Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz

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.


Introduction
clb Y .This considerably extends the main result of the paper [1] where the uniform continuity of the operator H is assumed.
The first paper concerning composition operators in the space of bounded variation functions was written by J. Miś and J. Matkowski in 1984 [2]; these results shown here have been verified by varying the hypothesis, in other contributions (see for example, [1] [3]- [7]).
Let us remark that the uniform boundedness of an operator (weaker than the usual boundedness) was introduced and applied in [8] for the Nemytskij composition operators acting between spaces of Hölder functions in the single-valued case and then extended to the set-valued cases in [6] for the operator with convex and compact values, in [7] for the operators with convex and closed values, and also, in [4] for the Nemytskij operator in the spaces of functions of bounded variation in the sense of Wiener.
The motivation for our work is due to the results of T. Ereú et al. [3] and Głazowska et al. [4], but only that our research is developed for some functions of bounded ϕ-variation in the sense of Riesz.

Preliminaries
Let  be the set of all convex functions : sup , where the supremum is taken over all finite and increasing sequences For ( ) ≥ condition 2 coincide with the classical concept of variation in the sense of Jordan [13] when 1 p = , and in the sense of Riesz [14] if 1 p > .The general Definition 2.2 was introduced by Medvedev [15].
Given ( ) and we introduce the operation defined as follows: ( ) where cl stands for the closure in Y.
 is an abstract convex cone, and this cone is complete provided Y is a Banach space (cf.[9] [12] [20]).
( ) . We say that F has bounded ϕ variation in the sense of Riesz, if where the supremum is taken over all finite and increasing sequences ; : : , for some 0 .
, ; where and where the supremum is taken over all finite and increasing sequences , ; and ϕ ∈  .Then for Let ( ) we consider the composition operator ( ) ( ) , , , .
A set-valued function ( ) and * Jensen if The for all x C ∈ .
For the normed spaces ( ) , we denote the normed space of all additive and continuous mappings Let C be a convex cone in a real normed space ( ) , X X ⋅ .From now on, let the set , C clb Y  can be equipped with the metric defined by . sup

Some Results and Its Consequences
For a set C X ⊂ , we put for some function γ +∞ → +∞ , then the left and right regularizations of h, i.e., the functions ( ) ( )  clb Y with respect to the Hausdorff metric implies the existence of the left regularization h − of h.Since H satisfies the inequality (21), by definition of the metric D ψ , we obtain According to Lemma 2.4, if ( ) Therefore, if For , , inf sup .
All this technique is based on [12].From the continuity of ψ and the definition of h − , passing to the limit in (27) when i s t ↑ , we obtain that 2 Hence, since m ∈  is arbitrary, we get, and, as ( )

D A t x A t x D A t x B t A t x B t D h t x h t x
Hence, the continuity of h with respect to the second variable implies the continuity of ( ) real normed spaces, C be a convex cone in X and I be an arbitrary real interval.Let ( ) clb Y denote the family of all non-empty bounded, closed and convex subsets of Y.For a given set-valued function bounded ϕ-variation in the sense of Riesz into the space functions of bounded ψ-variation in the sense of Riesz, and H is uniformly bounded, then the one-side regularizations h − and h + of h with respect to the first variable exist and are affine with respect to the second variable.In particular, mappings acting from C into ( )

Remark 2 . 1 .Definition 2 . 2 .
If ϕ ∈  , then ϕ is continuous and strictly increasing.An usually, I X stands for the set of Let ϕ ∈  and ( ) , X ⋅ be a normed space.A function I f X ∈ is of bounded ϕ-variation in the sense of Riesz in the interval I, if real vector space.Denote by ( ) clb Y the family of all nonempty closed bounded convex subset of Y equipped with the Hausdorff metric D generated by the norm in Y: Y → which are * additive and continuous (so positively homogeneous), i.e., the definitions of the operator H and the functional W ρ , imply

3 . 2 .
required claim.The representation of the right regularization h + can be obtained in a similar way.Remark If the function the assumption of the continuity of h with respect to the second variable can be omitted, as it follows from(2).Note that in the first part of the Theorem 3.1 the function is completely arbitrary.As in immediate consequence of Theorem 3.1 we obtain the following corollary Lemma 3.3.
following lemma was established for operators C with compact convex values in Y by Fifer([21], Theorem 2) (if K ) andNikodem ([22], Theorem 5.6) (if K is a cone).An abstract version of this lemma is due to W. Smajdor([9], Theorem 1).We will need the following result: Lemma 12.2) Let C be a convex cone be in a real linear space and let ( ) + =  )