Received 28 October 2015; accepted 16 January 2016; published 19 January 2016
A number of generalizations and extensions of variation of a function have been given in many directions since Camile Jordan in 1881 gave a first notion of bounded variation in the paper  devoted to the convergence of Fourier series. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis. Two well-known generalizations are the functions of bounded p-variation and the functions of bounded j-variation, due to N. Wiener  and L. C. Young  respectively. In 1924 N. Wiener  generalized the Jordan notion and introduced the notion of p-variation (variation in the sense of Wiener). Later, in 1937, L. Young  introduced the notion of j-variation of a function. The p-variation of a function f is the supremum of the sums of the pth powers of absolute increments of f over no overlapping intervals. Wiener mainly focused on the case, the 2-variation. For p-variations with, the first major work was done by Young  , partly with Love  . After a long hiatus following Young’s work, pth-variations were reconsidered in a probabilistic context by R. Dudley   , in 1994 and 1997, respectively. Many basic properties of the variation in the sense of Wiener and a number of important applications of the concept can be found in   . Also, the paper by V. V. Chistyakov and O. E. Galkin  , in 1998, is very important in the context of p-variation. They study properties of maps of bounded p-variation in the sense of Wiener, which are defined on a subset of the real line and take values in metric or normed spaces.
In 1997 while studying Poisson integral representations of certain class of harmonic functions in the unit disc of the complex plan B. Korenblum  introduced the notion of bounded k-variation and proved that a function f is of bounded k-variation if ot can be written as the difference of two k-decreasing functions. This concept differs from others due to the fact that it introduces a distortion function k that measures intervals in the domain of the function and not in the range. In 1986, S. Ki Kim and J. Kim  , gave the notion of the space of functions of kf-bounded variation on, which is a combination of the notion of bounded f-variation in the sense of Schramm and bounded k-variation in the sense of Korenblum, and J. Park et al.   proved some properties in this space. Considering for and, then it follows that this space generalized the space of functions of kp-bounded variation in the sense of Wiener-Korenblum. In 1990 S. Ki Kim and J. Yoon  showed the existence of the Riemann-Stieltjes integral of functions of bounded k-variation and in 2011 W. Aziz, J. Guerrero, J. L. Sánchez and M. Sanoja, in  , showed that the space of bounded k-variation satisfies the Matkowski’s weak condition. Also, in 2012, M. Castillo, M. Sanoja and I. Zea  presented the space of functions of bounded k-variation in the sense of Riez-Korenblum, denoted by, which is a combination of the notions of bounded p-variation in the sense of Riesz and bounded k-variation in the sense of Korenblum.
Recently, there has been an increasing interest in the study of various mathematical problems with variable exponents. With the emergency of nonlinear problems in applied sciences, standard Lebesgue and Sobolev spaces demonstrated their limitations in applications. The class of nonlinear problems with exponent growth is a new research field and it reflects a new kind of physical phenomena. In 2000 the field began to expand even further. Motivated by problems in the study of electrorheological fluids, L. Diening  raised the question of when the Hardy-Littlewood maximal operator and other classical operators in harmonic analysis are bounded on the variable Lebesgue spaces. These and related problems are the subject of active research nowadays. These problems are interesting in applications (see  - ) and give rise to a revival of the interest in Lebesgue and Sobolev spaces with variable exponent, the origins of which can be traced back to the work of W. Orlicz in the 1930’s  . In the 1950’s, this study was carried on by H. Nakano   who made the first systematic study of spaces with variable exponent. Later, Polish and Czechoslovak mathematicians investigated the modular function spaces (see for example J. Musielak   , O. Kovacik and J. Rakosnik  ). We refer to books  for the detailed information on the theoretical approach to the Lebesgue and Sobolev spaces with variable exponents. In 2015, R. Castillo, N. Merentes and H. Rafeiro  studied a new space of functions of generalized bounded variation. There the authors introduced the notion of bounded variation in the Wiener sense with the exponent p(×)-variable. In the same year, O. Mejia, N. Merentes and J. L. Sánchez in  , proved some properties in this space, for the composition operator and showed a structural theorem for mappings of bounded variation in the sense of Wiener with the exponent p(×)-variable.
The main purpose of this paper is threefold: First, we provide extension of the space of generalized bounded variation present in  and  in the sense Wiener-Korenblum and we give a detailed description of the new class formed by the functions of bounded variation in the sense of Wiener-Korenblum with the exponent p(×)- variable. Second, we prove a necessary and sufficient condition for the acting of composition operator
(Nemystskij) on the space and, third we show that any uniformly bounded composition operator that maps the space into itself necessarily satisfies the so called Matkowski’s weak condition.
We use throughout this paper the following notation: we will denote by
the diameter of the image (or the oscillation of f on) and by a number between.
The class of bounded variation functions exhibit many interesting properties that it makes them a suitable class of functions in a variety of contexts with wide applications in pure and applied mathematics (see  and  ). Since C. Jordan in 1881 (see  ) gave the complete characterization of functions of bounded variation as a difference of two increasing functions, the notion of bounded variation functions has been generalized in different ways.
Definition 2.1. Let be a function. For each partition of, we define
where the supremum is taken over all partitions of the interval. If, we say that f has bounded variation. The collection of all functions of bounded variation on is denoted by.
A generalization of this notion was presented by N. Wiener (see  ) who introduced the notion of p-variation as follows.
Definition 2.2. Given a real number, a partition of, and a function. The nonnegative real number
is called the Wiener variation (or p-variation in Wiener’s sense) of f on where the supremum is taken over all partitions.
In case that, we say that f has bounded Wiener variation (or bounded p-variation in Wiener’s sense) on. The symbol will denote the space of functions of bounded p-variation in
Wiener’s sense on.
Other generalized version was given by B. Korenblum in 1975  . He considered a new kind of variation, called k-variation, and introduced a function k for distorting the expression in the partition if self, rather than the expression in the range. On advantage of this alternative approach is that a function of bounded k-variation may be decomposed into the difference of two simpler functions called k-decreasing functions.
Definition 2.3. A function is called a distortion function (k-function) if k satisfies the following properties:
1) k is continuous with and;
2) k is concave and increasing;
B. Korenblum (see  ), introduced the definition of bounded k-variation as follows.
Definition 2.4. Let k be a distortion function, f a real function, and a partition of the interval. Let one consider
where the supremum is taken over all partitions of the interval. In the case one says that f has bounded k-variation on and one will denote by the space of functions of bounded k-variation on.
Some properties of k-function cab be found in    .
In 2013 R. Castillo, N. Merentes and H. Rafeiro  introduce the notation of bounded variation space in the Wiener sense with variable exponent on and study some of its basic properties.
Definition 2.5. Given a function, a partition of the interval, and a function. The nonnegative real number
is called Wiener variation with variable exponent (or p(×)-variation in Wiener’s sense) of f on where is a tagged partition of the interval, i.e., a partition of the interval together with a finite sequence of numbers subject to the conditions that for each i,.
In case that, we say that f has bounded Wiener variation with variable exponent (or bounded p(×)-variation in Wiener’s sense) on. The symbol will denote the space of functions of bounded p(×)-variation in Wiener’s sense with variable exponent on.
Remark 2.6. Given a function
1) If for all x in, then.
2) If for all x in and, then.
In  , O. Mejia, N. Merentes and J. L. Sánchez proved some properties in this space, for the composition operator and show a structural theorem for mappings of bounded variation in the sense of Wiener with the exponent p(×)-variable.
Now, we generalized the notion of bounded variation space in the sense of Wiener-Korenblum with variable exponent on. For this, we defined bellow the bounded p(×)-variation in the sense of Wiener-Korenblum with exponent variable.
Definition 2.7. Given a function, a partition of the interval, be a distortion function and a function. The nonnegative real number
is called Wiener-Korenblum variation with variable exponent (or p(×)-variation in the sense of Wiener-Korenblum) of f on where is a tagged partition of the interval, i.e., a partition of the interval together with a finite sequence of numbers subject to the conditions that for each i,.
In case that, we say that f has bounded Wiener-Korenblum variation with variable exponent (or bounded p(×)-variation in the sense of Wiener-Korenblum) on. The symbol
will denote the space of functions of bounded p(×)-variation in the sense Wiener-Korenblum with variable exponent on.
Remark 2.8. Given a function
1) If for all in, then.
2) If for all in and, then.
Example 2.9. Let be a function such that and for. Then, from mean value theorem, we have
3. Properties of the Space
Theorem 3.1. Let and be a distortion function then.
Proof. Let, and be a partition of the interval. Then, by the subadditivity, we have:
Then considering the supremum of the left side we get
therefore, and. W
Remark 3.2. From this result we deduce that every function of bounded p(×)-variation in of Wiener’s sense with variable exponent on the interval is a bounded p(×)-variation in the Wiener-Korenblum sense on the interval.
Now we will see that the class of function of bounded p(×)-variation in the sense of Wiener-Korenblum has a structure of vector space.
Theorem 3.3. Let, then the set is a vector space.
Proof. Let, then for each partition of, and is a tagged partition of the interval, we obtain:
Now adding from to we get
Since p(×) is bounded, then there is a such that for all, and we obtain
In other word, if, then the function is of bounded p(×)-variation in the sense of Wiener-Korenblum with variable exponent on and
On the other hand, since p(×) is bounded, there exists such that
therefore,. So, is a vector space. W
Proposition 3.4. Given a function, the variation is convex.
Proof. Let and. By Theorem 3.3. Since for the function is convex, then we get
Definition 3.5. (Norm in)
Let be a function that belongs to. Then
Theorem 3.6. is a normed space.
Proof. Let,. Then, we have that:
a) since and.
c) Fix and; then and. Now let. Then by convexity of
d) Let us now prove that if and only if. If, then for all, and so. Conversely, suppose that, i.e.,
then and, we get
without loss of generality, considering the partition we get
Hence, for all and, therefore. W
In the following, we show that endowed with the norm is a Banach space.
Theorem 3.7. Let be a function, then is a Banach space.
Proof. Let be a Cauchy sequence in, then given, there is such that for we have
Thus, for all and, we have that
by properties of function, we get
In consequence, the sequence, is a uniformly sequence of Cauchy, on the interval. Since is complete, there exists a function f defined on such that
We will show that converge on the norm.
Since the is a Cauchy sequence there is a such that
From the fact that converge uniformly to the function f on the interval, we get
Therefore, the sequence converge to the function f on the norm.
Thus is a Banach space. W
The following properties of elements of allow us to get characterizations of them.
Lemma 3.8. (General properties of the p(×)-variation) Let be a arbitrary map and be a distortion function. We have
(P1) Minimality: if, then
(P2) Change of variable: if and is a (not necessarily strictly) monotone function, then
Proof. (P1) Let,.
(P2) Let, a (not necessary strictly) monotone function, a tagged partition of the interval, and with, then
On the other hand, if a partition of is such that for, then there exist such that and again by the monotonicity of:
(P3) By monotonocity of we get
On the other hand, for any number there is a partition, with. We define a partition of the interval, then and . W
In the next section we will be dealing with the composition operator (Nemitskij).
4. Composition Operator between the Space
In any field of nonlinear analysis composition operators (Nemytskij), the superposition operators generated by appropriate functions, play a crucial role in the theory of differential, integral and functional equations. Their analytic properties depend on the postulated properties of the defining function and on the function space in which they are considered. A rich source of related questions is the monograph by J. Appell and P. P. Zabrejko  and J. Appell, J. Banas, N. Merentes  .
The composition operator problem refers to determining the conditions on a function, such that the composition operator, associated with the function h, maps a space of functions into itself   . There are several spaces where the composition operator problem has been resolved. In 1961, A. A. Babaev  showed that the composition H, associated with the function, maps the space of the Lipschitz functions into itself if and only if h is locally Lipschitz; in 1967, K. S. Mukhtarov  obtained the same result for the space of the Hölder functions of order .
The first work on the composition operator problem in the space of functions of bounded variation was made by M. Josephy in 1981,  . Other work of this type have been preformed over,
, , , , , , and (see  ).
Now, we define the composition operator. Given a function, the composition operator H, associated to a function f (autonomous case) maps each function into the composition function, given by
More generally, given, we consider the operator H, defined by
This operator is also called superposition operator or susbtitution operator or Nemytskij operator. In what follows, will refer (9) as the autonomus case and to (10) as the non-autonomus case.
In order to obtain the main result of this section, we will use a function of the zig-zag type such as the employed by J. Appell et al.   that the locally Lipschitz condition of the function h is a necessary and suffi-
cient condition such that and that in this situation H is bounded.
One of our main goals is to prove a result in the case when h is locally Lipschitz if and only if the composition operator maps the space of functions of bounded p(×)-variation into itself.
The following lemma, established in  , will be useful in the proof of our main Theorem (Theorem 4.2).
Lemma 4.1. Let, , then
Theorem 4.2. Let H be a composition operator associated to. H maps the space into itself if and only if h is locally Lipschitz.
Proof. We may suppose without loss generality that. First, let be locally Lipschitz on, and let. Then for some. Considering the local Lipschitz condition
for, for any partition we obtain the estimate
This shows that for, , and hence as claimed.
The proof of the only if direction will be by contradiction, that is we assume and h is not locally Lipschitz. Since the identity function belong to, then and therefore h is bounded in the interval. Without loss of generality we may assume that
Since h is not locally Lipschitz in 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 In this way for any increasing sequence of positive real numbers that converge to infinite, that we will define later, we can choose sequences, , such that
In addition choose such that
Considering subsequence if it necessary, we can assume without loss of generality that the sequence is monotone increasing.
Since is compact, from inequality (13) we have that exist subsequences of and that we will denote in the same way, and that converge to.
Since the sequence is a Cauchy sequence we can assume (taking subsequence if it is necessary) that
Again considering subsequences if needed and using the properties of the function we can assume that
Consider the new sequence defined by
From of inequalities (12) and (13) it follows that, therefore
Consider the sequence defined recursively by
This sequence is strictly increasing and from the relations (14) and (15), we get
Then to ensure that, is sufficient to suppose that.
We define the continuous zig-zag function, as shown below
We can write each interval, as the union of the family of non-overlapping intervals
And function u is defined on as follows
In all these situations the slopes of these segments of lines is 1.
Hence, we have for, the absolute value of the slope of the line segments in these ranges are bounded by 1, as shown below
We will show that.
Let, then there are the following possibilities for the location of s and t on.
Case 1: If are in the same interval
From relations (16), (17) and (18) follows.
Case 2: If are in two different intervals
There are several possibilities:
). By Lemma 4.1 and relations (16) and (17) we have
If proceed as).
If, again using the Lemma 4.1 and relations (16), (17) and (18) we obtain
Case 3: If.
From Lemma 4.1 and the second case, we conclude
Case 4: If.
Then from Lemma 4.1
Case 5: If.
From Lemma 4.1 and Case 4
Case 6: If
In this circumstance and the situation is trivial. Therefore we have that
So u is Lipschitz in. Moreover, for each partition of interval of the form
and, using the inequality (13), convexity of the function and definition of, we have
As the serie diverge, , which is a contradiction. W
5. Uniformly Continuous Composition Operator
In a seminal article of 1982, J. Matkowski  showed that if the composition operator H, associated with the function, maps the space of the Lipschitzian functions into itself and is a globally Lipschitzian map, then the function h has the form
There are a variety of spaces besides that verify this result  . The spaces of Banach that fulfill this property are said to satisfy the Matkowski property  .
In 1984, J. Matkowski and J. Miś  considered the same hypotheses on the operator H for the space of the function of bounded variation and concluded that (19) is true for the regularization of the function h with respect of the first variable; that is,
where. The spaces that satisfy this condition are said to verify weak Matkowski property,  .
In this section, we give the other main result of this paper, namely, we show that any uniformly bounded composition operator that maps the space into itself necessarily satisfies the so called Matkowski’s weak condition.
First of all we will give the definition of left regularization of a function.
Definition 5.1. Let, its left regularization of mapping f is the function given as
We will denote by the subset in which consists of those functions that are left continuous on.
Lemma 5.2. If, then.
Thus, if a function, then its left regularization is a left continuous function, i.e.,.
Also, we will denote by the subset in which consists of those functions that are left continuous on.
Lemma 5.3. If, then.
Proof. By Lemma 5.2, we have. Then, by Theorem 3.1,.
Thus, if a function, then its left regularization is a left continuous function, i.e.,. In consequence,.
Another lemma useful for the follow theorem is developed below:
Lemma 5.4. Let, be a distortion function, and. Then if and only if.
Proof. Let. Suppose that; then by definition of there exists such that and. Since, for the function is convex, we have:
Conversely, assume, then; hence.
Theorem 5.5. Suppose that the composition operator H generated by maps into itself and satisfies the following inequality
for some function. Then, there exist functions such that
where is the left regularization of for all.
Proof. By hypothesis, for fixed, the constant function belongs to. Since H maps into itself, we have. By Lemma 5.2 the left regularization for every.
From the inequality (20) and definition of the norm we obtain for,
From the inequality (22) and Lemma 5.2, if then
Let, and let be the equidistant partition defined by
Given with, define by
Then the difference satisfies
Consequently, by the inequality (20)
From the inequality (23) and the definition of p(×)-variation in the sense of Wiener-Korenblum we have
However, by definition of the functions and,
Since for all, , and passing to the limit as, then
So, we conclude that satisfies the Jensen equation in (see  , page 315). The continuity of with respect of the second variable implies that for every there exist such that
Because, and, for each, we obtain that. W
J. Matkowski  introduced the notion of a uniformly bounded operator and proved that any uniformly bounded composition operator acting between general Lipschitz function normed spaces must be of the form (21).
Definition 5.6. ( , Def. 1) Let and be two metric (or normed) spaces. We say that a mapping is uniformly bounded if, for any there exists a nonnegative real number such that for any nonempty set we have
Remark 5.7. Every uniformly continuous operator or Lipschitzian operator is uniformly bounded.
Theorem 5.8. Let and H be the composition operator associated with h. Suppose that H maps into itself and is uniformly continuous, then, there exist functions such that
where is the left regularization of for all.
Proof. Take any and such that
Since by the uniform boundedness of H, we have
and therefore, by the Theorem 5.5 we get