^{1}

^{2}

^{2}

We give a neccesary and sufficient condition on a function
such that the composition operator (Nemytskij Operator)
*H* defined by
acts in the space
and satisfies a local Lipschitz condition. And, we prove that every locally defined operator mapping the space of continuous and bounded Wiener
*p*(
·)-variation with variable exponent functions into itself is a Nemytskij com-position operator.

This paper lies in the field of variable exponent function spaces, exactly we will deal with the space

Variable exponent Lebesgue spaces appeared in the literature in 1931 in the paper by Orlicz [

for some

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

More generally, given

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 [

E. P. Sobolevskij in 1984 [

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

Throughout this paper, we use the following notation: Let a function

meter of the image

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

Definition 2.1 (See [

is called Wiener variation with variable exponent (or

In case that

Definition 2.2. (Norm in

where

Theorem 2.3 (See [

In 2015, O. Mejía, N. Merentes and J. L. Sánchez [

Lemma 2.4 (General properties of the

(P1) minimality: if

(P2) monotonicity: if

(P3) semi-additivity: if

(P4) change of a variable: if

strictly) monotone function, then

(P5) regularity:

The following structural theorem is taken from [

Theorem 2.5 (see [

Given

Proposition 2.6. Suppose that

Proof. Let

Afterwards, we choose

Then for these y, we have

Lemma 2.7. Let

Proof. Let

Thus

Proposition 2.8. Let

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

Proof. Let

By the pointwise convergence of

for all

therefore

hence

that is,

Passing the limit as

semicontinuous, i.e.,

if

Lemma 2.9 (Invariance Principle). Let

Proof. The function

is an affine homeomorphism with inverse the function

such that:

defines a 1-1 correspondence between all partitions

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

Theorem 3.1. Let

Proof. First let us assume that

The finiteness of

Fix

By the classical mean value theorem we find

Now, by definition of I we have

Making a simple calculation

Since

Again by the mean value theorem we find

and

By definition of J we have

Again a simple calculation shows that

Since

Summing up both partial sums and observing that

which proves the assertion.

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

is finite for each

This shows that h is locally Lipschitz, and so the derivative

Given

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

we may assume that

Since the composition operator H associate to h acts in the space

Now, we show that the sequences

in Wiener’s sense for all

Then,

obtain the estimates

Since the partition

holds for every

2.7, the definition of the function

hence

which shows that the sequence

Theorem 2.3 ensures the existence of a pointwise convergent subsequence of

verges pointwise on

Now setting

we note that

for almost all

It remains to prove that

where

whenever the sequence

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

Definition 4.1. Let

holds true.

Remark 4.1. For some pairs

Definition 4.2. (See [

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

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

From now on, let

Theorem 4.3. (See [

The locally defined operators have been the subject of intensive research and many applications of then can be found in the literature (See, for instance [

Theorem 4.4. Let

Proof. We begin by showing that for every

implies that

To this end choose arbitrary

belongs to

and

Since

Hence

Since, for all

the condition (4.2) implies that

according to Definition 4.2, we get

Therefore, by the continuity of

Suppose now that

The sequence of functions

for all

and

for all

so

Similar reasoning shows, that

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

Let us observe that

and for all

and for every

Put

From (4.7), (4.8) and (4.9) the function

and

To show that

Take an arbitrary

therefore, by (4.10) and (4.12)

in the case when

in the case when

By the lower semicontinuity of

and the convergence of series

Thus there exist a function

According to the first part of the proof, we have

Hence, by continuity of

When

To define the function

Of course

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

according to what has already been proved, we have

To prove the uniqueness of h, assume that

for all

which proves the uniqueness of h.

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

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. Also, we are grateful with the referees and editors for their comments and suggestions on this work.

Guerrero, J.A., Mejía, O. and Merentes, N. (2016) Locally Defined Operators and Locally Lipschitz Composition Operators in the Space . Advances in Pure Mathematics, 6, 727-744. http://dx.doi.org/10.4236/apm.2016.610059