Advances in Pure Mathematics, 2013, 3, 563-575
http://dx.doi.org/10.4236/apm.2013.36072 Published Online September 2013 (http://www.scirp.org/journal/apm)
Nemytskii Operator in the Space of Set-Valued Functions
of Bounded  -Variation
Wadie Aziz
Departamento de Fsica y Matemática, Universidad de Los Andes, Trujillo, RBV
Email: [email protected]
Received June 13, 2013; revised July 23, 2013, accepted August 19, 2013
Copyright © 2013 Wadie Aziz. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
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 N maps the space of functions bounded
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
 Nf  t   A  t  f  t   B  t  , where
A  t  is a linear continuous set-valued function and B is a set-valued
function of bounded 2 -variation in the sense of Riesz with respect to the weight.
Keywords: Bounded Variation; Function of Bounded Variation in the Sense of Riesz; Variation Space; Weight
Function; Banach Space; Algebra Space
1. Introduction
In [1], it was proved that every globally Lipschitz Nemytskii operator
 Nu  t   H  t , u  t  
mapping the space Lip  a, b  ; cc Y   into itself admits
the following representation:
 Nu  t   A  t  u  t   B  t  ,
u  Lip  a, b  ; cc Y   , t   a, b  ,
where A  t  is a linear continuous set-valued function
and B is a set-valued function belonging to the space
Lip  a, b  ; cc Y   . 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 p -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 p -variation in
the sense of Riesz into the space of set-valued functions
Copyright © 2013 SciRes.
of bounded q -variation in the sense of Riesz, where
1  q  p <  , and N is globally Lipschitz. In [7], they
showed a similar result in the case where the Nemytskii
operator N maps the space RV1  a, b  ; K  of setvalued functions of bounded 1 -variation in the sense of
Riesz into the space RW2  a, b  ; cc Y   of set-valued
functions of bounded 2 -variation in the sense of Riesz
and N 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 N maps the space RV1 ,  a, b  ; K  of setvalued functions of bounded 1 -variation in the sense of
Riesz with a weight  into the space
RW2 ,  a, b  ; cc Y   of set-valued functions of bounded
2 -variation in the sense of Riesz with a weight  and
N 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  :  0,     0,  
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such that   x   0 if and only if x  0 , and
  x    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 V  f  
V  f , I ,   of the function f : I  X with respect to
the weight function  in two steps as follows (cf. [9]).
If I   a, b  is a closed interval and π is a partition
π : a  t0 < t1 <  < tn  b of the interval I (i.e., n   ),
we set
n
 f  ti   f  ti 1 
V  f , π,   :  
i 1    ti     ti 1 


   ti     ti 1  .


Denote by  the set of all partitions of  a, b  , we
set
V  f   V  f ,  a, b  ,   : sup V  f , π,   : π   .
If I is any interval in  , we put


V  f   V  f , I ,   : sup V  f ,  a, b  ,   :  a, b   I .
The set of all functions of bounded generalized  variation with weight  will be denoted by
RV  I   RV  I ,     f :  a, b   X V  f , I ,     .
I f   t   id  t   t , t  I   a, b  , a n d       q ,
  0 , q > 1 , the  -variation V  f , I ,   , also
written as Vq  f  , is the classical q -variation of f in
the sense of Riesz [10], showing that Vq  f  <  if and
only if f  AC  I  (i.e., f : I   is absolutely continuous) and its almost everywhere derivative f  is
Lebesgue q -summable on I . Recall that, as it is well
known, the space RV  I  with I,  and  as above
and endowed with the norm
f
 f  a   Vq  f  
1q
q
is a Banach algebra for all q  1 .
Riesz’s criterion was extended by Medvedev [11]: if
   , then f  RV  I  if and only if f  AC  I 
and   f   t  dt <  . Functions of bounded generalI


ized  -variation with    and   id (also called functions of bounded Riesz-Orlicz  -variation) were
studied by Cybertowicz and Matuszewska [12]. They
showed that if f  RV  I  , then


and that the space

GV  I   f   I such that lim  0V   f   0
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p  f   inf r  0 V  f r   1 .
Later, Maligranda and Orlicz [15] proved that the
space GV  I  equipped with the norm
f

 sup tI f  t   p  f 
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 1 if
lim sup
t 
 t 
t
 .
(1)
For φ convex, (1) is just lim t    t  t   . Clearly,
for id  1 the space RV  f ,  a, b  , id  coincides with
the classical space BV  f ,  a, b  of functions of bounded
variation. In the particular case when X   and 1 <
p <  , we have the space RV p ,  f ,  a, b  ; X  of functions of bounded Riesz p -variation. Let
 a, b , ,  
be a measure space with the Lebesgue-Stieltjes measure
defined in  -algebra  and
L p ,  a, b  :
 f : a, b   f is 

integrable and
b
a
f
p

d   .
Moreover, let  be a function strictly increasing and
continuous in  a, b  . We say that E   a, b  has  measure 0, if given  > 0 there is a countable cover
 an , bn  n   by open intervals of E , such that
 n 1   bn     an  <  .

Since  is strictly increasing, the concept of “ 
measure 0 ” coincides with the concept of “measure 0”
of Lebesgue. [cf. [16], § 25].
Definition 3.2 (Jef) A function f :  a, b    is said
to be absolutely continuous with respect to  , if for
every  > 0 , there exists  > 0 such that
 j 1  f  b j   f  a j     ,
n
for every finite number of nonoverlapping intervals
 a j , b j  , j  1, , n with  a j , b j    a, b and
 j 1   b j     a j    .
n
V  f   I f   t  dt ,

is a semi-normed linear space with the LuxemburgNakano (cf. [13,14]) seminorm given by
The space of all absolutely continuous functions
f :  a, b    , with respect to a function  strictly increasing, is denoted by   AC . Also the following
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characterization of [17,18] is well-known:
Lemma 3.3 Let f    AC  a, b  . Then f exists
and is finite in  a, b  , except on a set of  -measure
0.
Lemma 3.4 Let f    AC  a, b  . Then f is integrable in the sense Lebesgue-Stieltjes and
f  x   f  a    L  S   f  t  d  t  , x   a, b  .
x
a
Lemma 3.5 Let    such that satisfies the 1
condition. If f  RV  f ,  a, b  ,   , then f is  -ab-
solutely continuous in  a, b  , i.e.,
Also the following is a generalization of Medvedev
Lemma [11]:
Theorem 3.6 (Generalization a Medvedev Lemma)
Let    such that satisfies the 1 condition,
f :  a, b   X . Then
1) If f is  -absolutely continuous on  a, b  and
a  f  x   d  x  <  ,
b
then

f   t  d  t 
t1 d  t 
t2

  t2     t1 

 t 2 f   t  d  t  .
t
1
Let π : a  t0 <  < tn  b be any partition of interval
a
 , b ; then
 f  t2   f  t1  
n
    t     t     t2     t1 
i 1 
2
1 




  t j  f   t  d  t   a f   t  d  t   ,
i 1
t
i 1
b
and we have


V  f ,  a, b  ,    a f  x  d  x  .
b
Thus f  RV  f ,  a, b  ,   .
2 ) Let f  RV  a, b  . Then f is  -absolutely
continuous on  a, b  by Lemma 3.5 and f exist a.e.
on  a, b  .
For every n   , we consider
a partition of the interval  a, b  define by
and


π n :a  t0, n  t1, n    tn, n  b
f  RV  f ,  a, b  ,  

RV  f ,  a, b  ,    a f  x  d  x  .
b
t2
n
RV  f ,  a, b  ,      AC  a, b  .
t1  
ti , n  a  i
ba
, i  0,1, , n .
n
2) If f  RV  f ,  a, b  ,   (i.e., RV  f  < ), then
f is  -absolutely continuous on  a, b  and
Let  f n n be a sequence of step functions, defined by
f n :  a,b   
a  f  x   d  x   RV  f ,  a, b ,   .
 f  ti 1, n   f  ti , n 
, ti , n  t  ti 1, n

t  f n  t      ti 1, n     ti , n 

t b
0,
b
Proof. 1 ) Since f is  absolutely continuous,
there exists f  a.e. in  a, b  by Lemma 3.3. Let
t1 , t2   a, b  , t1 < t2
 f  t2   f  t1  

   t2     t1 
   t2     t1  


t
 2 f t d t 
 t       
 1
   t2     t1 
   t2     t1  


by Lemma 3.4 and  is strictly increasing
 t2 f  t  d  t  
 t  

 1
   t2     t1 

t

t





2
1




t2

f  t  d  t  
 t1  


t2
   t2     t1 
 t d  t 



1
using the generalized Jenssen’s inequality
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 f n n converge to f a.e. on  a, b  . It is sufficient
to prove  f n   f in those points where f is  differentiable and different from ti , n , i  0,, n for
n   , i.e., in
  t   a, b  f  t  exists
 ti ,n n   , i  0,1, , n
For t   , and each n   , there exists k  0, , n
such that tk , n  t < tk 1, n , so
f  tk 1,n   f  tk ,n 
fn t  

  tk 1,n     tk ,n 
  tk 1,n     t  f  tk 1,n   f  t 
  tk 1,n     tk ,n    tk 1,n     t 

  t     tk , n 
f  t   f  tk , n 
  tk 1,n     tk ,n    t     tk ,n 
.
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Therefore, f n  t  is a convex combination of points
f  tk 1, n   f  t 
  tk 1, n     t 
and
f  t   f  tk , n 
  t     tk , n 
.
  tk 1, n     t 
and
f  t   f  tk , n 
  t     tk , n 
b
4. Set-Valued Function
Now if n   , then tk , n  t and tk 1, n  t and
since f is  -differentiable for t , the expressions
f  tk 1, n   f  t 
a  f  x   d  x   RV  f ,  a, b ,   .
Let cc  X  be the family of all non-empty convex
compact subsets of X and D be the Hausdorff metric
in cc  X  , i.e.,
D  A, B  : inf t  0 : A  B  tS , B  A  tS  ,
.
tend f  t  to which is  -differentiable from f in
t . So results
lim f n  t   f  t   t   a.e. in  a, b  .
where S   y  X : y  1 , or equivalently,
D  A, B   max e  A, B  , e  A, B  : A, B  cc  X  ,
where
n 
Since  is continuous, we have





lim f n  t    lim f n  t    f  t 
n 
n 

t  .
Using the Fatou’s Lemma and definition of
sequence, results that
f n
  f n  t   d  t 
a  f  t   d  t   a nlim

b
b
n 

n 1 t



 lim inf  t i 1,n f n  t  d  t  .
n 
i 0
i ,n
By definition from f n
Definition 4.1 Let    ,  a fixed continuous
strictly increasing function and F :  a, b   cc  X  . We
say that F has bounded  -variation in the sense of
Riesz if
 f  ti 1, n   f  ti , n  
   ti 1, n     ti , n 
n  i  0 i ,n
   ti 1, n     ti , n  


 t
n 1  f  t
i 1, n   f  ti , n 
  i1,nd  t 
 lim inf  
n  i  0    t
 ti ,n
i 1, n     ti , n  


n 1  f  t
i 1, n   f  ti , n 
   ti 1, n     ti , n 
 lim inf  
n  i  0    t
   ti , n  

i

1,
n


 V  f ,  a, b   .
n
(3)
 D  F  ti  , F  ti 1   
: sup 
   ti     ti 1   ,


π i 1
   ti     ti 1  
where the supremum is taken over all partitions π of
 a, b  .
Definition 4.2 Denote by
RW*  F ,  a, b  ,  
n 1 t
i 1,n
 lim inf  t

which is what we wished to demonstrate.
Corollary 3.7 Let    such that satisfies the 1
condition, then f  RV  I  if and only if f is
 -absolutely continuous on  a, b  and
a  f  x   d  x  <  .
b
Also
a  f  x   d  x   RV  f ,  a, b ,   .
b
Corollary 3.8 Let    such that satisfies the 1
condition. If f  RV  I  , then f is  -absolutely
continuous on  a, b  and
Copyright © 2013 SciRes.
(2)
W  F ,  a, b  ,  
 lim inf a f n  t  d  t 
b
e  A, B   sup d  x, B  : x  A ,

d  x, B   inf d  x, y  : y  B .

(4)

: F :  a, b   cc  X  : W  F ,  a, b  ,    
and
RW  F ,  a, b  ,   :  F :  a, b 
(5)

 cc  X  : RW*   F    for some   0 ,
both equipped with the metric
D  F1 , F2  : D  F1  a  , F2  a  
(6)
 inf   0 : W  F1  , F2    1 ,
where
W  F1 , F2 
n
 D  F1  ti   F2  ti 1  , F1  ti 1   F2  ti   
 sup 


  ti     ti 1 
π i 1 


   ti     ti 1  .
Now, let  X ,   , Y ,   be two normed spaces and
K be a convex cone in X . Given a set-valued function
H :  a, b   K  cc Y  we consider the Nemytskii
a ,b
a ,b
operator N : K    Y   generated by H , that is
APM
W. AZIZ
the composition operator defined by:
 Nf  t  : H  t , f  t   ,
lity, we get
f :  a, b   K ;t   a, b  .

D  A  C , B  C   D  A, B  .
D  F  t  , F  t0   
t  t0
 lim
(7)
t  t0
(8)
such that satisfies the 1
F  RW  a, b  ; cc Y  ,   ,
i 1
for all partitions of
t , t0   a, b  , we have
 a, b 
Copyright © 2013 SciRes.
1
2
 x, y  X  .
(13)
(14)
Proof. 1) Since N is globally Lipschitz, there exists a
constant M   0,   such that
D2  Nf1 , Nf 2 
 M f1  f 2
(10)
1
f,f
1
2

 RV1  a, b  ,  ; K  .
(15)
Using the definitions of the operator N and metric
D2 we have

f,f
 
(12)
 0.
H  t , x   A  t  x  B  t   t   a, b  , x  K 
(9)

 D ht ,t N f , f , ht ,t N f , f
n

i i 1
1 2
i 1 i
1 2
D  Nf1  a  , Nf 2  a    inf   0 : sup 2 





t
t
π




i 1
i
i 1


1
M
2) There are functions A :  a, b   L  K , cc Y   and
B  RW2  f ,  a, b  ,  ; cc Y   such that
M >0
Since  is convex  -function, from the last inequa-
 M f1  f 2
 
D  H t, x  , H t, y   M t  x  y
, in particular given
 D  F  t  , F  t0   
   t     t0   M .

   t     t0  


lim
RW2  f ,  a, b  ,  ; cc Y   and if it is globally Lipschitz,
then the set-valued function H satisfies the following
conditions:
1) For every t   a, b  there exists M  t    0,   ,
such that
5. Main Results
 D  F  ti  , F  ti 1   
   ti     ti 1   M ,

   ti     ti 1  




This proves the continuity of F at t0 . Thus F is
continuous on  a, b  .
Now, we are ready to formulate the main result of this
work.
Main Theorem 5.2 Let  X ,   , Y ,   be normed
spaces, K be a convex cone in X and 1 , 2 be two
convex  -functions in X , strictly increasing, that
satisfy 1 condition and such that there exists constants
c and T0 with 2  t   1  ct  for all t  T0 . If the
Nemitskii operator N generated by a set-valued function
H :  a, b   K  cc Y  maps the space
RV1  f ,  a, b  ,  ; K  into the space
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  .
n

M
   t     t0 

1
  t     t0 
 1 
F  x   A t   B , x  K .
 
(11)
limD  F  t  , F  t0  
if and only if there exists an additive set-valued function
A : K  cc Y  and a set B  cc Y  such that
then F :  a, b   cc  X  is continuous.
Proof. Since F  RW  a, b  ,   , exists
such that



.
By (1),
Lemma 4.4 ([20]) Let  X ,   , Y ,   be normed
spaces and K be a convex cone in X . A set-valued
function F : K  cc Y  satisfies the Jensen equation
 x y 1
F
   F  x   F  y   , x, y  K ,
 2  2
M
   t     t0 

1
  t     t0 
 1 
We denote by L  K ; cc Y   the space of all setvalued function A : K  cc Y  , i.e., additive and positively homogeneous, we say that A is linear if
A  L  K ; cc Y   .
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  X ,   be a normed space
and let A, B, C be subsets of X . If A, B are convex
compact and C is non-empty and bounded, then
Lemma 5.1 If   
condition and
567

 RV1  a, b  ,  ; K  ,
    t     t


i


i 1   1

APM
W. AZIZ
568
where hs ,t N f1 , f2 :  Nf1  s    NF2  t  . In particular,


 D d f , f  H , t, t  , d f , f  H , t , t 

1 2
1 2
inf   0 : 2 





t
  t 


for all f1 , f 2  RV1  a, b  ,  ; K 
t  t , where
and
  
t , t   a, b  ,





 1  M f1  f 2

 t    t 

1
   t    t 

i   i 


d f1 , f2  H , s, t   H  s, f1  s    H  t , f 2  t   .
Since 1 and  2 satisfy
1
,

    t     t   1, i  1, 2, (16)


we obtain


 D d f , f  H , t, t  , d f , f  H , t , t 

1 2
1 2
inf   0 :  2 

   t    t 


    t     t   1  D




d
f1 , f 2
 H , t, t  , d f , f  H , t , t .
1
2
Therefore


D d f1 , f2  H , t , t  , d f1 , f 2  H , t , t   M f1  f 2

1
Define the auxiliary function  :  a, b    0,1 by:
      a 
, a   t

   :    t     a 
1,
t    b.

1
   t    t 

  t     t  2 
Let us fix
f i :  a, b   K
(18)
f1  f 2
1
 i  1, 2 
(17)
and define the functions
by:
f1   : x,
f 2   :    y  x   x,
Then   RV1  a, b  ,   and

1
V1  ,  a, b  ,    1 
  t     a 

x, y  K

.


(19)
   a, b  .

  t     a  .


Then the functions fi  RV1  a, b  ,  ; K 
 i  1, 2 
and
 f1  a   f 2  a 

n

 inf   0 : sup1 

π i 1


 f1  f 2   ti    f1  f 2   ti 1  
   ti     ti 1 

   ti     ti 1 


 1

(20)
From the definition of f1 and f 2 , we have
f1  f 2
1


x y
 inf  > 0 : 1 
   t     a 




   t     a   1 .



(21)
From (16), we get


x y
inf  > 0 : 1 
   t     a 




x y
  t     a   

1

   t     a   1 
1
  t     a 





(22)
Hence,
Copyright © 2013 SciRes.
APM
W. AZIZ

D d f1 , f2  H , t , t  , d f1 , f 2  H , t , t 

569


1
M   t     t   21 

   t    t  




1
  t     a  11 
  t     a 

Hence, substituting in inequality (5) the particular functions fi
 i  1, 2 
x y
(23)




defined by (19) and taking   t     a 
in (23), we obtain

1
  t     a 

D  H  t , x   H  a, x  , H  a, x   H  t , y    M

1
11 
  t     a 

21 
for all t   a, b  , x, y  K .
1   :
By Lemma 4.3 and the inequality (24), we have

1
  t     a 

D  H t, x  , H t, y   M

1
11 
  t     a 

21 



 x y ,




for all t   a, b  , x, y  K .
Now, we have to consider the case   t     b  .
1
      a 
,    a, b  .
 b    a 

1
V1 1   1 
  b     a 

f1   : x, f 2   : 1   x  y   y;    a, b  .


   b     a   1




1
  b    a 








1
 x  y 1 

1

1

b
a








1

  b    a 


Substituting   t     a  and   t     b  , and
consider     b     a  , we obtain

 MK a, b, x, y,  , 
for all x, y  K , where
Copyright © 2013 SciRes.

(26)
x y
  b     a  11 
1
2

  b    a  .


Then the functions fi  RV1  a, b  ,  ; K  (i = 1,2) and
 x y 
1
1
(25)
Let us fix x, y  K and define the functions
fi :  a, b   K  i  1, 2  by


x y
 x  y  inf   0 : 1 
   b    a 


D  H  b, x   H  a, y  , H  a, x   H  b, x  
(24)
Then the function 1  RV1  a, b  ,   and
Define the function 1 :  a, b    0,1 by
f1  f 2



 x y ,




(27)



.





K a, b, x, y, 11 ,  21
 1
   21 







1

x  y 1 

  11  1








 

APM
W. AZIZ
570
By Lemma 4.3 and the above inequality, we get

1
 1

   a, b  ;
 f1   : 2 2   x   1  2  2    y,


(29)

 f   : 1 1     x  1 1     y,    a, b .
 2  2 2 
 2
2
D  H  a, y  , H  a, x  




 1 
1


1
 M   2   x  y 1 




 
  11  1  
  

 

The functions fi  RV1  a, b  ,  ; K 
f1  f 2
for all x, y  K . Define the function M :  a, b    by


1
 21 
  t     a 


M

1

1


1



  t     a 
M t   


 M   1  1  x  y
 
2 

 





 ,a  t  b








1


1


 , t  a.


1

1

 1   


  

Hence
 M  t  x  y  x, y  X , t   a, b   ,
and, consequently, for every t   a, b  the function
H :  a, b   K  cc Y  is continuous.
This completes the proof of part 1).
Now we shall prove that H satisfies equality 2).
Let us fix t , t0   a, b  such that t0 < t . Since the
Nemytskii operator N is globally Lipschitzian, there
exists a constant M , such that
D  d u , v  H , t , t 0  , d u , v  H , t0 , t  
where

,


(28)
du ,v  H , s, t   H  s, u  s    H  t , v  t   . Define
the function 2 :  a, b    0,1 by
       a 
, a    t0 ,

   t0     a 

   t     
2    
, t0    t ,
   t     t0 
0,
t    b.


The function 2  RV1  a, b  ,   .
Let us fix x, y  K and define the functions
fi :  a, b   K by
Copyright © 2013 SciRes.

x y
2
and
.
Hence, substituting in the inequality (28) the particular
functions fi  i  1, 2  defined by (29), we obtain

 x y
 x  y 
D  H  t 0 , x   H  t , y  , H  t0 ,
  H  t,

2 
2 




1
1
 M   t     t0   21 
   t     t0 
2

(30)

 x y .


Since N maps
RV1  a, b  ,  ; K  into RW2  a, b  ,  ; cc Y   ,
then H  , z  is continuous for all z  K . Hence letting
t0  t in the inequality (30), we get

 x y
 x  y 
D  H t, x   H t, y  , H  t,
  H  t,

2 
2   (31)



 0,
D  H t, x  , H t, y 

1
 M u  v    t0     t   2 
1
   t0     t 

1
 i  1, 2 
for all t   a, b  and x, y  K .
Thus for all t   a, b  , x, y  K , we have
 x y
 x y
H  t,
  H t, x   H t, y  .
  H  t,
2 
2 


(32)
Since H is convex, we have
 x y 1
H  t,
   H  t , x   H  t , y   ,
2  2

(33)
for all t   a, b  , x, y  K . Thus for all t   a, b  , the
set-valued function H  t ,  : K  cc Y  satisfies the
Jensen Equation (33). Now by Lemma 4.4, there exists
an additive set-valued function A  t  : K  cc Y  and a
set B  t   cc Y  , such that
H  t , x   A  t  x  B  t  ,  x  K , t   a, b   .
(34)
Substituting H  t , x   A  t  x  B  t  into inequality
(13), we deduce that for all t   a, b  there exists
M  t    0,   , such that
D  A  t  x, A  t  y   M  t  x  y
 x, y  K  ,
consequently, for every t   a, b  the set-valued function A  t  : K  cc Y  is continuous, and
A  t    L  K , cc Y   .
Since A  t   is additive and 0  K , then
A  t   0 for all t   a, b  , thus H  , 0   B   .
APM
W. AZIZ
The Nemytskii operator N maps the space
RV1  a, b  ,  ; K  into the space RW2  a, b  ; cc Y   ,
then
H  , 0   B    RW2
 a, b  ,  ; K  .
Consequently the set-valued function H has to be of
the form
H  t , x   A  t  x  B  t  , t   a, b  , x  K ,
where A  t   L  K , cc Y   and
 a, b , ; cc Y   .
 X ,   , Y ,   be normed spaces,
B  RW2
Theorem 5.3 Let
K a convex cone in X and 1 , 2 be two convex
 -functions in X , strictly increassing, satisfying 1
condition and lim t 21 1  t   t   . If the Nemytskii
571
D1  Nf1 , Nf 2 
 M f1  f
H  t , x   H  t , 0   t   a, b  , x  K  ;
i.e., the Nemytskii operator is constant.
Proof. Since the Nemytskii operator N is globally
Lipschizian between RV1  a, b ,  ; K  and the space
RW1  a, b  ,  ; cc Y   , then there exists a constant M ,
such that
f1  f 2
2
f,f
1
2
(35)

 RV2  a, b  ,  ; K  .
Let us fix t , t0   a,b  such that t0 < t . Using the
definitions of the operator N and of the metric D1 ,
we have

D H  t , f1  t    H  t0 , f 2  t0   ,
H  t0 , f1  t0    H  t , f 2  t  


 M   t     t0  f1  f 2
2
f,f

1
2
 RV2  a, b  ,  ; K 
1
   t     t0 

11 
 (36)



Define the auxiliary function 3 :  a, b    0,1 . by
a    t0 ,
1,

       t 
3   : 
, t0    t ,
   t     t0 
0,
t    b.

operator N generated by a set-valued function
H :  a, b   K  cc Y  maps the space RV2  a, b  ; K 
into the space RW1  a, b  ,  ; cc Y   and if it is
globally Lipschizian, then the set-valued function H
satisfies the following condition
2 2
The function 3  RV2  a, b  ,   and

1
V2 3 ;  a, b    t     t0  21 
   t     t0 

Let us fix x  K and
fi :  a, b   K  i  1, 2  by
f1   : x,
define
the
n
  f1  f 2   ti    f1  f 2   ti 1 

 f1  a   f 2  a   inf   0 : sup2 

   ti     ti 1 
π i 1





x

   t     t0   1
 inf   0 :  2 
    t     t0  




x
.



1
  t     t0  21 

   t     t0  


Hence, substituting in the inequality (36) the auxiliary functions fi
D  H  t , x   H  t0 , x  , H  t0 , x   H  t , 0    M   t     t0 
 i  1, 2 
functions
f 2   : 3   x,    a, b  .
The functions fi  RV2  a, b  ,  ; K 

.


 i  1, 2 
(37)
and


   ti     ti 1   1



defined by (37), we obtain


1

   t     t0  



1
  t     t0   21 

   t     t0 
11 




x.
By Lemma 4.3 and the above inequality, we get
Copyright © 2013 SciRes.
APM
W. AZIZ
572

D  H  t , x  , H  t , 0    M   t     t0 
Since lim t 21 1  t   t   , letting t0  t in the
above inequality, we have
D  H  t , x  , H  t , 0    0.
Thus for all t   a, b  and for all x  K , we get
H t, x   H t, 0 .
Theorem 5.4 Let  X ,   , Y ,   be normed spaces,
K a convex cone in X and  be a convex  function in X satisfying the 1 condition. If the
Nemytskii operator N generated by a set-valued function H :  a, b   K  cc Y  maps the space
RV  a, b ,  ; K  into the space BW  a, b  ; cc Y  
and if it is globally Lipschizian, then the left regularization H  :  a, b   K  cc Y  of the function
H defined by
 H  t , x  ,
H *  t , x  : 
limH  s, x  ,
 s  a
t   a, b  , x  K ;
t  a, x  K ,
satisfies the following conditions:
 for all t   a, b  there exists M  t  , such that
f1  f 2

n


 f1  a   f 2  a   inf   0 : sup  
π i 1 



1

   t     t0  



1
  t     t0  21 
   t     t0 

11 





x.

D H * t, x  , H * t, y   M t  x  y
 x, y  K  .
H *  t , x   A  t  x  B  t   t   a, b  x  K  , where
A  t  is a linear continuous set-valued function, and
B  BW  a, b  ; cc Y   .
Proof. We take t   a, b  , and define the auxiliary
function 4 :  a, b    0,1 by:

a    t,
1,

4   :       b 
, t    b.

  t    b 
The function 4  RV  a, b  ,  ; K  and

1
V 4 ,  a, b    
  b    t 


  b   t  .


Let us fix x, y  K and define the functions
fi :  a, b   K  i  1, 2  by
f1   : x, f 2   :  4   y  x   x,    a, b  .
The functions fi  RV  a, b  ,  ; K 
 i  1, 2 

 f1  f 2   ti    f1  f 2   ti 1  
   ti     ti 1   1

   ti     ti 1 


(38)
and
(39)
From the definition of f1 and f 2 , we obtain
f1  f 2




1
 x  y 1 

1

  b     t   1 

  b   t 


Since the Nemytskii operator N is globally Lipschitzian between



.




(40)
RV  a, b  ,  ; K  and BW  a, b  ; cc Y   ,
then there exists a constant M , such that


D H  b, f1  b    H  t , f 2  t   , H  t , f1  t    H  b, f 2  b    M f1  f 2  .
for f1 , f 2  RV  a, b  ,  ; K  . By Lemma 4.3, substituting the particular functions fi
above inequality, we obtain
 i  1, 2 
defined by (38) in the
D H  b, f1  b    H  t , f 2  t   , H  t , f1  t    H  b, f 2  b    M  t  x  y  ,
(41)

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
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W. AZIZ
for all x, y  K , t   a, b  . By Lemma 4.3, we get
D  H t, x  , H t, y   M t  x  y
(42)

for all t   a, b  and x, y  K .
In the case where t  b , by a similar reasoning as
573
above, we obtain that there exists a constant M  b  ,
such that
D  H  b, x  , H  b, y    M  b  x  y

 x, y  K  .
(43)
Define the function M :  a, b    by
 
 
 
1
 M 1 

M (t )   
1
  b     t   1 
 
  b    t 
 


M b  ,



 , a  t < b,




t  b.
(44)
fine the partition π n of the interval t0 , t  by
Hence,
D  H t, x  , H t, y   M t  x  y  ,
π n : a  t0  t1    t2 n 1  t2 n  t ,
t   a , b  , x, y  K .
where ti  ti 1 
By passing to the limit in the inequality (41) by the
inequality (43) and the definition of H * we have for all
t   a,b  that there exists M  t  , such that


D H * t, x  , H * t, y   M t  x  y
The Nemytskii operator N is globally Lipschitzian
between RV  a, b  ,  ; K  and BW  a, b  ; cc Y   ,
then there exists a constant M > 0 , such that
n
D  d f , f  H , t2i , t2i 1  , d f , f  H , t2i 1 , t2i  

 t   a , b  x, y  K 
i 1
Now we shall prove that H * satisfies the following
equality
H t, x   A t  x  B t 
*
 t   a, b  , x  K  ,
where A  t  is a linear continuous set-valued functions,
and
B  BW  a, b  ; cc Y   .
Let us fix t , t0   a, b  , n   such that t0 < t . De0,

       ti 1  ,
  ti     ti 1 
5   : 
       ti  ,
  t    t 
i
i 1

0,
t  t0
, i  1, 2, , 2n.
2n
1
2
1
 M f1  f 2
2
(45)
,

where
f1 , f 2  RV  a, b  ,  ; K 
and
d f1 , f2  H , s, t   H  s, f1  s    H  t , f 2  t   .
We define the function 5 :  a, b    0,1 in the following way:
a    t0 ;
ti 1    ti , i  1,3, , 2n  1;
ti 1    ti , i  2, 4, , 2n;
t    b.
The function 5  RV  a, b  ,  ; K  and

2n
V 5 ,  ;  a, b     t     t0   
   t    ( t0 


.


Let us fix x, y  K and define the functions fi :  a, b   K by:
Copyright © 2013 SciRes.
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W. AZIZ
574

1
 1

   a, b  ;
 f1   : 2 5   x  1  2 5    y,



 f  : 1 1      x  1 1      y,   a, b .
 
5
5


 2   2 
2
(46)
The functions fi  RV  a, b  ,  ; K   i  1, 2  and
f1  f 2

x y

2
Substituting in the inequality (45) the particular functions fi
n

i1



D  H  t2i 1 , x   H  t2i , y  , H  t2i 1 ,
x  y 

 x y
*
D  H * t, x   H * t, y  , H *  t,
  H  t,

2 
2 



M

x y .
2n
for all x, y  K and n   . By passing to the limit
when n   , we get
x y
 x y
*
*
*
H *  t,
  H  t,
  H  t, x   H t, y  ,
2 
2 


t   a , b  , x, y  K .
Since H *  t , x  is a convex function, then
 x y 1 *
*
H *  t,
  H  t , x   H  t , y  
2  2

 t   a , b  , x, y  K  .
Thus for every t   a, b  , the set-valued function
H *  t ,  : K  cc Y  satisfies the Jensen equation. By
Lemma 4.4 and by the property (a) previously
established, we get that for all t   a, b  there exist an
additive set-valued function A   : K  cc Y  and a set
B  t   cc Y  , such that
defined in (46), we obtain
 t   a, b  , x  K  .
A  t    L  K , cc Y   and B  BW  a, b  ; cc Y   .
6. Acknowledgements
This research was partly supported by CDCHTA of
Universidad de Los Andes under the project NURR-C-

x, y  K .
(47)
547-12-05-B.
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By the same reasoning as in the proof of Theorem 5.2,
we obtain that
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 i  1, 2 
x y
 x  y  1
  H  t2 i ,
  M x  y
2 
2  2

Since the Nemytskii operator N maps the spaces
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H * t, x   A t  x  B t 
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