Nemytskii Operator in the Space of Set-Valued Functions of Bounded-Variation

In this paper we consider the Nemytskii operator, i.e., the composition operator defined by        , Nf t H t f t  , where H is a given set-valued function. It is shown that if the operator maps the space of functions bounded N 1  -variation in the sense of Riesz with respect to the weight function  into the space of set-valued functions of bounded 2  -variation in the sense of Riesz with respect to the weight, if it is globally Lipschitzian, then it has to be of the form , where Nf         t A t f t B t     A t is a linear continuous set-valued function and is a set-valued function of bounded B 2  -variation in the sense of Riesz with respect to the weight.


Introduction
In [1], it was proved that every globally Lipschitz Nemytskii operator a b cc Y  into itself admits the following representation: where   A t is a linear continuous set-valued function and is a set-valued function belonging to the space a b cc of bounded -variation in the sense of Riesz, where q < 1 q p   N , and is globally Lipschitz.In [7], they showed a similar result in the case where the Nemytskii operator maps the space is globally Lipschitz.N While in [8], we generalize article [6] by introducing a weight function.Now, we intend to generalize [7] in a similar form we did in [8], i.e., the propose of this paper is proving an analogous result in which the Nemytskii operator maps the space of setvalued functions of bounded 1  -variation in the sense of Riesz with a weight  into the space f set-valued functions of bounded  .The first such theorem for single- valued functions was proved in [2] on the space of Lipschitz functions.A similar characterization of the Nemytskii operator has also been obtained in [3] on the space of set-valued functions of bounded variation in the classical Jordan sense.For single-valued functions it was proved in [4].In [5,6], an analogous theorem in the space of set-valued functions of bounded -variation in the sense of Riesz was obtained.Also, they proved a similar result in the case in which that the Nemytskii operator N maps the space of functions of bounded -variation in the sense of Riesz into the space of set-valued functions such that if and only if , and as x   .Let  be the set of all convex continuous functions that satisfy Definition 2.1.

Definition 2.2 Let  ,
X   be a normed space and  be a  -function.Given I   be an arbitrary (i.e., closed, half-closed, open, bounded or unbounded) fixed interval and : I    a fixed continuous strictly increasing function called a it is weight.If    , we define the (total) generalized  -variation  of the function : f I  X with respect to the weight function  in two steps as follows (cf. [9]).
is a closed interval and is a partition of the interval I (i.e., , : .
Denote by the set of all partitions of    , a b , we set The set of all functions of bounded generalized variation with weight  will be denoted by , is absolutely continuous) and its almost everywhere derivative Recall that, as it is well known, the space

 
RV I  with I,  and  as above and endowed with the norm is a Banach algebra for all . 1 q  Riesz's criterion was extended by Medvedev [11]: if . Functions of bounded generalized  -variation with    and id   (also called functions of bounded Riesz-Orlicz  -variation) were studied by Cybertowicz and Matuszewska [12].They showed that if , and that the space is a semi-normed linear space with the Luxemburg-Nakano (cf. [13,14])seminorm given by Later, Maligranda and Orlicz [15] proved that the space is a Banach algebra.

Generalization of Medvedev Lemma
We need the following definition: For φ convex, (1) is just , , BV f a b of functions of bounded variation.In the particular case when X   and 1 < < p  , we have the space Moreover, let  be a function strictly increasing and Since  is strictly increasing, the concept of "   measure " coincides with the concept of "measure 0" of Lebesgue.[cf.[16], 25].
The space of all absolutely continuous functions   : , f a b   , with respect to a function  strictly in- creasing, is denoted by AC   . Also the following characterization of [17,18] is well-known: Also the following is a generalization of Medvedev Lemma [11]: Theorem 3.6 (Generalization a Medvedev Lemma) π : f be a sequence of step functions, defined by . .
which is what we wished to demonstrate.
cc X be the family of all non-empty convex compact subsets of X and be the Hausdorff metric in where , inf , : Definition 4.1 Let    ,  a fixed continuous strictly increasing function and . We say that F has bounded  -variation in the sense of where the supremum is taken over all partitions of π   and for some 0 , both equipped with the metric where Copyright © 2013 SciRes.APM lity, we get the composition operator defined by: We denote by the space of all setvalued function , i.e., additive and positively homogeneous, we say that A is linear if By ( 1), In the proof of the main results of this paper, we will use some facts which we list here as lemmas.
satisfies the Jensen equation This proves the continuity of F at .Thus Now, we are ready to formulate the main result of this work.
  be two convex  -functions in X , strictly increasing, that satisfy 1  condition and such that there exists constants and 0 T with c if and only if there exists an additive set-valued function and a set such that We will extend the results of Aziz, Guerrero, Merentes and Sánchez given in [8] and [21] to set-valued functions of  -bounded variation with respect to the weight function  .

Main Results
Lemma 5.1 If    such that satisfies the 1  condition and for all partitions of   , a b , in particular given and if it is globally Lipschitz, then the set-valued function H satisfies the following conditions: 1) For every 2) There are functions Proof. 1) Since is globally Lipschitz, there exists a constant Using the definitions of the operator and metric , , ; , : , , , , , inf 0 : 1 , , , , , we obtain and 2  satisfy , , ,  , , .
Define the auxiliary function From the definition of From ( 16), we get Hence, Copyright © 2013 SciRes., , , , 1 Hence, substituting in inequality (5) the particular functions i f   1, 2 i  defined by (19) and taking     for all

 
, , , t a b x y K   .
By Lemma 4.3 and the inequality (24), we have Now, we have to consider the case Then the function , and consider , we obtain , , , , , where Let us fix , x y K  and define the functions equality (28) the particular fu Hence, substituting in the in nctions i f   defined by (29), we obtain is continuous for all .Hence letting z K  0 t t  in the inequality (30), we get x y Since H is convex, we have  ,

H t Htx Hty
is continuous, and The Nemytskii operator maps th Consequently the set-valued function H has to e of b the form   be two convex  -functions in X , strictly increassing, satisfying 1  condition and i.e., the Nemytskii operator is constant.

Nemytskii oper Li
Proof.Since the ator N is globally pschizian between

Using e definitions o
N and of the metric x K  Let us fix and define the functions The functions Hence, substituting in the inequality (36) the auxiliary functions defined by (37), we obtain By Lemma 4.3 and the above inequality, we get The functions π From the definition of 1 f and 2 f , we obtain

2 
valued functions of bounded  -variation in the sense of Riesz into the space of set-valued functions of bounded 2 RW a  -variation in the sense of Riesz and


-variation in the sense of Riesz with a weight  and N is globally Lipschitz.
space with the Lebesgue-Stieltjes measure defined in  -algebra  and 

,
a b by Lemma 3.5 and f   exist a.e. on   , a b .For every n   , we consider  where  41)  .