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In this paper we present the notion of the space of bounded p(·)-variation in the sense of Wiener-Korenblum with variable exponent. We prove some properties of this space and we show that the composition operator
*H*, associated with
, maps the
into itself, if and only if
*h* is locally Lipschitz. Also, we prove that if the composition operator generated by
maps this space into itself and is uniformly bounded, then the regularization of
*h* is affine in the second variable, i.e. satisfies the Matkowski’s weak condition.

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 [^{th}-variations were reconsidered in a probabilistic context by R. Dudley [

In 1997 while studying Poisson integral representations of certain class of harmonic functions in the unit disc of the complex plan B. 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 [

The main purpose of this paper is threefold: First, we provide extension of the space of generalized bounded variation present in [

(Nemystskij) on the space

We use throughout this paper the following notation: we will denote by

the diameter of the image

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 [

Definition 2.1. Let

where the supremum is taken over all partitions

A generalization of this notion was presented by N. Wiener (see [

Definition 2.2. Given a real number

is called the Wiener variation (or p-variation in Wiener’s sense) of f on

In case that

Wiener’s sense on

Other generalized version was given by B. Korenblum in 1975 [

Definition 2.3. A function

1) k is continuous with

2) k is concave and increasing;

3)

B. Korenblum (see [

Definition 2.4. Let k be a distortion function, f a real function

where the supremum is taken over all partitions

Some properties of k-function cab be found in [

In 2013 R. Castillo, N. Merentes and H. Rafeiro [

Definition 2.5. Given a function

is called Wiener variation with variable exponent (or p(×)-variation in Wiener’s sense) of f on

In case that

Remark 2.6. Given a function

1) If

2) If

In [

Now, we generalized the notion of bounded variation space in the sense of Wiener-Korenblum with variable exponent on

Definition 2.7. Given a function

is called Wiener-Korenblum variation with variable exponent (or p(×)-variation in the sense of Wiener-Korenblum) of f on

In case that

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

2) If

Example 2.9. Let

Therefore,

Theorem 3.1. Let

Proof. Let

Thus,

Then considering the supremum of the left side we get

therefore,

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

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

Proof. Let

Now adding from

Since p(×) is bounded, then there is a

In other word, if

On the other hand, since p(×) is bounded, there exists

therefore,

Proposition 3.4. Given a function

Proof. Let

Then,

W

Definition 3.5. (Norm in

Let

where

Theorem 3.6.

Proof. Let

a)

b)

Therefore,

c) Fix

Hence

Thus,

d) Let us now prove that

then

i.e.,

without loss of generality, considering the partition

then

we get

Hence,

In the following, we show that

Theorem 3.7. Let

Proof. Let

i.e.

Then

Thus, for all

then

therefore

by properties of function

then

hence

In consequence, the sequence

We will show that

Since the

From the fact that

Therefore, the sequence

Thus

The following properties of elements of

Lemma 3.8. (General properties of the p(×)-variation) Let

(P1) Minimality: if

(P2) Change of variable: if

(P3) Regularity:

Proof. (P1) Let

(P2) Let

On the other hand, if a partition

(P3) By monotonocity of

On the other hand, for any number

In the next section we will be dealing with the composition operator (Nemitskij).

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 [

The composition operator problem refers to determining the conditions on a function

The first work on the composition operator problem in the space of functions of bounded variation

Now, we define the composition operator. Given a function

More generally, given

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. [

cient condition such that

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 [

Lemma 4.1. Let

Theorem 4.2. Let H be a composition operator associated to

Proof. We may suppose without loss generality that

for

This shows that for

The proof of the only if direction will be by contradiction, that is we assume

Since h is not locally Lipschitz in

In addition choose

Considering subsequence if it necessary, we can assume without loss of generality that the sequence

Since

Since the sequence

Again considering subsequences if needed and using the properties of the function

Consider the new sequence

From of inequalities (12) and (13) it follows that

Consider the sequence defined recursively

This sequence is strictly increasing and from the relations (14) and (15), we get

Then to ensure that

We define the continuous zig-zag function

Put

We can write each interval

And function u is defined on

and

In all these situations the slopes of these segments of lines is 1.

Hence, we have for

We will show that

Let

Case 1: If

From relations (16), (17) and (18) follows

Case 2: If

There are several possibilities:

a)

b)

If

If

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

So u is Lipschitz in

and

As the serie

In a seminal article of 1982, J. Matkowski [

for some

There are a variety of spaces besides

In 1984, J. Matkowski and J. Miś [

where

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

First of all we will give the definition of left regularization of a function.

Definition 5.1. Let

We will denote by

Lemma 5.2. If

Thus, if a function

Also, we will denote by

Lemma 5.3. If

Proof. By Lemma 5.2, we have

Thus, if a function

Another lemma useful for the follow theorem is developed below:

Lemma 5.4. Let

Proof. Let

Conversely, assume

Theorem 5.5. Suppose that the composition operator H generated by

for some function

where

Proof. By hypothesis, for

From the inequality (20) and definition of the norm

From the inequality (22) and Lemma 5.2, if

Let

Given

and

Then the difference

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

Then

Since

hence,

So, we conclude that

Because

J. Matkowski [

Definition 5.6. ([

Remark 5.7. Every uniformly continuous operator or Lipschitzian operator is uniformly bounded.

Theorem 5.8. Let

where

Proof. Take any

Since

that is,

and therefore, by the Theorem 5.5 we get

This research has been partially supported by the Central Bank of Venezuela. We want to give thanks to the library staff of B.C.V for compiling the references.

O.Mejía,N.Merentes,J. L.Sánchez,M.Valera-López, (2016) The Space of Bounded p(·)-Variation in the Sense Wiener-Korenblum with Variable Exponent. Advances in Pure Mathematics,06,21-40. doi: 10.4236/apm.2016.61004