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Journal of Mathematics
Volume 2015, Article ID 925091, 4 pages
http://dx.doi.org/10.1155/2015/925091
Research Article

Locally Defined Operators in the Space of Functions of Bounded Riesz-Variation

1Universidad de Los Andes, Departamento de Física y Matemática, Trujillo, Venezuela
2Universidad Nacional Experimental del Táchira, Departamento de Matemática y Física, San Cristóbal, Venezuela
3Universidad Central de Venezuela, Escuela de Matemáticas, Caracas, Venezuela

Received 14 October 2014; Revised 11 January 2015; Accepted 16 January 2015

Academic Editor: Abdellatif Agouzal

Copyright © 2015 W. Aziz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the locally defined operator on the spaces of bounded Riesz -variation functions and we prove that those operators are the Nemytskii operator.

1. Introduction

We have an closed interval of the real line and let be function spaces . An operator is called a locally defined operator, or -local operator, briefly, a local operator [1], if for every open interval and for all functions , the implicationis true, where denotes the restriction of to .

There is a vast literature on the problem treated here, mainly compiled of definitions of locally defined operators involving a measure space (cf., e.g., [25]). Also we proved that, in general, is a composition (or Nemytskii) operator of the form for a two-variable function . Assuming additionally that is continuous in measure, the generating function can be replaced by a function satisfying the Caratheodory conditions (cf. [6]). The present paper concerns topological aspects of locally defined operators (cf. [1, 710]). For more knowledge on theory of the composition operators, see Appell and Zabrejko [11]. In [7] it was done is the case when and or . Subsequently, this result has been extended by several authors: [8, 9, 12] (for spaces of Whitney differentiable functions), [10, 13] (for space of Hölder functions), [14] (for continuous and monotone functions), and [1] (for functions of bounded -variation in the sense Wiener). In the present paper we are interested in such operators in the context of bounded Riesz-variation functions. In particular, we show that if the operator maps the space into itself and is locally defined, then is a Nemytskii composition operator.

2. Notation and Preliminaries

In this section we present some necessary notations and definitions and recall some knowledge concerning the bounded Riesz-variation.

In the sequel, , , and denote, respectively, the set of positive integers, nonnegative integers, and the set of real numbers.

Let ; be partition of , defined by . As usually, denote the family of all functions .

Given and a partition of , we definewhere the supremum is taken over all partitions of . is the classical -variation of in the sense of Riesz [15] in . A function is said to be of bounded -variation if . By we denote the Banach space of all functions of bounded -variation equipped with the norm

Lemma 1. Let be an interval and let , be fixed. Then for every sequence satisfying the conditionthere exists a function such that, for all ,

Proof. Take an arbitrary sequence satisfying (4) and define a sequence of functions , , byLet us observe thatand for every there exist such thatPutFrom (7) and (8), the function is well defined. Moreover, is nondecreasing andand by (9), for each , we obtainso . Thus the sequence tends uniformly to .
Now as for all and tends uniformly to , thenthus and therefore .

Similarly, we can get the following.

Remark 2. If , where and is a sequence satisfying the condition then there exists a function such that, for all ,

3. Locally Defined Operators

Now we can introduce the definition of the local defined operators of type .

Definition 3 (see [1]). An operator is said to be locally defined, if, for every two functions and for every open interval ,

Theorem 4. Let . If a locally defined operator maps into , then there exists a unique function such that, for all ,

Proof. We begin by showing that, for every and, for every , the conditionimplies thatTo this end choose arbitrary and take an arbitrary pair of functions which fulfil (17) (i.e., ). The function , defined bybelongs to . Indeed, define the functions bySince , and . Let be a partition of such that for some . ThenHence . By a similar reasoning, we have . Finally , as is a linear space. ThusSince, for all the condition (22) implies that .
Asby definition of a local operator, we getTherefore, by the continuity of , and at , we obtainSuppose now that is the left endpoint of the interval (i.e., . There exists a sequence such that , , and by the continuity of and at By Lemma 1 there exists a function such that for all .
There is no loss of generality in supposing that , and , .
According to the first part of the proof, we haveHence, by continuity of , , and at , letting , we getWhen is the right endpoint of , the argument is similar.
To define the function and fix arbitrarily an , let us define a function byOf course , as a constant function, belongs to . For , , put Since, by (30), for all functions ,according to what has already been proved, we haveTo prove the uniqueness of , assume that is such thatfor all and . To show that let us fix arbitrarily , and take with . From (33), we havewhich proves the uniqueness of .

Definition 5. Let and a function be fixed. The mapping , given byis said to be composition (Nemytskii or superposition) operator. The function is referred to as the generator of the operator .

As an immediate consequence of Theorem 4 we get the following.

Corollary 6. Let . If a local operator maps into , then it is a Nemytskii operator.

Note that if a local operator maps into itself then, obviously, maps into . Therefore, by Theorem 4, we get the following.

Theorem 7. Let . If a local operator maps into itself, then there exists a unique function such that, for all ,

Corollary 8. Let . If a local operator maps into itself, then it is a Nemytskii operator.

Under the additional assumption that the locally defined operator is uniformly continuous, we get a complete characterization of its generating function . Namely, we have the following.

Theorem 9. Let . If a local operator is uniformly continuous, then there exists such that

Proof. From Theorem 7 there exists a unique function such that for all , . Fix , take an arbitrary sequence with , and let be defined by , . Since ,applying the continuity of at , we get the continuity of with respect to the first variable. Thus, by [16, Theorem 1] (with ),for some . Since and , , the functions .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referee for his valuable comments and suggestions. W. Aziz wants to mention that this research was partly supported by CDCHTA of Universidad de Los Andes under the project NURR-C-584-15-05-B. J. A. Guerrero and K. Maldonado were partially funded by the Decanato de Investigación, Universidad Nacional Experimental del Táchira-Venezuela, under the Project 04-011-2015. This research has been partly supported by Central Bank of Venezuela.

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