Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz ()
1. Introduction
Let
,
be real normed spaces, C be a convex cone in X and I be an arbitrary real interval. Let
denote the family of all non-empty bounded, closed and convex subsets of Y. For a given set-valued
function
we consider the composition (superposition) Nemytskij operator
defined by
for
. It is shown that if H maps the space
of function of bounded j-variation in the sense of Riesz into the space
of closed bounded convex valued functions of bounded y-variation in the sense of Riesz, and H is uniformly bounded, then the one-side regularizations
and
of h with respect to the first variable exist and are affine with respect to the second variable. In particular,
(1)
for some functions
and
, where
stands for the space of all linear mappings acting from C into
. 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.
Some ideas due to W. Smajdor [9] and her co-workers [10] [11] , V. Chistyakov [12] , as well as J. Matkoswki and M. Wróbel [6] [7] are applied.
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 j-variation in the sense of Riesz.
2. Preliminaries
Let
be the set of all convex functions
such that
and
for
.
Remark 2.1. If
, then
is continuous and strictly increasing. An usually,
stands for the set of all functions
.
Definition 2.2. Let
and
be a normed space. A function
is of bounded j-variation in the sense of Riesz in the interval I, if
(2)
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
, and in the sense of Riesz [14] if
. The general Definition 2.2 was introduced by Medvedev [15] .
Denote by
the set of all functions
such that
for some
.
is a normed space endowed with the norm
(3)
where
and
.
For
the linear normed space
was studied by Ciemnoczołowski and Orlicz [16] and Merentes et al. [5] . The functional
is called Luxemburg-Nakano-Orlicz seminorm (see [17] -[19] ).
Let
be a normed real vector space. Denote by
the family of all nonempty closed bounded convex subset of Y equipped with the Hausdorff metric D generated by the norm in Y:
(4)
Given
, we put
and we introduce the operation
in
defined as follows:
(5)
where
stands for the closure in Y. The class
with the operation
is an Abelian semigroup, with {0} as the zero element, which satisfies the cancelation law. Moreover, we can multiply elements of
by nonnegative number and, for all
and
, the following conditions hold:
(6)
Since,
(7)
is an abstract convex cone, and this cone is complete provided Y is a Banach space (cf. [9] [12] [20] ).
Definition 2.3. Let
and
. We say that F has bounded j variation in the sense of Riesz, if
(8)
where the supremum is taken over all finite and increasing sequences
,
,
.
Let
(9)
For
put
(10)
where
(11)
and
(12)
where the supremum is taken over all finite and increasing sequences
.
Lemma 2.4. ([12] , Lemma 4.1 (c)) The
and
. Then for ![]()
(13)
Let
,
be two real normed spaces. A subset
is said to be a convex cone if
for all
and
. It is obvious that
. Given a set-valued function
we consider the composition operator
generated by h, i.e.,
(14)
A set-valued function
is said to be *additive, if
(15)
and *Jensen if
(16)
The following lemma was established for operators C with compact convex values in Y by Fifer ( [21] , Theorem 2) (if
) and Nikodem ( [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 2.5. ([12] , Lemma 12.2) Let C be a convex cone be in a real linear space and let
be a Banach space. A set-valued function
is *Jensen, if and only if, there exists an *additive set-valued function
and a set
such that
(17)
for all
.
For the normed spaces
,
by
, briefly
, we denote the normed space of all additive and continuous mappings
.
Let C be a convex cone in a real normed space
. From now on, let the set
consists of all set-valued function
which are *additive and continuous (so positively homogeneous), i.e.,
(18)
The set
can be equipped with the metric defined by
(19)
3. Some Results and Its Consequences
For a set
, we put
(20)
Theorem 3.1. Let
be a real normed space,
a real Banach space,
a convex cone,
an arbitrary interval and let
. Suppose that set-valued function
is such that, for any
the function
is continuous with respect to the second variable. If the composition operator H generated by the set-valued function h maps
into
, and satisfies the inequality
(21)
for some function
, then the left and right regularizations of h, i.e., the functions
and
defined by
![]()
exist and
![]()
for some functions
,
,
and
, where
,
, and
.
Proof. For every
, the constant function
,
belongs to
. Since H maps
into
, the function
belongs to
. By
( [12] , Theorem 4.2), the completeness of
with respect to the Hausdorff metric implies the existence of the left regularization
of h. Since H satisfies the inequality (21), by definition of the metric
, we obtain
(22)
According to Lemma 2.4, if
, the inequality (22) is equivalent to
(23)
Therefore, if
,
,
,
, the definitions of the operator H and the functional
, imply
(24)
For
, we define the function
by
(25)
Let us fix
. For an arbitrary finite sequence
and
, the functions
defined by
(26)
belongs to the space
. It is easy to verify that
![]()
whence
![]()
and, moreover
![]()
Applying (24) for the functions
and
we get:
(27)
All this technique is based on [12] . From the continuity of
and the definition of
, passing to the limit in (27) when
, we obtain that
(28)
that is
(29)
Hence, since
is arbitrary, we get,
![]()
and, as
only if
, we obtain
![]()
Therefore
(30)
for all
and all
.
Thus, for each
, the set-valued function
satisfies the *Jensen functional equation.
Consequently, by Lemma 2.5, for every
there exist an *additive set--valued function
and a set
such that
(31)
which proves the first part of our result.
To show that
is continuous for any
, let us fix
. By (7) and (31) we have
(32)
Hence, the continuity of h with respect to the second variable implies the continuity of
and, consequently, being *additive,
for every
. To prove that
let us note that the *additivity of
implies
. Therefore, putting
in (31) we get
(33)
which gives the required claim.
The representation of the right regularization
can be obtained in a similar way.
Remark 3.2. If the function
is right continuous at 0 and
, then 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.
Lemma 3.3. Let
be a real normed space,
a real Banach space, C a convex cone in X and suppose that
. If the composition operator H generated by a set-valued function
maps
into
, and there exists a function
right continuous at 0 with
, such that
(34)
then
![]()
for some
,
,
and
.
4. Uniformly Bounded Composition Operator
Definition 4.1. ([8] , Definition 1) Let X and Y be two metric (normed) spaces. We say that a mapping
is uniformly bounded if for any
there is a real number
such that for any nonempty set
we have
(35)
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
be a real normed space,
be a real Banach space,
be a convex cone,
be an arbitrary interval and suppose
. If the composition operator
generated by a set-valued function
maps
into
, and is uniformly bounded, then
![]()
for some functions
,
,
and
, where
,
, and
.
Proof. Take any
and arbitrary
such that
(36)
Since
, by the uniform boundedness of H, we have
(37)
that is
(38)
and the result follows from Theorem 3.1.
Acknowledgements
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.