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

We give a neccesary and sufficient condition on a function : f →   such that the composition operator (Nemytskij Operator) H defined by Hf f h =  acts in the space ( ) [ ] ( ) , p WBV a b ⋅ 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 composition operator.


Introduction
This paper lies in the field of variable exponent function spaces, exactly we will deal with the space 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 : u Ω →  such that set in n  .This space is denotated by L ϕ 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 W 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 composition operators.Next we define such operators.
Definition 1. 1.Given a function : h →   , the composition operator H, associated to a function f (autonomous case) maps each function 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 : Lip a b if and only if the derivative h′ 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 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 ( ) p ⋅ -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

Preliminaries
Throughout this paper, we use the following notation: Let a function  : , , : sup the conditions that for each j, where , , , , .
, , , , (P5) regularity: The following structural theorem is taken from [2], this gives us a characterization of the members of , consider the ( ) , , , Afterwards, we choose δ such that ( ) ( ) Then for these y, we have .
be a sequence such that n f converges to f almost everywhere, with And by the Minkowski's inequality, we get ( ) ( ) ( ) Passing the limit as α tends ( ) ,

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 Fix a b , we split the index set {1, …, m} into a union I J ∪ of disjoint sets I and J by defining the following: By the classical mean value theorem we find j α between ( ) j g t and ( ) Now, by definition of I we have 3 2 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j

Hf t Hg t Hf t Hg t h f t g t h f t g t
Again by the mean value theorem we find j β between ( ) j f t and ( ) 1 j f t − and j γ between ( ) j g t and ( ) By definition of J we have Again a simple calculation shows that 4 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j

Hf t Hg t Hf t Hg
1, p a b → ∞ and adding on j J ∈ we get that which proves the assertion.
Conversely, suppose that H satisfies a Lipschitz condition.By assumption, the constant is finite for each 0 r > .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 h′ exists almost everywhere in  .It remains to prove that h′ exists everywhere in  and is locally Lipschitz.For the proof of the first claim we show that h′ exists in any closed interval ; , 1.

H H h z t h z t h z t h z t H H h u t h v t h u t h v t H H Hu Hv Hu Hv
, , , m t t t π =


was arbitrary, the inequality holds for every n ∈  and each . By Lemma 2.7, we conclude that which shows that the sequence { }  the invariance principle (Lemma 2.9), we deduce that the derivative h′ of h exists on any interval, and so everywhere in  .
It remains to prove that h′ satisfies a local Lipschitz condition.Denoting by F the composition operator associate to the function f from (3.5), we claim that, for where ( ) K r is the Lipschitz constant from (3.1).In fact, by Theorem 2.3 we conclude that , a b to some function f.Combining this with (3.4) and the observation that ( ) ( ) as n → ∞ we obtain (3.6).We conclude that the composition operator F maps the space

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 ( ) is called a locally defined, or ( ) ,   -local operator, briefly, a local operator, if for every open interval J ⊂  and for all functions , f g ∈  , the implication ( ) ( ) , 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 ϕ -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 : K →   is said to be 1) left-hand defined, if and only if for every 0 s I ∈ and for every two functions 2) right-hand defined, if and only if for every 0 s I ∈ and for every two functions .
From now on, let , where ( ) C I stands for the space of continuous functions defined on I.We begin this section with some definitions.
Theorem 4.3.(See [13]) The operator : K →   is locally defined if and only if it is left and right defined operator.
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 [22], [23] and the references therein).for , . Indeed, define the functions 1 1 , : for , be a partition of I such that 1 0 .
. By a similar reasoning, we have .
Since, for all , t t I ′ ∈ ( )( ) ( )( ) ( ) ( ) according to Definition 4.2, we get Therefore, by the continuity of ( ) ( ) Suppose now that 0 s is the left endpoint of the interval I (i.e., 0 s a = ).By the con- tinuity of f and g at 0 s , there exist a sequence ( ) n n t ∈ such that: ( ) 3) The sequence of functions for all k ∈  , belong to the space Similar reasoning shows, that p ⋅ -variation in Wiener's sense with vari- λ > and ϕ satisfying some natural assumptions, where Ω is an open How to cite this paper: Guerrero, J.A., Mejía, O. and Merentes, N. (2016) Locally Defined Operators and Locally Lipschitz Composition Operators in the Space we enunciate and prove our second main result related to the composition operator: If a locally defined operator K maps . Castillo, N. Merentes and H. Rafeiro [1] introduced the notion of bounded variation space in the Wiener sense with variable exponent on [ ] , a b and present a result of compactness (Helly principle) in this space.Definition 2.1 (See [1]).Given a function Wiener variation with variable exponent (or ( ) p ⋅ -variation in Wiener's sense) of f on [ ] , a b where * π is a tagged partition of the interval [ ] , a b , i.e., a partition of the interval [ ] , a b together with a finite sequence of numbers 0

∈⋅
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 ( ) p ⋅ -variation).Let[ ] : , f a b →  be an ar- bitrary map.We have (P1) minimality: if →  is of bounded ( ) p ⋅ -variation if and only if there exists a bounded nondecreasing function [ ]

9 (⋅
Invariance Principle).Let : h →   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 c d < , the space

⋅) 1 K 2 K
variation in the Wiener's sense with variable exponent.Theorem 3.1.Let : h →   be a function.If the composition operator H generated by h maps the space into itself then H is locally Lipschitz if and only if h′ exist and is locally Lipschitz in  .Proof.First let us assume that h′ is locally Lipschitz in  .For 0 r > we denote by ( r the minimal Lipschitz constant of h′ and by r 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 ( ) p ⋅ -norm (2.1).We do this by applying twice the mean value theorem.
→ ∞ and adding on j I ∈ we get that

8 K
r do not de- pend on the partition π we conclude that . From Lemma 2.7, the definition of the function , n z h α in (3.2), and the definition of the functions n u and v in (3.3), we further get

(
of generality we assume that the whole sequence { } have the same derivative on [ ] , a b , we conclude that they differ only by some constant on [ ] , a b , and so h′ exists everywhere on [ ] , a b .From

⋅
into itself, and so the corresponding function f is locally Lipschitz on  .By (3.5), the same is true for the function h′ .

p
⋅ -variation in Wiener's sense functions into itself is a composition operator (Nemytskij operator)

Remark 4 . 1 .
For some pairs ( ) ,   of function spaces the forms local : K →   (or their representation theorems) have been established.For instance in[13] it was done is the case when Proof.We begin by showing that for every ( ) ( ) Finally by the definition of 2 , k k γ ∈  , the triangle inequality, (4.1) and (4.3), we have