Locally Defined Operators and Locally Lipschitz Composition Operators in the Space WBVp(·)([a, b])

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1. Introduction

This paper lies in the field of variable exponent function spaces, exactly we will deal with the space of bounded -variation in Wiener’s sense with vari- able exponent (see [1] , [2] ).

Variable exponent Lebesgue spaces appeared in the literature in 1931 in the paper by Orlicz [3] . He was interested in the study of function spaces that contain all measurable functions such that

for some and satisfying some natural assumptions, where is an open set in. This space is denotated by and it is now called Orlicz space. However, we point out that in [3] the case corresponding to variable exponents is not included. In the 1950’s, these problems were systematically studied by Nakano [4] , who developed the theory of modular function spaces. Nakano explicitly mentioned variable exponent Lebesgue spaces as an example of more general spaces he considered, see Nakano [4] p. 284. In 1991, Kováčik and Rákosník [5] established several basic properties of spaces and with variable exponents. Their results were extended by Fan and Zhao [6] in the framework of Sobolev spaces.

With the emergence of nonlinear problems in applied sciences, standard Lebesgue and Sobolev spaces demostrated their limitations in applications. The class of nonlinear problems with variable exponents growth is a new research field and it reflects a new kind of physical phenomena.

It is well known that the class of nonlinear operator equations of various types has many useful applications in describing numerous problems of the real world. A number of equations which include a given operators have arisen in many branches of science such as the theory of optimal control, economics, biological, mathematical physics and engineering. Among nonlinear operators, there is a distinguished class called composi- tion operators. Next we define such operators.

Definition 1.1. Given a function, the composition operator H, associated to a function f (autonomous case) maps each function into the composi- tion function given by

(1.1)

More generally, given we consider the operator H, defined by

(1.2)

This operator is also called superposition operator or susbtitution operator or Nemytskij operator. The operator in the form (1.1) is usually called the (autonomous) composition operator and the one defined by (1.2) is called non-autonomos.

A rich source of related questions are the excellent books by J. Appell and P. P. Zabrejko [7] and J. Appell, J. Banas, N. Merentes [8] .

E. P. Sobolevskij in 1984 [9] proved that the autonomous composition operator associate to is locally Lipschitz in the space if and only if the derivative exists and is locally Lipschitz. In recent articles J. Appell, N. Merentes, J. L. Sánchez [10] , N. Merentes, S. Rivas, J. L. Sánchez [11] and O. Mejía, N. Merentes, B. Rzepka [12] , obtained several results of the Sobolevskij type. According to the authors mentioned above the importance of these results lies in the fact that in most applications to many nonlinear problems it is sufficient to impose a local Lipschitz condition, instead of a global Lipschitz condition. In fact, they proved that Sobolevskij’s result is valid in the spaces, , , and

.

In this paper, we obtained two main results. The organization of this paper is as follows. Section 2, we gather some notions and preliminary facts, and necessary back- ground about the class of functions of bounded -variation in Wiener’s sense with variable exponent, also we expose some new properties of this space. In Section 3, we establish our first main result of the Sobolevskij type which is also valid in some spaces of functions of generalized bounded variations such as. In Section 4, we enunciate and prove our second main result related to the composition operator: If a locally defined operator maps into then it is composition operator.

2. Preliminaries

Throughout this paper, we use the following notation: Let a function and we will denote by the dia-

meter of the image (or the oscillation of f on), by a number be- tween and.

In 2013 R. Castillo, N. Merentes and H. Rafeiro [1] introduced the notion of bounded variation space in the Wiener sense with variable exponent on and present a result of compactness (Helly principle) in this space.

Definition 2.1 (See [1] ). Given a function, a partition

of the interval and a function. The nonnegative real number

is called Wiener variation with variable exponent (or -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 j,.

In case that, we say that f has bounded Wiener variation with variable exponent (or bounded -variation in Wiener’s sense) on. The symbol will denote the space of functions of bounded -variation in Wiener’s sense with variable exponent on.

Definition 2.2. (Norm in) The functional

defined by

(2.1)

where is a norm on.

Theorem 2.3 (See [1] ). Every sequence in has a subsequence conver- gent pointwise to a function

In 2015, O. Mejía, N. Merentes and J. L. Sánchez [2] showed the following properties of elements of that allow us to get characterizations of them.

Lemma 2.4 (General properties of the -variation). Let be an ar- bitrary map. We have

(P1) minimality: if, then

(P2) monotonicity: if and, then

, and

.

(P3) semi-additivity: if, then

(P4) change of a variable: if and is a (not necessarily

strictly) monotone function, then.

(P5) regularity:.

The following structural theorem is taken from [2] , this gives us a characterization of the members of.

Theorem 2.5 (see [2] ). The map is of bounded -variation if and only if there exists a bounded nondecreasing function a Hölderian map of exponent and such that on.

Given, consider the -variation function in Wiener’s sense defined by

(2.2)

Proposition 2.6. Suppose that is continuous at some point

; then, the function (2.2) is also continuous at.

Proof. Let and suppose that is continuous function at, without loss of generality we can assume that. Consider the difference

. Choose partitions and

such that

Afterwards, we choose such that for which is possible by the continuity of f at. By definition of there exist a partition and such that

Then for these y, we have

Lemma 2.7. Let. Then

Proof. Let is a tagged partition of the interval, take. Then

Thus

Proposition 2.8. Let be a sequence such that converges to f almost everywhere, with. Then

that is, the Luxemburg norm is lower semi-continuous on.

Proof. Let such that for. By the Definition 2.1, for any with exist a tagged partition of

such that

By the pointwise convergence of to exist such that

for all and,. And by the Minkowski’s in- equality, we get

therefore

hence

that is,

Passing the limit as tends, we get that is sequentially lower

semicontinuous, i.e.,

if and for all. By the Definition 2.1 it fol- lows that

Lemma 2.9 (Invariance Principle). Let be a function. Then, the com- position operator (1.1) maps the space into itself if and only if it maps, for any other choice of, the space into itself.

Proof. The function defined by

is an affine homeomorphism with inverse the function defined by

such that: and. Thus, defined by

defines a 1-1 correspondence between all partitions of and all par- titions of since v is strictly increasing. Consequently, for

, we obtain

3. Locally Lipschitz Composition Operators

In this section, we expose one of the main results of this paper. We demonstrate that a result of the Sobolevskij type is also valid in the space of bounded -variation in the Wiener’s sense with variable exponent.

Theorem 3.1. Let be a function. If the composition operator H gene- rated by h maps the space into itself then H is locally Lipschitz if and only if exist and is locally Lipschitz in.

Proof. First let us assume that is locally Lipschitz in. For we denote by the minimal Lipschitz constant of and by the supremum of on the bounded set

The finiteness of implies that H satisfies a local Lipschitz condition in the norm (norm of supremum), so we only have to prove a local Lipschitz condition for H with respect to the -norm (2.1). We do this by applying twice the mean value theorem.

Fix with. Given a partition

of, we split the index set {1, …, m} into a union of disjoint sets I and J by defining the following:

if

if

By the classical mean value theorem we find between and such that

Now, by definition of I we have

Making a simple calculation

Since and adding on we get that

Again by the mean value theorem we find between and and between and such that

and

By definition of J we have

Again a simple calculation shows that

Since and adding on we get that

Summing up both partial sums and observing that and do not de- pend on the partition we conclude that

which proves the assertion.

Conversely, suppose that H satisfies a Lipschitz condition. By assumption, the constant

(3.1)

is finite for each. Considering, in particular, both functions u and v in (3.1) constant, we see that

This shows that h is locally Lipschitz, and so the derivative exists almost every- where in. It remains to prove that exists everywhere in and is locally Lipschitz. For the proof of the first claim we show that exists in any closed interval.

Given, consider with. Let be a de-

creasing sequence of positive real numbers converging to 0; without loss of generality,

we may assume that for all. Define a sequence of functions

by

(3.2)

Since the composition operator H associate to h acts in the space, by assumption, the functions given by (3.2) belong to.

Now, we show that the sequences have uniformly bounded -variation

in Wiener’s sense for all with. In fact, let

be a partition of the interval of. For each define fun- ctions and v by

(3.3)

Then, and. Furthermore, from Lemma 2.7, (3.2) and (3.3), we

obtain the estimates

Since the partition was arbitrary, the inequality

holds for every and each with. From Lemma

2.7, the definition of the function in (3.2), and the definition of the functions and v in (3.3), we further get

hence. By Lemma 2.7, we conclude that

(3.4)

which shows that the sequence satisfies the hypotheses of Theorem 2.3.

Theorem 2.3 ensures the existence of a pointwise convergent subsequence of

; without loss of generality we assume that the whole sequence con-

verges pointwise on to some function.

Now setting, where small enough such that. By (3.3)

we note that

(3.5)

for almost all. Since the primitive of f and the function are both absolutely continuous and have the same derivative on, we conclude that they differ only by some constant on, and so exists everywhere on. From the invariance principle (Lemma 2.9), we deduce that the derivative of h exists on any interval, and so everywhere in.

It remains to prove that satisfies a local Lipschitz condition. Denoting by F the composition operator associate to the function from (3.5), we claim that, for

with, we have

(3.6)

where is the Lipschitz constant from (3.1). In fact, by Theorem 2.3 we conclude that

whenever the sequence of functions converges pointwise on to some function f. Combining this with (3.4) and the observation that

as we obtain (3.6). We conclude that the composition opera- tor F maps the space into itself, and so the corresponding function is locally Lipschitz on. By (3.5), the same is true for the function.

4. Locally Defined Operators

In this section, we present our second main result, which is related to the notion of locally defined operator. We prove that every locally defined operator mapping the space of continuous and bounded -variation in Wiener’s sense functions into itself is a composition operator (Nemytskij operator).

Definition 4.1. Let be a closed interval of the real line, and let, be function spaces. An operator is called a locally defined, or -local operator, briefly, a local operator, if for every open interval and for all functions, the implication

holds true.

Remark 4.1. For some pairs of function spaces the forms of local operators (or their representation theorems) have been established. For instance in [13] it was done is the case when and or, in [14] - [16] in the case when and are the spaces of n-times (k-times, respectively) Whitney differentiable functions, in [17] , [18] in the case when is the space of Hölder functions and, in [19] for continuous and monotone functions, in [20] in the case when for functions of bounded -variation in the sense of Wiener and and in [21] in the case when for functions of bounded Riesz-variation and.

Definition 4.2. (See [13] ) An operator is said to be

1) left-hand defined, if and only if for every and for every two functions,

2) right-hand defined, if and only if for every and for every two functions,

From now on, let, where stands for the space of continuous functions defined on I. We begin this section with some definitions.

Theorem 4.3. (See [13] ) The operator is locally defined if and only if it is left and right defined operator.

Theorem 4.4. Let. If a locally defined operator K maps

into then there exist a unique function such that, for all,

Proof. We begin by showing that for every and for every

the condition

(4.1)

implies that

To this end choose arbitrary and take an arbitrary pair of functions

which fulfil (4.1). The function defined by

belongs to. Indeed, define the functions by

and

Since, are continuous in and

. Let be a partition of I such that for some

. Then

Hence. By a similar reasoning, we have. Finally

, as is a linear space. Thus

(4.2)

Since, for all

the condition (4.2) implies that. As

according to Definition 4.2, we get

Therefore, by the continuity of and en, we obtain

Suppose now that is the left endpoint of the interval I (i.e.,). By the con- tinuity of f and g at, there exist a sequence such that:

and

(4.3)

The sequence of functions, defined by

for all, belong to the space. Indeed, by the definition of, the triangle inequality, (4.1) and (4.3), we have

and

for all. Therefore

so

(4.4)

Similar reasoning shows, that

(4.5)

From (4.4) and (4.5), we obtain that and

(4.6)

Let us observe that

(4.7)

and for all,

(4.8)

and for every there exist such that

(4.9)

Put

From (4.7), (4.8) and (4.9) the function is well defined and

(4.10)

and

(4.11)

To show that is continuous at, fix an. By the continuity of f and g at, there exist such that

(4.12)

Take an arbitrary. There exist such that and either or. Since, by triangle inequality and (4.7)

therefore, by (4.10) and (4.12)

in the case when, and by (4.11) and (4.12)

in the case when. As the continuity of at the remaining points is obvious, is continuous.

By the lower semicontinuity of (Proposition 2.8) and (4.6)

and the convergence of series implies that.

Thus there exist a function and sequence such that

According to the first part of the proof, we have

Hence, by continuity of and at, letting, we get

When is the right endpoint of I, the argument is similar.

To define the function, fix arbitrarily an, let us define a fun- ction by

(4.13)

Of course, as a constant function, belongs to. For, put

Since, by (4.13), for all functions f,

according to what has already been proved, we have

(4.14)

To prove the uniqueness of h, assume that is such that

for all and. To show that let us fix arbitrarily

and take with. From (4.14), we have

which proves the uniqueness of h.

5. Conclusion

In this paper, we get two important results. In Theorem 3.1, we show that the result of the Sobolevkij type is valid for the space of functions of bounded -variation in Wiener’s sense () on. And the Theorem 4.4, we show that if a locally defined operator K maps into then it is composition operator.

Acknowledgements

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