Nemytskii Operator in the Space of Set-Valued Functions of Bounded φ-Variation ()
1. Introduction
In [1], it was proved that every globally Lipschitz Nemytskii operator
![](https://www.scirp.org/html/1-5300290\ce3f1ef6-9266-45db-abb2-3809a27521ec.jpg)
mapping the space
into itself admits the following representation:
![](https://www.scirp.org/html/1-5300290\4414aa91-8768-4c63-8b35-83002824356e.jpg)
where
is a linear continuous set-valued function and
is a set-valued function belonging to the space
. The first such theorem for singlevalued 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 of bounded
-variation in the sense of Riesz, where
, and
is globally Lipschitz. In [7], they showed a similar result in the case where the Nemytskii operator
maps the space
of setvalued functions of bounded
-variation in the sense of Riesz into the space
of set-valued functions of bounded
-variation in the sense of Riesz and
is globally Lipschitz.
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
-variation in the sense of Riesz with a weight
into the space
of set-valued functions of bounded
-variation in the sense of Riesz with a weight
and
is globally Lipschitz.
2. Preliminary Results
In this section, we introduce some definitions and recall known results concerning the Riesz
-variation.
Definition 2.1 By a
-function we mean any nondecreasing continuous function
such that
if and only if
, and
as
.
Let
be the set of all convex continuous functions that satisfy Definition 2.1.
Definition 2.2 Let
be a normed space and
be a
-function. Given
be an arbitrary (i.e., closed, half-closed, open, bounded or unbounded) fixed interval and
a fixed continuous strictly increasing function called a it is weight. If
, we define the (total) generalized
-variation
of the function
with respect to the weight function
in two steps as follows (cf. [9]). If
is a closed interval and
is a partition
of the interval I (i.e.,
), we set
![](https://www.scirp.org/html/1-5300290\15f3801e-85aa-4503-b133-5a323038b891.jpg)
Denote by
the set of all partitions of
, we set
![](https://www.scirp.org/html/1-5300290\9b496004-498e-44b8-9915-3faa77e8d03c.jpg)
If
is any interval in
, we put
![](https://www.scirp.org/html/1-5300290\5e8d944a-6d31-4098-b107-af9ae74cf1cc.jpg)
The set of all functions of bounded generalized
- variation with weight
will be denoted by
.
If
, and
,
,
, the
-variation
, also written as
, is the classical
-variation of
in the sense of Riesz [10], showing that
if and only if
(i.e.,
is absolutely continuous) and its almost everywhere derivative
is Lebesgue
-summable on
. Recall that, as it is well known, the space
with I,
and
as above and endowed with the norm ![](https://www.scirp.org/html/1-5300290\270bc8f2-ee71-4e53-b2f6-619ba0120627.jpg)
is a Banach algebra for all
.
Riesz’s criterion was extended by Medvedev [11]: if
, then
if and only if ![](https://www.scirp.org/html/1-5300290\205f7d3e-234b-41de-8efa-1b2541817fcf.jpg)
and
. Functions of bounded generalized
-variation with
and
(also called functions of bounded Riesz-Orlicz
-variation) were studied by Cybertowicz and Matuszewska [12]. They showed that if
, then
and that the space
![](https://www.scirp.org/html/1-5300290\f14ac06e-c535-4d6e-896d-5d76bc832a57.jpg)
is a semi-normed linear space with the LuxemburgNakano (cf. [13,14]) seminorm given by
.
Later, Maligranda and Orlicz [15] proved that the space
equipped with the norm
![](https://www.scirp.org/html/1-5300290\2c1539ac-b962-4d61-a478-242a80b09bc1.jpg)
is a Banach algebra.
3. Generalization of Medvedev Lemma
We need the following definition:
Definition 3.1 Let
be a
-function. We say
satisfies condition
if
(1)
For φ convex, (1) is just
. Clearlyfor
the space
coincides with the classical space
of functions of bounded variation. In the particular case when
and ![](https://www.scirp.org/html/1-5300290\47b668eb-838b-4062-9eac-8ffb390f5685.jpg)
, we have the space
of functions of bounded Riesz
-variation. Let ![](https://www.scirp.org/html/1-5300290\b6b3f58e-b650-4cb5-9a7a-0dd39ea200d8.jpg)
be a measure space with the Lebesgue-Stieltjes measure defined in
-algebra
and
![](https://www.scirp.org/html/1-5300290\fbb28f34-292b-4338-81ba-258c6e8ffbca.jpg)
Moreover, let
be a function strictly increasing and continuous in
. We say that
has
- measure 0, if given
there is a countable cover
by open intervals of
, such that
.
Since
is strictly increasing, the concept of “
measure
” coincides with the concept of “measure 0” of Lebesgue. [cf. [16],
25].
Definition 3.2 (Jef) A function
is said to be absolutely continuous with respect to
, if for every
, there exists
such that
![](https://www.scirp.org/html/1-5300290\ec6d7df9-4d35-421e-8de3-cbaacd92c5e5.jpg)
for every finite number of nonoverlapping intervals
,
with
and
.
The space of all absolutely continuous functions
, with respect to a function
strictly increasing, is denoted by
. Also the following characterization of [17,18] is well-known:
Lemma 3.3 Let
. Then
exists and is finite in
, except on a set of
-measure
.
Lemma 3.4 Let
. Then
is integrable in the sense Lebesgue-Stieltjes and
![](https://www.scirp.org/html/1-5300290\bb31fc35-d042-4ddf-9d71-13875c26b108.jpg)
Lemma 3.5 Let
such that satisfies the ![](https://www.scirp.org/html/1-5300290\46fa3272-3bbb-467e-8067-db236c3a2f53.jpg)
condition. If
, then
is
-absolutely continuous in
, i.e.,
![](https://www.scirp.org/html/1-5300290\eaf0efb4-ab69-465c-9a1b-50bddc1b25f9.jpg)
Also the following is a generalization of Medvedev Lemma [11]:
Theorem 3.6 (Generalization a Medvedev Lemma) Let
such that satisfies the
condition,
. Then 1) If
is
-absolutely continuous on
and
then
![](https://www.scirp.org/html/1-5300290\4a127d7b-c88a-48bb-ae5d-129b10d556df.jpg)
and
.
2) If
(i.e.,
), then
is
-absolutely continuous on
and
.
Proof.
) Since
is
absolutely continuous, there exists
a.e. in
by Lemma 3.3. Let
, ![](https://www.scirp.org/html/1-5300290\575fcc80-187c-4294-ba64-40e07cdd10cf.jpg)
![](https://www.scirp.org/html/1-5300290\fec6c7e8-1354-45a4-bb2a-855449be2d55.jpg)
by Lemma 3.4 and
is strictly increasing
![](https://www.scirp.org/html/1-5300290\27f264ff-94b4-416e-bc40-4f5456e4df84.jpg)
using the generalized Jenssen’s inequality
![](https://www.scirp.org/html/1-5300290\dd8b3193-ecf0-45db-aa53-59e34f010e05.jpg)
Let
be any partition of interval
; then
![](https://www.scirp.org/html/1-5300290\76afa175-1967-4734-b201-f67794822f1f.jpg)
and we have
.
Thus
.
) Let
. Then
is
-absolutely continuous on
by Lemma 3.5 and
exist a.e. on
.
For every
, we consider
![](https://www.scirp.org/html/1-5300290\4295ff51-4a07-48c5-bdda-5ba8d7276ef4.jpg)
a partition of the interval
define by
,
.
Let
be a sequence of step functions, defined by ![](https://www.scirp.org/html/1-5300290\d529f05b-6946-4b56-9e03-4947a6d20dff.jpg)
![](https://www.scirp.org/html/1-5300290\6f5293b4-61c9-411c-8569-3a7b23982671.jpg)
converge to
a.e. on
. It is sufficient to prove
in those points where
is
- differentiable and different from
,
for
, i.e., in
![](https://www.scirp.org/html/1-5300290\d5f5dc26-9722-481d-926c-1b99f16e04f1.jpg)
For
, and each
, there exists
such that
, so
![](https://www.scirp.org/html/1-5300290\341dc67f-a029-4e7c-b652-2ed8d990ae6a.jpg)
Therefore,
is a convex combination of points
![](https://www.scirp.org/html/1-5300290\1d0d34f6-1f2d-4fcb-8b52-91b07192bf66.jpg)
Now if
, then
and
and since
is
-differentiable for
, the expressions
![](https://www.scirp.org/html/1-5300290\60f32acb-d0c5-4eae-adfd-a94a25f6f5f4.jpg)
tend
to which is
-differentiable from
in
. So results
![](https://www.scirp.org/html/1-5300290\ffc41f52-9735-47b4-aff2-21159dc93e84.jpg)
Since
is continuous, we have
![](https://www.scirp.org/html/1-5300290\7f6bb7b8-e9ba-4b8a-8c50-31331c0ad127.jpg)
Using the Fatou’s Lemma and definition of
sequence, results that
![](https://www.scirp.org/html/1-5300290\10281499-bd76-4322-945b-c8698f6a9012.jpg)
By definition from ![](https://www.scirp.org/html/1-5300290\b9d5d6a1-feda-4a4a-91a8-0494e86b25cd.jpg)
![](https://www.scirp.org/html/1-5300290\21c45377-c997-453e-9aef-2838ad463cd5.jpg)
which is what we wished to demonstrate.
Corollary 3.7 Let
such that satisfies the
condition, then
if and only if
is
-absolutely continuous on
and
.
Also
![](https://www.scirp.org/html/1-5300290\d74a9f22-3371-4652-adc6-a12ee8c1226c.jpg)
Corollary 3.8 Let
such that satisfies the
condition. If
, then
is
-absolutely continuous on
and
![](https://www.scirp.org/html/1-5300290\47dd76f8-5b1b-4f16-a4f7-6604621c88f0.jpg)
4. Set-Valued Function
Let
be the family of all non-empty convex compact subsets of
and
be the Hausdorff metric in
, i.e.,
![](https://www.scirp.org/html/1-5300290\8bd52175-a27b-462d-bf48-ca4975d29c06.jpg)
where
, or equivalently,
![](https://www.scirp.org/html/1-5300290\879686df-2fa8-422b-9f48-bc54b19409d7.jpg)
where
(2)
Definition 4.1 Let
,
a fixed continuous strictly increasing function and
. We say that
has bounded
-variation in the sense of Riesz if
(3)
where the supremum is taken over all partitions
of
.
Definition 4.2 Denote by
(4)
and
(5)
both equipped with the metric
(6)
where
![](https://www.scirp.org/html/1-5300290\301f9bae-f2f3-4aca-92e9-a1e179882a30.jpg)
Now, let
,
be two normed spaces and
be a convex cone in
. Given a set-valued function
we consider the Nemytskii operator
generated by
, that is the composition operator defined by:
![](https://www.scirp.org/html/1-5300290\ac3270b7-1a37-484d-a8eb-0b756cd52dc6.jpg)
We denote by
the space of all setvalued function
, i.e., additive and positively homogeneous, we say that
is linear if
.
In the proof of the main results of this paper, we will use some facts which we list here as lemmas.
Lemma 4.3 ([19]) Let
be a normed space and let
be subsets of
. If
are convex compact and
is non-empty and bounded, then
(7)
Lemma 4.4 ([20]) Let
,
be normed spaces and
be a convex cone in
. A set-valued function
satisfies the Jensen equation
(8)
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
.
5. Main Results
Lemma 5.1 If
such that satisfies the
condition and
then
is continuous.
Proof. Since
, exists
such that
(9)
for all partitions of
, in particular given
, we have
(10)
Since
is convex
-function, from the last inequality, we get
(11)
By (1),
(12)
This proves the continuity of
at
. Thus
is continuous on
.
Now, we are ready to formulate the main result of this work.
Main Theorem 5.2 Let
,
be normed spaces,
be a convex cone in
and
be two convex
-functions in
, strictly increasing, that satisfy
condition and such that there exists constants
and
with
for all
. If the Nemitskii operator
generated by a set-valued function
maps the space
into the space
and if it is globally Lipschitz, then the set-valued function
satisfies the following conditions:
1) For every
there exists
, such that
(13)
2) There are functions
and
such that
(14)
Proof. 1) Since
is globally Lipschitz, there exists a constant
such that
(15)
Using the definitions of the operator
and metric
we have
![](https://www.scirp.org/html/1-5300290\7af6066e-1472-49ac-895f-4a8b60266ef7.jpg)
where
. In particular,
![](https://www.scirp.org/html/1-5300290\ded471d3-2566-4856-9f88-58278b992ea4.jpg)
for all
and
,
, where
.
Since
and
satisfy
(16)
we obtain
![](https://www.scirp.org/html/1-5300290\22b9f05d-40fc-46cf-b828-c78f45cc5b4f.jpg)
Therefore
(17)
Define the auxiliary function
by:
(18)
Then
and
![](https://www.scirp.org/html/1-5300290\fa5b1938-8a26-4156-b1d8-eb98795304f4.jpg)
Let us fix
and define the functions
by:
(19)
Then the functions
and
(20)
From the definition of
and
, we have
(21)
From (16), we get
(22)
Hence,
(23)
Hence, substituting in inequality (5) the particular functions
defined by (19) and taking
in (23), we obtain
(24)
for all
.
By Lemma 4.3 and the inequality (24), we have
![](https://www.scirp.org/html/1-5300290\8f9331cd-f12e-4093-bbe3-f8d453c372f1.jpg)
for all
.
Now, we have to consider the case
. Define the function
by
(25)
Then the function
and
![](https://www.scirp.org/html/1-5300290\bb60311e-0355-4592-a66a-8add60c00eca.jpg)
Let us fix
and define the functions
by
(26)
Then the functions
(i = 1,2) and
![](https://www.scirp.org/html/1-5300290\3a38bcd8-2473-42a9-b754-b085a6bc6d5b.jpg)
Substituting
and
, and consider
, we obtain
(27)
for all
, where
![](https://www.scirp.org/html/1-5300290\f5c2bbfa-2c32-498f-bebb-75da62902557.jpg)
By Lemma 4.3 and the above inequality, we get
![](https://www.scirp.org/html/1-5300290\5c107fc6-e0e8-4879-a023-d9ad2b739e80.jpg)
for all
. Define the function
by
![](https://www.scirp.org/html/1-5300290\32ce4bcd-6810-458d-9945-9a820195da40.jpg)
Hence
![](https://www.scirp.org/html/1-5300290\d8f18e7c-180a-4c9c-809c-5492c04c393f.jpg)
and, consequently, for every
the function
is continuous.
This completes the proof of part 1).
Now we shall prove that
satisfies equality 2).
Let us fix
such that
. Since the Nemytskii operator
is globally Lipschitzian, there exists a constant
, such that
(28)
where
. Define the function
by
![](https://www.scirp.org/html/1-5300290\dd89f91f-0533-4fad-98da-3d2b0b770722.jpg)
The function
.
Let us fix
and define the functions
by
(29)
The functions
and
.
Hence, substituting in the inequality (28) the particular functions
defined by (29), we obtain
(30)
Since
maps
into
then
is continuous for all
. Hence letting
in the inequality (30), we get
(31)
for all
and
.
Thus for all
, we have
(32)
Since
is convex, we have
(33)
for all
. Thus for all
, the set-valued function
satisfies the Jensen Equation (33). Now by Lemma 4.4, there exists an additive set-valued function
and a set
, such that
(34)
Substituting
into inequality (13), we deduce that for all
there exists
, such that
![](https://www.scirp.org/html/1-5300290\431e0112-fe3e-4411-a1d0-e1a307392e65.jpg)
consequently, for every
the set-valued function
is continuous, and
.
Since
is additive and
, then
for all
, thus
.
The Nemytskii operator
maps the space
into the space
, then
.
Consequently the set-valued function
has to be of the form
![](https://www.scirp.org/html/1-5300290\f3251ea3-6809-42b2-a687-2c1140bb3dc1.jpg)
where
and
.
Theorem 5.3 Let
,
be normed spaces,
a convex cone in
and
be two convex
-functions in
, strictly increassing, satisfying ![](https://www.scirp.org/html/1-5300290\19a0316e-cec6-4b8b-9769-2e1b459be0a4.jpg)
condition and
. If the Nemytskii operator
generated by a set-valued function
maps the space ![](https://www.scirp.org/html/1-5300290\79c4bc87-fc8c-4eb7-92d3-ce9555805fef.jpg)
into the space
and if it is globally Lipschizian, then the set-valued function
satisfies the following condition
![](https://www.scirp.org/html/1-5300290\4e236b9d-9dea-48a5-b09d-65c1581a626f.jpg)
i.e., the Nemytskii operator is constant.
Proof. Since the Nemytskii operator
is globally Lipschizian between
and the space
, then there exists a constant
, such that
(35)
Let us fix
such that
. Using the definitions of the operator
and of the metric
, we have
(36)
Define the auxiliary function
by
![](https://www.scirp.org/html/1-5300290\da73b8bc-339d-4711-8609-5ce9f1ed6091.jpg)
The function
and
![](https://www.scirp.org/html/1-5300290\7b403add-0b80-4e27-907b-16d2bbd6f57d.jpg)
Let us fix
and define the functions
by
(37)
The functions
and
![](https://www.scirp.org/html/1-5300290\c7efbb60-4c6e-4384-a2f4-ee1c8a1b1123.jpg)
Hence, substituting in the inequality (36) the auxiliary functions
defined by (37), we obtain
![](https://www.scirp.org/html/1-5300290\2c87047a-8a26-43c8-8e50-0b5c66989eb5.jpg)
By Lemma 4.3 and the above inequality, we get
![](https://www.scirp.org/html/1-5300290\d06d1d46-fac9-4f5a-bcfc-60d48e89ad9c.jpg)
Since
, letting
in the above inequality, we have
![](https://www.scirp.org/html/1-5300290\436bf54a-72d4-486f-99ff-ba4192f8a59e.jpg)
Thus for all
and for all
, we get
![](https://www.scirp.org/html/1-5300290\417345e6-156c-4bba-b735-790c54961e17.jpg)
Theorem 5.4 Let
,
be normed spaces,
a convex cone in
and
be a convex
- function in
satisfying the
condition. If the Nemytskii operator
generated by a set-valued function
maps the space
into the space
and if it is globally Lipschizian, then the left regularization
of the function
defined by
![](https://www.scirp.org/html/1-5300290\d2a3f0c1-84cb-4792-9409-f2aaab0ec5d3.jpg)
satisfies the following conditions:
• for all
there exists
, such that
• ![](https://www.scirp.org/html/1-5300290\91761f58-4fd7-4e20-83c2-a16ee8ab90c3.jpg)
•
, where
is a linear continuous set-valued function, and
.
Proof. We take
, and define the auxiliary function
by:
![](https://www.scirp.org/html/1-5300290\8aff8f03-63f8-445b-9c32-2c434752a3fd.jpg)
The function
and
![](https://www.scirp.org/html/1-5300290\2d6255c3-9b7b-42a3-97de-86787b6e77b3.jpg)
Let us fix
and define the functions
by
(38)
The functions
and
(39)
From the definition of
and
, we obtain
(40)
Since the Nemytskii operator
is globally Lipschitzian between
and
then there exists a constant
, such that
![](https://www.scirp.org/html/1-5300290\d96efaa2-b818-4618-a867-17a7bdbc052b.jpg)
for
. By Lemma 4.3, substituting the particular functions
defined by (38) in the above inequality, we obtain
(41)
for all
. By Lemma 4.3, we get
(42)
for all
and
.
In the case where
, by a similar reasoning as above, we obtain that there exists a constant
, such that
(43)
Define the function
by
(44)
Hence,
![](https://www.scirp.org/html/1-5300290\e5ddba0f-8dfd-464f-b20d-bd10db6c3fc4.jpg)
By passing to the limit in the inequality (41) by the inequality (43) and the definition of
we have for all
that there exists
, such that
![](https://www.scirp.org/html/1-5300290\c4c87e47-4b66-4ec3-8a9d-74d3a0b06b96.jpg)
Now we shall prove that
satisfies the following equality
![](https://www.scirp.org/html/1-5300290\cc7d042a-30d7-46a2-8a77-3cfbb012c786.jpg)
where
is a linear continuous set-valued functions, and
.
Let us fix
such that
. Define the partition
of the interval
by
![](https://www.scirp.org/html/1-5300290\1cbab4ab-8fad-42a9-89df-59abef18c378.jpg)
The Nemytskii operator
is globally Lipschitzian between
and
, then there exists a constant
, such that
(45)
where
![](https://www.scirp.org/html/1-5300290\c0396845-9701-4857-9e2e-f5eb810b4c7a.jpg)
and
.
We define the function
in the following way:
![](https://www.scirp.org/html/1-5300290\60447822-a6b3-41c2-8427-33297b6f0b52.jpg)
The function
and
.
Let us fix
and define the functions
by:
(46)
The functions
and
.
Substituting in the inequality (45) the particular functions
defined in (46), we obtain
(47)
Since the Nemytskii operator
maps the spaces
into
, then for all
, the function
. Letting
in the inequality (47), we get
![](https://www.scirp.org/html/1-5300290\3f523482-b008-4767-b19d-d164185e5f88.jpg)
for all
and
. By passing to the limit when
, we get
![](https://www.scirp.org/html/1-5300290\346b6c55-c524-4abd-a9b6-5cc34238da1a.jpg)
Since
is a convex function, then
![](https://www.scirp.org/html/1-5300290\f08e04d2-5700-411a-b3d6-82bdef8c5bda.jpg)
Thus for every
, the set-valued function
satisfies the Jensen equation. By Lemma 4.4 and by the property (a) previously established, we get that for all
there exist an additive set-valued function
and a set
, such that
![](https://www.scirp.org/html/1-5300290\34d38120-3ce2-4d74-85e2-822bdb36986b.jpg)
By the same reasoning as in the proof of Theorem 5.2, we obtain that
and
.
6. Acknowledgements
This research was partly supported by CDCHTA of Universidad de Los Andes under the project NURR-C- 547-12-05-B.