|𝑢(0)|+𝐿¹₀(𝑢)≤𝑟. (72)
Without loss of generality assume that 𝑟=1. As ℎ is locally Lipschitz, there exists 𝑘(1)>0, such that ℎ(𝑥)−ℎ𝑦≤𝑘(1)𝑥−𝑦,|𝑥|≤1,𝑦≤1. (73)
As the identity function 𝐼_𝑑∶[0,1]→[0,1] belongs to Lip[0,1], then ℎ∘𝐼_𝑑∈𝜅BV_𝜙[0,1]. By Theorem 9 we have that ‖ℎ∘𝐼_𝑑‖_∞<∞.
Let {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[0,1] and choose 𝜆 such that 𝑘(1)𝜅𝑉_𝜙(𝐼_𝑑)<𝜆, then ∑^∞_𝑛=1𝜑_𝑛ℎ𝑢𝑏_𝑛−ℎ𝑢𝑎_𝑛/𝜆∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛≤^∞_𝑛=1𝜑_𝑛𝑘(1)𝑢𝑏_𝑛−𝑢𝑎_𝑛𝜆≤^∞_𝑛=1𝜑_𝑛𝑘(1)𝑏_𝑛−𝑎_𝑛𝜆≤^∞_𝑛=1𝑘(1)𝜆𝜑_𝑛𝑏_𝑛−𝑎_𝑛≤𝑘(1)𝜆𝜅𝑉_𝜙𝐼_𝑑<1. (74)
From Lemma 6 we have 𝜇(ℎ∘𝑢)<𝜆(𝑟). Thus we conclude that ‖𝐻(𝑢)‖_𝜅𝜙=|ℎ(𝑢(𝑎()≤‖‖))|+𝜇ℎ∘𝑢ℎ∘𝑢_∞+𝜆(𝑟). (75)
And so operator 𝐻 is bounded. In the case that 𝐻(Lip[0,1])⊂𝜅BV[0,1], proceed similarly.

In the following result we give a Lemma of invariance.

Lemma 18. Let 𝜅 be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, and 𝜈∶[0,1]→[𝑎,𝑏] affine function that maps [0,1] on [𝑎,𝑏](𝑣(𝑡)∶=(𝑏−𝑎)𝑡+𝑎,𝑡∈[0,1]). (1)𝑢∈𝜅BV_𝜙[𝑎,𝑏] if and only if 𝑢∘𝜈∈𝜅BV_𝜙[0,1].(2)𝑢∈𝜅BV[𝑎,𝑏] if and only if 𝑢∘𝜈∈𝜅BV_𝜙[0,1].

Proof. (1) Let 𝑢∈𝜅BV_𝜙[𝑎,𝑏] and {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[0,1], then {[𝜈(𝑎_𝑛),𝜈(𝑏_𝑛)]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏] and ∑^∞_𝑛=1𝑢𝜈𝑏_𝑛−𝑢𝜈𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛=∑^∞_𝑛=1𝑢𝜈𝑏_𝑛−𝑢𝜈𝑎_𝑛∑^𝑛−1_𝑛=1𝜅𝜈𝑏_𝑛−𝜈𝑎_𝑛/(𝑏−𝑎)≤𝜅𝑉_𝜙(𝑢;[𝑎,𝑏]). (76)
Hence 𝑢∘𝜈∈𝜅BV_𝜙[0,1].
Reciprocally, let us suppose that 𝑢∘𝜈∈𝜅BV_𝜙[0,1] and let {[𝑎_𝑛,𝑏_𝑛]}_𝑛≥1∈𝐹_ℕ[𝑎,𝑏] be a partition of the interval [𝑎,𝑏]. Define 𝐼_𝑛∶=𝜈⁻¹𝑏_𝑛,𝜈⁻¹𝑎_𝑛(𝑛∈ℕ). (77) We have {𝐼_𝑛}_𝑛≥1∈𝐹_ℕ[0,1] and ∑^𝑛−1_𝑗=1𝑢𝑏_𝑛−𝑢𝑎_𝑛∑^∞_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛/()=𝑏−𝑎_∞𝑛=1𝑢𝜈𝑏_𝑛/(𝜈𝑎−𝑎𝑏−𝑎)−𝑢_𝑛/()×−𝑎𝑏−𝑎_∞𝑛=1𝜅𝑏_𝑛/()−𝑎−𝑎𝑏−𝑎_𝑛−𝑎/(𝑏−𝑎)⁻¹≤𝜅𝑉_𝜙(𝑢∘𝜈;[0,1]), (78) so 𝑢∈𝜅BV_𝜙[𝑎,𝑏].
(2) The proof is similar from part (1).

As consequence of Lemma 18 we have the following results

Lemma 19. Let 𝜅 be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, ℎ∶ℝ→ℝ, and 𝐻 the composition operator associated to the function ℎ. (1)𝐻(Lip[𝑎,𝑏])⊂𝜅BV_𝜙[𝑎,𝑏] if and only if 𝐻(Lip[0,1])⊂𝜅BV_𝜙[0,1]. (2)𝐻(Lip[𝑎,𝑏])⊂𝜅BV[𝑎,𝑏] if and only if 𝐻(Lip[0,1])⊂𝜅BV[0,1].

Corollary 20. Let 𝜅 be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, and ℎ∶ℝ→ℝ. Then the composition operator 𝐻, associated with function ℎ, maps the space Lip[𝑎,𝑏] on 𝜅BV_𝜙[𝑎,𝑏] or on 𝜅BV[𝑎,𝑏] if and only if ℎ is locally Lipschitz.

Corollary 21. Let 𝜅 be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, ℎ∶ℝ→ℝ, and 𝕏,𝕐 normed spaces such that Lip[𝑎,𝑏]⊂𝕏⊂𝕐, where 𝕐⊂𝜅BV_𝜙[𝑎,𝑏] or 𝕐⊂𝜅BV[𝑎,𝑏]. Then the composition operator 𝐻 associated with the function ℎ maps the space 𝕏 in the space 𝕐 if and only if ℎ is locally Lipschitz.

Some particular cases of Corollary 21 are the following(1)𝕏=𝕐, where 𝕏 is one of the following spaces Lip[𝑎,𝑏] [3], 𝐻_𝛼[𝑎,𝑏](0<𝛼<1) [4], BV_𝜑[𝑎,𝑏] [6], BV[𝑎,𝑏] [5], HBV [7], AC[𝑎,𝑏] [8], RV_𝜑[𝑎,𝑏] [9], 𝜙BV [23], ΛBV [11], and 𝜅BV[𝑎,𝑏] [12]. (2)𝕏=RV_𝜑[𝑎,𝑏], 𝕐=BV[𝑎,𝑏]. See [10]. (3)𝕏=Lip[𝑎,𝑏], 𝕐=BV[𝑎,𝑏]. See [1].

From Corollary 21 we get the following new cases(1)𝕏 is one of the following spaces: Lip[𝑎,𝑏], 𝐻_𝛼(0<𝛼<1), RV_𝜑[𝑎,𝑏],AC[𝑎,𝑏],BV[𝑎,𝑏], BV_𝜙[𝑎,𝑏], and 𝜅RV_𝜑[𝑎,𝑏]; 𝕐 is one of the following spaces: 𝜅BV[𝑎,𝑏], 𝜅BV_𝜙[𝑎,𝑏]. (2)𝕏=𝜅RV_𝑝[𝑎,𝑏],𝕐=𝜅RV_𝑞[𝑎,𝑏], 1<𝑝,𝑞<∞. (3)𝕏=𝜅BV[𝑎,𝑏], 𝕐=𝜅BV_𝜙[𝑎,𝑏],

More generally (1)𝕏=𝜅₁BV[𝑎,𝑏],𝕐=𝜅₂RV_𝜙[𝑎,𝑏]; (2)𝕏=𝜅₁RV_𝜑[𝑎,𝑏],𝕐=𝜅₂RV_𝜓[𝑎,𝑏]; (3)𝕏=𝜅₁RV_𝜙[𝑎,𝑏],𝕐=𝜅₂RV_𝜙[𝑎,𝑏];

where 𝜅₁,𝜅₂ are distortion functions.

4. Uniformly Continuous Composition Operator in the Space 𝜅BV_𝜙[𝑎,𝑏]

In many problems solving equation where the composition operator appears to guarantee the existence of solution it is necessary to apply a Fixed Point Theorem. To ensure the application this type of results is necessary to request the condition of global Lipschitz operator 𝐻. In several works Matkowski and Mís have shown that this condition implies that the function ℎ has the form (1) or (2) (see, e.g., [16, 27]). This means that we may apply the Banach contraction mapping principle only if the underlying problems are actually linear and therefore are not interesting.

More recently, Matkowski and other researchers have replaced the condition of global Lipschitz by uniform continuity conditions or uniform boundedness composition operator (see e.g., [14]).

In this section we present results in this direction for the space 𝜅BV_𝜙[𝑎,𝑏].

Theorem 22. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, and ℎ∶[𝑎,𝑏]×ℝ→ℝ such that the function ℎ(𝑡,⋅)∶ℝ→ℝ is continuous with respect to the second variable, for each 𝑡∈(𝑎,𝑏]. If the composition operator H, associated with ℎ, maps 𝜅BV_𝜙[𝑎,𝑏] into itself and satisfies the inequality ‖𝐻(𝑢)−𝐻(𝑣)‖_𝜅𝜙≤𝛾‖𝑢−𝑣‖_𝜅𝜙,𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏] (79) then there exists 𝛼,𝛽∈𝜅BV⁻_𝜙[𝑎,𝑏], such that ℎ⁻(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),(𝑡∈[𝑎,𝑏],𝑥∈ℝ). (80)

Proof. As for each 𝑥∈ℝ fixed the function 𝑢(𝑡)=𝑥,𝑡∈[𝑎,𝑏] is in 𝜅BV_𝜙[𝑎,𝑏], then 𝐻(𝑢)=ℎ(⋅,𝑥)∈𝜅BV_𝜙[𝑎,𝑏]. There is left regularization of ℎ⁻(⋅,𝑥).
From inequality (79) and Lemma 16, we get 𝜅𝑉_𝜙𝐻(𝑢)−𝐻(𝑣)𝛾‖𝑢−𝑣‖_𝜅𝜙<1,𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏]. (81)
Let 𝑎<𝑟<𝑎₁<𝑏₁<𝑎₂<⋯<𝑎_𝑚<𝑏_𝑚=𝑠<𝑏,𝑥₁,𝑥₂∈ℝ,𝑥₁≠𝑥₂ and put the zig-zag continuous functions 𝑢_𝑘∶[𝑎,𝑏]→ℝ,𝑘=1,2, as𝑢_𝑘(𝑡)=𝑥_𝑘+𝑥₂2,𝑎≤𝑡≤𝑎₁or𝑡=𝑎_𝑖𝑥,𝑖=1,…,𝑚_𝑘+𝑥₁2,𝑏_𝑚≤𝑡≤𝑏or𝑡=𝑏_𝑖,𝑖=1,…,𝑚,𝑘=1,2affine,othercase. (82)
The functions 𝑢_𝑘,𝑘=1,2 are Lipschitz and therefore belongs to 𝜅BV_𝜙[𝑎,𝑏]. Furthermore 𝑢₁−𝑢₂(𝑡)=𝑥₁−𝑥₂2,(𝑡∈[𝑎,𝑏]). (83)
From inequality (81) and the definition of 𝑘𝜙-variation, we have ^𝑚_𝑛=1𝜑_𝑛ℎ𝑏_𝑛,𝑢₁𝑏_𝑛−ℎ𝑏_𝑛,𝑢₂𝑏_𝑛−ℎ𝑎_𝑛,𝑢₁𝑎_𝑛+ℎ𝑎_𝑛,𝑢₂𝑎_𝑛×𝛾𝑢₁−𝑢_2𝜅𝜙⁻¹×𝜅𝑎₁−𝑎+𝑏−𝑎^𝑚_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛+𝑏−𝑎_{𝑚−1𝑛=1}𝜅𝑎_𝑛+1−𝑏_𝑛𝑏−𝑎+𝜅𝑏−𝑏_𝑚𝑏−𝑎⁻¹≤1. (84)
By the construction of the 𝑢_𝑘,𝑘=1,2, we get ^𝑚_𝑛=1𝜑_𝑛2ℎ𝑏_𝑛,𝑥₁−ℎ𝑏_𝑛,𝑥₁+𝑥₂2−ℎ𝑎_𝑛,𝑥₁+𝑥₂2+ℎ𝑎_𝑛,𝑥₂×𝛾2⁻¹𝑥₁−𝑥₂⁻¹×𝜅𝑎₁−𝑎+𝑏−𝑎^𝑚_𝑛=1𝜅𝑏_𝑛−𝑎_𝑛+𝑏−𝑎_{𝑚−1𝑛=1}𝜅𝑎_𝑛+1−𝑏_𝑛𝑏−𝑎+𝜅𝑏−𝑏_𝑚𝑏−𝑎⁻¹≤1. (85)
Let 𝑟 tend to 𝑠 in the above inequality; we obtain ^𝑚_𝑛=1𝜑_𝑛ℎ⁻𝑠,𝑥₁−2ℎ⁻𝑠,𝑥₁+𝑥₂/2+ℎ⁻𝑠,𝑥₂𝛾2⁻¹𝑥₁−𝑥₂≤𝜅𝑠−𝑎𝑏−𝑎+𝜅𝑏−𝑠,((.𝑏−𝑎𝑠∈𝑎,𝑏]) (86)
Passing the limit as 𝑚→∞, ^∞_𝑛=1𝜑_𝑛ℎ⁻𝑠,𝑥₁−2ℎ⁻𝑠,𝑥₁+𝑥₂/2+ℎ⁻𝑠,𝑥₂𝛾2⁻¹𝑥₁−𝑥₂≤𝜅𝑠−𝑎𝑏−𝑎+𝜅𝑏−𝑠,((.𝑏−𝑎𝑠∈𝑎,𝑏]) (87)
As the series ∑^∞_𝑛=1𝜑_𝑛(𝑥) is divergent for each 𝑥>0, necessarily ℎ⁻𝑠,𝑥₁−2ℎ⁻𝑥𝑠,¹+𝑥₂2+ℎ⁻𝑠,𝑥₂((.=0,𝑠∈𝑎,𝑏]) (88)
So we conclude that ℎ⁻(𝑠,⋅) satisfies the Jensen equation in ℝ (see [28], page 315). The continuity of ℎ with respect of the second variable implies that for every 𝑡∈[𝑎,𝑏] there exist 𝛼,𝛽∶[𝑎,𝑏]→ℝ, such that ℎ⁻(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),(𝑡∈[𝑎,𝑏],𝑥∈ℝ). (89)
Since 𝛽(𝑡)=ℎ⁻(𝑡,0),𝑡∈[𝑎,𝑏],𝛼(𝑡)=ℎ⁻(𝑡,1)−𝛽(𝑡) and ℎ⁻(⋅,𝑥)∈𝜅BV⁻_𝜙[𝑎,𝑏], for each 𝑥∈ℝ, we obtain that 𝛼,𝛽∈𝜅BV_𝜙[𝑎,𝑏].

Corollary 23. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, and ℎ∶[𝑎,𝑏]×ℝ→ℝ. If the composition operator 𝐻, associated with ℎ, maps 𝜅BV_𝜙[𝑎,𝑏] in itself and satisfies the inequality ‖𝐻(𝑢)−𝐻(𝑣)‖_𝜅𝜙≤𝛾‖𝑢−𝑣‖_𝜅𝜙,𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏], (90) for any function 𝛾∶[0,∞)→[0,∞), verifying 𝛾(𝑡)→𝛾(0)=0, when 𝑡↓0, then there exists 𝛼,𝛽∈𝜅BV⁻_𝜙[𝑎,𝑏], such that ℎ⁻(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),(𝑡∈[𝑎,𝑏],𝑥∈ℝ). (91)

Proof. Let us fix 𝑥,𝑦∈ℝ,𝑥≠𝑦, and define 𝑢(𝑡)∶=𝑥,𝑣(𝑡)∶=𝑦,𝑡∈[𝑎,𝑏]. Then 𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏] and ‖𝑢−𝑣‖_𝜅𝜙=|𝑥−𝑦|. Then from inequality (90) and Lemma 16, we have 𝜑₁ℎ(𝑡,𝑥)−ℎ𝑡,𝑦−ℎ(𝑎,𝑥)+ℎ𝑎,𝑦𝛾𝑥−𝑦≤𝜅𝑡−𝑎𝑏−𝑎+𝜅𝑏−𝑡,((.𝑏−𝑎𝑡∈𝑎,𝑏]) (92)
And therefore ℎ(𝑡,𝑥)−ℎ𝑡,𝑦−ℎ(𝑎,𝑥)+ℎ𝑎,𝑦≤𝜑₁⁻¹(2)𝛾𝑥−𝑦. (93)
Proceeding as in the proof of Theorem 22 we have that there is left regularization of ℎ⁻(⋅,𝑥), for each 𝑥∈ℝ and if
𝑎≤𝑟<𝑎₁<𝑏₁<𝑎₂<⋯<𝑎_𝑚<𝑏_𝑚=𝑠≤𝑏,  𝑥,𝑦∈ℝ,𝑥≠𝑦 we defined the zig-zag continuous functions 𝑢,𝑣∶[𝑎,𝑏]→ℝ, as show below 𝑢(𝑡)=𝑥,𝑎≤𝑡≤𝑎₁or𝑡=𝑎_𝑖,𝑖=1,…,𝑚𝑦,𝑏_𝑚≤𝑡≤𝑏or𝑡=𝑏_𝑖𝑢(𝑡)=,𝑖=1,…,𝑚affine,othercase,𝑦,𝑎≤𝑡≤𝑎₁or𝑡=𝑎_𝑖,𝑖=1,…,𝑚𝑥,𝑏_𝑚≤𝑡≤𝑏or𝑡=𝑏_𝑖,𝑖=1,…,𝑚affine,othercase, (94)
then 𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏], (𝑢−𝑣)(𝑡)=𝑥−𝑦,𝑡∈[𝑎,𝑏] and for 𝑠∈(𝑎,𝑏]𝜑_𝑚2ℎ⁻(𝑠,𝑥)−ℎ⁻𝑠,𝑦𝛾≤𝑥−𝑦_𝑚𝑛=1𝜑_𝑛2ℎ⁻(𝑠,𝑥)−ℎ⁻𝑠,𝑦𝛾𝑥−𝑦≤𝜅𝑠−𝑎𝑏−𝑎+𝜅𝑏−𝑠𝑏−𝑎≤2. (95)
Hence, ℎ⁻(𝑠,𝑥)−ℎ⁻≤𝜑𝑠,𝑦^𝑚−1(2)2𝛾,((.𝑥−𝑦𝑠∈𝑎,𝑏]) (96)
And so ℎ⁻(⋅,𝑥) is continuous in (𝑎,𝑏]. Now the result is a consequence of Theorem 22.

Corollary 24. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, ℎ∶[𝑎,𝑏]×ℝ→ℝ, and 𝐻 the composition operator associated with ℎ. Suppose that 𝐻 maps 𝜅BV_𝜙[𝑎,𝑏] into itself and is uniformly continuous, then there exist 𝛼,𝛽∈𝜅BV⁻_𝜙[𝑎,𝑏], such that ℎ⁻(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),(𝑡∈[𝑎,𝑏],𝑥∈ℝ), (97) where ℎ⁻(⋅,𝑥) is the left regularization of ℎ(⋅,𝑥) for all 𝑥∈ℝ.

Proof. Consider the modulus of continuity associated with 𝐻; that is, 𝛾(𝑡)∶=sup‖𝐻(𝑢)−𝐻(𝑣)‖_𝜅𝜙∶‖𝑢−𝑣‖_𝜅𝜙≤𝑡,𝑢,𝑣∈𝜅BV_𝜙[,().𝑎,𝑏]𝑡≥0 (98)
Then 𝛾(𝑡)≥0,𝛾(0)=0. Furthermore, if 𝑡>0 and ‖𝑢−𝑣‖_𝜅𝜙≤𝑡, we obtain ‖𝐻(𝑢)−𝐻(𝑣)‖_𝜅𝜙≤𝛾(𝑡). (99)
Particularly, in the case where 𝑡=‖𝑢−𝑣‖_𝜅𝜙, we have ‖𝐻(𝑢)−𝐻(𝑣)‖_𝜅𝜙≤𝛾‖𝑢−𝑣‖_𝜅𝑣,𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏]. (100)
From Corollary 23 we obtain the conclusion.

Matkowski [27] introduced the notion of a uniformly bounded operator and proved that the generator of any uniformly bounded composition operator acting between general Lipschitz function normed spaces must be affine with respect to the function variable.

Definition 25. Let 𝕐 and ℤ be two metric (or normed) spaces. We say that a mapping 𝐹∶𝕐→ℤ is uniformly bounded, if for any 𝑡>0, there exists a nonnegative real number 𝛾(𝑡) such that for any nonempty set 𝐵⊂𝕐 we have diam𝐵≤𝑡⟹diam𝐹(𝐵)≤𝛾(𝑡). (101)

Remark 26. Every uniformly continuous operator or Lipschitzian operator is uniformly bounded.

Corollary 27. Let 𝜅∶[0,1]→[0,1] be a distortion function, 𝜙={𝜑_𝑛}_𝑛≥1 a 𝜙-sequence, and ℎ∶[𝑎,𝑏]×ℝ→ℝ, such that function ℎ(𝑡,⋅)∶ℝ→ℝ is continuous in respect to the second variable, for each 𝑡∈(𝑎,𝑏]. If the composition operator 𝐻, associated with ℎ, maps 𝜅BV_𝜙[𝑎,𝑏] in itself and is uniformly bounded, then exists 𝛼,𝛽∈𝜅BV⁻_𝜙[𝑎,𝑏], such that ℎ⁻(𝑡,𝑥)=𝛼(𝑡)𝑥+𝛽(𝑡),(𝑡∈[𝑎,𝑏],𝑥∈ℝ). (102)

Proof. Take any 𝑡>0 and 𝑢,𝑣∈𝜅BV_𝜙[𝑎,𝑏] such that ‖𝑢−𝑣‖_𝜅𝜙≤𝑡. Since diam{𝑢,𝑣}≤𝑡, by uniform boundedness of 𝐻, we have diam𝐻({𝑢,𝑣})≤𝛾(𝑡); that is, ‖𝐻(𝑢)−𝐻(𝑣)‖_𝜅𝜙=diam𝐻({𝑢,𝑣})≤𝛾‖𝑢−𝑣‖_𝜅𝜙. (103)
From Theorem 22 we have (102).

Acknowledgment

This research has been party supported by the Central Bank of Venezuela. The authors want to thank the library staff of BCV for compiling the references.