^{1}

^{*}

^{2}

We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.

Let

function

for some functions

The first paper concerning composition operators in the space of bounded variation functions was written by J. Miś and J. Matkowski in 1984 [

Let us remark that the uniform boundedness of an operator (weaker than the usual boundedness) was introduced and applied in [

Some ideas due to W. Smajdor [

The motivation for our work is due to the results of T. Ereú et al. [

Let

Remark 2.1. If

Definition 2.2. Let

where the supremum is taken over all finite and increasing sequences

For

Denote by

where

For

Let

Given

where

Since,

Definition 2.3. Let

where the supremum is taken over all finite and increasing sequences

Let

For

where

and

where the supremum is taken over all finite and increasing sequences

Lemma 2.4. ([

Let

A set-valued function ^{*}additive, if

and ^{*}Jensen if

The following lemma was established for operators C with compact convex values in Y by Fifer ( [

Lemma 2.5. ([^{*}Jensen, if and only if, there exists an ^{*}additive set-valued function

for all

For the normed spaces

Let C be a convex cone in a real normed space^{*}additive and continuous (so positively homogeneous), i.e.,

The set

For a set

Theorem 3.1. Let

for some function

exist and

for some functions

Proof. For every

( [

According to Lemma 2.4, if

Therefore, if

For

Let us fix

belongs to the space

whence

and, moreover

Applying (24) for the functions

All this technique is based on [

that is

Hence, since

and, as

Therefore

for all

Thus, for each^{*}Jensen functional equation.

Consequently, by Lemma 2.5, for every ^{*}additive set--valued function

which proves the first part of our result.

To show that

Hence, the continuity of h with respect to the second variable implies the continuity of ^{*}additive, ^{*}additivity of

which gives the required claim.

The representation of the right regularization

Remark 3.2. If the function

Note that in the first part of the Theorem 3.1 the function

As in immediate consequence of Theorem 3.1 we obtain the following corollary Lemma 3.3.

Lemma 3.3. Let

then

for some

Definition 4.1. ([

Remark 4.2. Obviously, every uniformly continuous operator or Lipschitzian operator is uniformly bounded. Note that, under the assumptions of this definition, every bounded operator is uniformly bounded.

The main result of this paper reads as follows:

Theorem 4.3. Let

for some functions

Proof. Take any

Since

that is

and the result follows from Theorem 3.1.

The author would like to thank the anonymous referee and the editors for their valuable comments and suggestions. Also, Wadie Aziz want to mention this research was partly supported by CDCHTA of Universidad de Los Andes under the project NURR-C-584-15-05-B.

WadieAziz,NelsonMerentes, (2015) Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz. International Journal of Modern Nonlinear Theory and Application,04,226-233. doi: 10.4236/ijmnta.2015.44017