this file 1728 4397 1 SM

J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 221

Available online at http://journal.walisongo.ac.id/index.php/jnsmr

P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type
Yulia Romadiastri
Mathematics Education Departement, Faculty of Sains and Technology, Universitas Islam Negeri Walisongo
Central Java,Indonesia

Abstracts
Corresponding
author:
yulia_romadiastri@waliso
ngo.ac.id
Recived: 05 May 2017,
Revised : 10 June 2017
Accepted: 30 June 2017.

In this paper, we described about Musielak-Orlicz function spaces of Bochner
type. It has been obtained that Musielak-Orlicz function space L  , X  of
Bochner type becomes a Banach space. It is described also about P-convexity of

Musielak-Orlicz function space L  , X  of Bochner type. It is proved that the
Musielak-Orlicz function space
if both spaces

L  , X  of Bochner type is P-convex if and only

L and X are P-convex..©2017 JNSMR UIN Walisongo. All rights

reserved.
Key words: Musielak-Orlicz Function; Musielak-Orlicz Functio; Bochner type; PConvex.

1. Introduction
The Orlicz space was first introduced in
1931 by W. Orlicz [1]. Orlicz space theory has a
very important role and has been widely
applied to various branches of mathematics,
one of them on the issue of Optimal Control.
The development and refinement of the Orlicz
space itself is also progressing very rapidly,
one them were Musielak and Orlicz [2] which

develops a functional space generated by a
modular that having convex properties. In this
~
case, I   f    t , f t  X d the modular

 



convex generates the Musielak-Orlicz function
spaces of Bochner-type, which is also the
Banach space [3,4].
The convexity and reflexive properties of
the Banach space also has been widely
developed by many mathematicians. Yining,
et.al [5,6] in their paper entitled "P-convexity
and reflexivity of Orlicz spaces" proved that
for Orlicz spaces reflexivity is equivalent to Pconvexity. The same result for the MusielakOrlicz sequence and function spaces were
obtained by Kolwicz and Pluciennik [7-9].

T

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J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 222

2. Auxiliary Lemmas
Definition 1
Given
f : T  R is  -measurable
function. Function f said to be μ-integrable, if
there exist sequence function  f n  such that

f n  x   f  x  a.e. x  T and for every   0
there exist natural number



n such that
fi  x   f j  x    dx    , for every i, j  n.

T

Hence, the finite value of lim f n  x   dx  , is
n 



T

called Lebesgue integral of function f and
denoted by f  x    dx  or f d  .





T

T

Given measurable set X . The set of all  measurable functions from X to R* denoted
by M  X  . It can be proved that M  X  is a
linear space. For 1  p   , defined





Lp ( X )  f  M ( X )  f d   .
p

X

In another word, Lp (X ) is a set of all
measurable functions f  M (X ) , such that

f

p

 -integrable in X.

Given f extended real valued function on
measurable set X . Supremum essensial f on E
defined by
ess sup  f  x  : x  X   M  A  X ,   A  0  sup  f  x  : x  X  A  M .
Then, we defined L (X ) the set of all
measurable functions by the formula





 

L  X   f : ess sup f  x  : x  X   .
Definition 2
Given linear space T
Non-negatif function  : T  0,  is called

modular on T if for every x, y  T this
conditions below apply
(M1)  x   0  x   ,
(M2)   x    x  ,

(M3)  x  y    x     y  if  ,   0,1
with     1 .
Linear space T that completed with modular
is called modular space and denoted by
T ,   .
A set B  Y , with Y linear space, is called
convex set if
for each x, y  B and

 ,   0,1 with     1 , is applicable
x  y  B . Function f : B  R is called

convexs, if B convexs and for every x, y  B

and  ,   0,1 with     1 be valid

f x  y   f x   f  y  . Furthermore, the

modular  is called comply character convexs
if  is fungtion of convexs. On the next
discussion, the meaning modular is the
modular that comply of character convexs,
Unless otherwise stated.
Theorem 3
(i)
If 1 ,  2  R with 0  1   2 , then

 1 x     2 x  for every x  X .
If  x    for every   0 ,then x   .
Definition 4
Given linear space T.

Function  : T  R  0,  is called Musielak
– Orlicz function if:
(1)  t , u   0  u  0 , for every t  T ,

(2)  (t ,u)   (t , u)
(3)  (t ,.) continu
(4)  (t ,.) increase on (0, )
(5)  , u  measurable, for each u  R
(6)  t , convex,
(7)

 t , u 
u

 0 if u  0 on T.

For the function of Musielak-Orlicz  , is

defained of function I  : L0  0,  with

I   f     t , f t d ,
T

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J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 223

f  L0 . So, can be shown that

for every

function I  is modular convexs. Then, defined
space of function Musielak–Orlicz L , with

L   f  L0 : I  cf    for some c  0.

Forthermore, for every of the function
Musielak–Orlicz  , is defained of function

 * : T  R  0,   , with

3. Main Result



The set S  X   x  X : x

 t , v   supu v   t , u ,
u 0

Definition 6 Function of Musielak – Orlicz  is

i  j ; i , j n

called comply condition -  2 , writes    2 ,if

u0  0 that
 t ,2u   k t , u  ,for every t  T and u  u0 .
k  0 and

Fathermore,

defined
~
0
I  : L T , X   0,  ,



~
I   f     t , f t  X d ,

function
with
for

every

T

f  L T , X  . That, can be shown that
~
function I  is modular. For the function of
0

defained
,
L  , X   f  L T , X  : f . X  L .

Musielak-Orlicz

0

The defained

f  f . X



. : L   , X   R , with

, for every f  L  , X  , can





be shown that L  , X , . is normed space.





X



1

is called disebut closed unit ball.

,for x, y U  X  with





.

 1 is called area

with center O and set U  X   x  X : x

for every v  R and t  T . Can be shown that
function   is function of Musielak-Orlicz.
Theorem 5 For every function of MusielakOrlicz  , be valid uv   t , u     t , v  ,
u, v  0, for every t  T .

constanta

X

X

Definition 9 The space Banach X is called
reflexive if for a   0 there  so as x  y  

*

that



Definition 8 Given the space norm X , .



Theorem 7 Space norm L  , X , . is space
Banach.
Furthermore, the space function L  , X  is

1
x  y   1   .
2

Definition 10 The space norm linear X is called
P–convexs, if that   0 and n  N so for
every
x1 , x2 , ...., xn  S  X  , be valid

min xi  x j

X

 21    .

Lemma 11 The space Banach X P–convexs if
and only if that n0  N and  0  0 so for
every

x1 , x2 , ...., xn0  X \ 0that

i0 , j0 so be valid

xi0  x j0
2



xi0

X

X

 x j0
2

X

integers



 2 0 min xi , x j
0 X
0
1 

xi0  x j0
X
X


Theorem 12 To every of function MusielakOrlicz  be valid, if  *   2 then for any

   
a  1 that   1 so   t , u  
  t , u  for
 a  a
every u  R, t  T .
1

Lemma 13 Be discovered  and   meet the
conditions  2 .

  0,1 that of function is
measurable h : T  R
with I  h    ,
number a   0,1 and    a   0,1
such that for
 h.d. t  T be valid
For every

called space of function Musielak – Orlicz
type Bochner.

Copyright @2017, JNSMR, ISSN: 2460-4453

X





J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 224

 u  v  1 
 t , u    t , v  for

2 
2

v
u  h t  and
 a.
u

  t,

every

to theorem 9, obtained L reflexive.

c  d 

Lemma 14 If    2 , that for every   0,1
there is an undeveloped sequence of measurable
sets of up to

B ,


m

m N,

for every




Bm   0 and

 m1 
there
k m  2 so

so   T \



 t ,2u   k m  t , u  for  a.e., t  Bm and for
every u  f t  ,there f of condition  2 .
.Lemma 15 If    2 ,then for every   0,1
there is a measurable function g :T  R
and

k  2

so I  g    

Be discovered L and X P-convex. According

and

  t , 2u   k   t , u  , for  a.e., t  T ,and
u  g  t  .

Theorem 16 If space Banach X P-convexs, then
X reflextif.
Theorem 17 Be discovered
 function
Musielak – Orlicz and X space Banach.
Then the following statements are equivalent:
(a) L  , X  P-convex,

The characteristics of reflexsive on the space
funcion of Musielak-Orlicz L ekuivalen with

   2 and  *   2 .

d   a 

Be discovered

X

defined
t T




f  t   max h 1  t  , g 1  t   ,
4 n0
 4 n0



there of function h 1 and g 1 is function
Then,

for

every

4n0

1
.
4n 0
1
. Can be chosen
as a result, I   f  
2n0
  a . Because    2 , then obtained
1 
I  f    .
a 
Bma 0
Selected
set
so
fulfilling






  t,

f t  
1
.
d 
a 
2n 0

(c) L reflectif dan X P-convex,

Selected l  R so

k ma 0  2 so

Be discovered L  , X  P-convex. Because of

l
 1 
0
 a 
when v  a f t  .

the space L dan X embedded isometrically

Then,

well apply on subspace, then obtained L and

k 

to L  , X  and characteristics P-convex as

X P-convex.

b  c

1
 2 l . There are numbers
a

  t , v   k ma   t , v  for

a   b

4n0

measureble with  

T \ Bma 0

Evidence:

and

   2 . Can be chosen n0  N .

(b) L and X P-convex,
(d) X P-convex,    2 and     2 .

P-convex,    2



1

a l
m0

chosen

  a, m0    m0 .

 t , au    m  t , u  ,
0

  a.e. t  Bma ,
0

v
u
a

and
Obtained

For   h.d . t  Bma 0 , and for every u  f t  .
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J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 225

 1 
 a 
  a.e. t T , and v  f t  , then k   k

So,

  t , v   k l  t , v 

obtained

 n0
n0  1 

1 
  t , x i
2  n0 n0  1  i 1
for   a.e. t  TM .

1
4 n0

.

In the same way, obtained  t , au    t , u  ,





for

II. It

f t 
For   h.d . t  T , and for every u 
.
a

is

assumed

i  k apply

xi1

X

xk

X

that

X

,

for

all

a.

Furthermore, it will be shown that there are
so
for
every
numbers
r1  0,1

Then xi  0 , for every i  k . Can be taken

x1 , x2 , ..., xn0 space group Banach X and to

i0 , j0 . and assumed that

  a.e. t  TM obtained
 xi  x j
 t,



2
i 1 j i 
n0

 n0  1 n0

r1   t , xi

2
i 1




n0

X

X

a


On

the

,


f t  
With TM  t  T : max1i n0 xi X 
.
a 

Taken x1 , x2 , ..., xn0  X . For example k is



index so xk

 max1i n0

X

I. Suppose
so

xi1

X

xk

X



 x .
i X

i1  1, 2, ..., n0  \ k

there

xk 

f t 
 f t 
a

for

  a.e. t  TM , so obtained

 x
 xk
 xi  x k 
    t , i1 X
  t, 1



2
2
X 


1
 1     t , xi1
  t , xk X .
X
2



 

n0

n0



xi  x j



2

i 1 j i

 n0  1 n0

  t , xi

2 i 1




X



X

n  1 n0
 0
  t , xi
2 i 1





X



X


max  x

min xi0

a

xi0  x j0
2

X





  t,

i1 X

, x j0

X

i0 X

, x j0

X

haved

  min  x
x


i0 X

So
X

, x j0

X



k X

obtained

 x j0

X

2

k X



X

 1
  1     t , xi
0
 2


.

X

  t, x 
j0

X

n0

n0

2a
 0,1 . so, obtained
1 a



xi  x j



2

 n0  1 n0

  t , xi

2 i 1




X

n0  1 n0

  t , xi
2 i 1



n0  1 n0

  t , xi
2 i 1



0

  2n  t, x 
n0

0 i 1

X

2a  xi0

 1 

 1 a 

2

  t,

k X

  2n n  t, x 
X

hand,

contradiction.

xi0  x j0

With  

  2  t, x   t, x 

0

x j0

X

other



it's

i 1 j i



n0  1
  t , xi
2 i 1
n0

X

X

x j0

1
.
a



,






Based on characteristics convex of  t , . for
  a.e. t T , obtained

  t,

.
So

xi1

X

Based on characteristics convex of  t , . for
  a.e. t T , obtained

a.

Because

a

xi1

i X

Copyright @2017, JNSMR, ISSN: 2460-4453

X

X

X

  2  t, x

   t, a x 
k X

  
n   x 
n
0

0

k X

i0

X

  t, x 
j0

X

J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 226

Obtained,



n0  1
  t , xi
2 i 1
n0



 f i t   f j t 
  t,


2
i 1 j i 

  
 t, x 
n
n0

X

i X

i 1

0

n0

n 1
2  n0
1 
  t , x i
 0
2  n0 n0  1  i 1



X

,

1
I f   2 .

X

 f t  .

n0 ~







n0 t , max1i n0

X

every t  E . Furthermore E divided into the
following
two
subsets:

  t , f t  d
i

X

 f  t   d 
i



 f t  
n0  t ,
d
 a 
E2 \ Bma 0



1
 f t  
n0  t ,
d  .
2
 a 
T \ Bma 0

X





Obtained
n0 ~





n0

n0 ~

~

i 1

.
because f i  1 for i  1, 2, ..., n0 and    2 ,
then I   f i   1 for i  1, 2, ..., n0 . as a result,
~

n0 ~



0





0



E 21

dan

0



E 22 as follows
a
E21  E2  Bm0 , and E22  E2 \ Bma 0 .
set

0

1

.
Defined



i 1

I  f i  E1  E21  n0  1 .


f t  
i 1
E1  t  T :max1i n0 fi  t  X 
 , and
obtained
a 

n
n ~
n
n ~
1

1
I    f i  f j    I    f i  f j  T \  E  E


f i 1t ji  2
 i 1 j i  2
E2  t  T : f  t   max1i n0 fi  t  X 

a 






 I  f i  T \E1 E21   I   f i  T \ E    I  f i  E22  1
i 1

 f t    f t  for
i

n0

i 1 E \ B a
2
m0

E2 \ Bma0

defained

Obviously that, max1i n0





Then proved true



E  t  T :   t , f i t  X  n0 t , f t  .
i 1





i 1





2 m0 


,1 
r2  max 1 
.
n0  n0  1 
 n0  n0  1


f1 , f 2 , ..., f n0  S L  , X 
Taken
and



 I  f i  E22  

with

n0

T \E

taken t  E22 , then obtained

Banach X and for   h.d . t  Bma 0 comply



i

i 1

 xi  x j  n0  1 n0

r2   t , xi X
  t,



2
2
i 1 j i 
i 1
X 
, for every x1 , x2 , ..., xn0 component space

maks1i n0 xi



n0 ~

Then the number can be determined r2  0,1
so
be
valid



X

obviously r  0,1 . Furthermore, from the
define set E and function f , obtained



2 


r1  max 1 
,1 
.
n0  n0  1 
 n0  n0  1



n0



X

,
for   a.e. t  E1  E2 , with r  max r1 , r2  .

for   a.e. t  TM .
Defined

n0

 1  n0  n0
     t , f i t 
 n0  2  i 1


n0



1
n0

 n0  n0 ~
r
  I  f i  T \ E1  E21  
n0
 2  i 1



Copyright @2017, JNSMR, ISSN: 2460-4453







21 

 I   2  f
n0

n0 ~

i 1 j i

1

i


 f j  E1  E21 


 n0  n0 ~
  I  f i  E1 E21
 2  i 1





J. Nat. Scien. & Math. Res. Vol. 3 No.1 (2017) 221-227, 227

n0 ~
1  n  n0 ~
  0   I   f i    I  f i  E1 E21
n0  2  i 1
i 1



 r  n0  n0 ~
    I  f i  E1  E21
 n0  2  i 1





 n0  1  r n0 ~
  1 
 I  f i  E1 E21
n0 i 1
 2 





as

a







result,

because

   2 , obtained



1
f i  f j1  1  q p  , with 0  q p   1 .
2 1
obtained that space L  , X  P-convexs.

4. Conclusion
Modular I   f    t , f t d generates



T

the

Musielak-Orlicz

function

space

L   f  L : I  cf    for some c  0.
0

Furthermore, for every Musielak - Orlicz
function
we
define
function
,

 * t , v   supu v   t , u , which is also a
u 0

Musielak - Orlicz function and it is called the
complementary function in the sense of Young.
~
Modular I   f    t , f t  X d generates

 



T

the Musielak-Orlicz function spaces of Bochner
type L  , X   f  L0 T , X  : f . X  L .



Acknowledgement
Acknowledgments are submitted to the
Department of Mathematics,UGM for support
in this research.






 n  1  r n0  1  n0 
  1  p  , with
  0 1 
n0
2
 2 


1 r
.
p
2
So, be found i1 , j1  1, 2, ..., n0  so that
~
1

I   f i1  f j1   1  p .
2




is proved that reflexivity is equivalent to Pconvexity.

Furthermore, L  , X  is a Banach space.



References
[1]

W. Orlicz, Linear Functional Analysis,
World Scientific, London, 1992.
[2] H. Hudzik, A. Kaminska, and W. Kurc,
Uniformly
non-ln 1 Musielak-Orlicz
Spaces, Bull. Acad. Polon. Sci. Math., 35,
7-8, pp. 441-448, 1987.
[3] H. Hudzik, and S. Chen, On Some
Convexities of Orlicz and Orlicz-Bochner
Spaces, Comment. Math., Univ. Carolinae
029 No. 1, pp. 13-29, 1988.
[4] P. Kolwicz, and R. Pluciennik, P-convexity
of Musielak-Orlicz Function Spaces of
Bochner Type, Revista Matematica
Complutense Vol. 11 No. 1. 1998.
[5] Y. Yining, and H. Yafeng, P-convexity
property in Musielak-Orlicz sequence
spaces, Collect. Math. 44, pp. 307-325,
2008.
[6] D.P. Giesy, On a Convexity Condition in
Normed Linear Spaces, Transactions of
the American Mathematical Society Vol.
125 No. 1, pp. 114-146, 1966.
[7] P. Kolwicz, and R. Pluciennik, On P-convex
Musielak-Orlicz Spaces, Comment. Math.,
Univ . Carolinae 36, pp. 655-672, 1995.
[8] P. Kolwicz, and R. Pluciennik, P-convexity
of Bochner-Orlicz Spaces, Proc. Amer.
Math. Soc.,
[9] P. Kolwicz and R. Pluciennik, P-convexity
of Musielak-Orlicz Sequence Spaces of
Bochner Type, Collect. Math. 1997.

The Musielak-Orlicz function spaces of
Bochner type L  , X  is P-convex if and only
if both L and X are P-convex. Furthermore, it
Copyright @2017, JNSMR, ISSN: 2460-4453