this file 1728 4397 1 SM

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

Corresponding author: yulia_romadiastri@waliso ngo.ac.id

Recived: 05 May 2017, Revised : 10 June 2017 Accepted: 30 June 2017.

Abstracts

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 L_





,X



of Bochner type is P-convex if and only if both spaces

L

_ and

X

are P-convex..©2017 JNSMR UIN Walisongo. All rights reserved.

Key words: Musielak-Orlicz Function; Musielak-Orlicz Functio; Bochner type; P-Convex.

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,

 







 



T

X

d

t

f

t

f

I

_



,



~

the modular

convex generates the Musielak-Orlicz function spaces of Bochner-type, which is also the Banach space [3,4].

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

 

 

. .

n

f

x



f x a e x T



and for every



0

there exist natural number n such that

 

   

,

i j

T

f x



f

x



dx







for every

i j

,



n

.

Hence, the finite value of

lim

_n

   

,

n T

f

x



dx





is

called Lebesgue integral of function f and

denoted by

   

T

f x



dx



or

T

f d





.

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





_









X p p

d

f

X

M

f

X

L

(

)

(

)



.

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

p

f



-integrable in X.

Given f extended real valued function on measurable set X . Supremum essensial f on E

defined by

 





 



 



sup : , 0 sup : .

ess f x xX M A X  A   f x x X A M

Then, we defined L(X) _{the set of all}

measurable functions by the formula

 



:

sup



 

:





.

L



X



f ess

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

B

y

x









. Function

f B

:



R

is called

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



,





 

0

,

1

with









1

be valid



x

y



f

 

x

f

 

y

f















. 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)

 

, 0

u u t



if u0 on T.

For the function of Musielak-Orlicz



, is

defained of function

I

_

:

L

0





0

,





with

 



_



 



T

d

t

f

t

f

for every f L0. So, can be shown that function I_ is modular convexs. Then, defined

space of function Musielak–Orlicz L_, with

 



0



:

0 .

L

_



f



L I

_

cf

 

for some c



Forthermore, for every of the function Musielak–Orlicz



, is defained of function





*

:

T

0,



 

R



, with

 

t v



uv

 

t u



u

, sup

,

0

*





 



,

for every vR and tT . Can be shown that function



 is function of Musielak-Orlicz.

Theorem 5 For every function of Musielak-Orlicz



, be valid uv



 

t,u 





 

t,v ,

, 0,

u v for every tT.

Definition 6 Function of Musielak – Orlicz



is

called comply condition -



₂, writes







₂,if

that constanta k 0and

u

₀



0

that

 

t

,

2

u

k



 

t

,

u





,for everytTand uu₀.

Fathermore, defined function



,

  



0

,



:

~

0

X

T

L

I

_ , with

 



_



 



T

X

d

t

f

t

f

I

_



,



~

, for every



T X



L

f  0 , . That, can be shown that function I~_ is modular. For the function of Musielak-Orlicz



, defained









 









X

f

L

T

X

f

L

L

X







,

:

.

,

0 .

The defained

. :

L

_





,

X





_R

, with





X

f

f  . , for every f L_





,X



, can be shown that



L

_





,

X



,

.



is normed space.

Theorem 7 Space norm



L

_





,

X



,

.



is space Banach.

Furthermore, the space function

L

_





,

X



is called space of function Musielak – Orlicz type Bochner.

3.

Main Result

Definition 8 Given the space norm





X

X

,

.

.

The set

 







:



1



X

x

X

x

X

S

is called area

with center O and set

 







:



1



X

x

X

x

X

U

is called disebut closed unit ball.

Definition 9 The space Banach X is called reflexive if for a



0 there



so as

x



y





,for

x y U X

,



 

with







1



2

1

y

x .

Definition 10 The space norm linear X is called P–convexs, if that



0 and nN so for every x₁,x₂,....,x_n S

 

X , be valid









 



minj;i,j n i j X 21

i x x .

Lemma 11 The space Banach

X

P–convexs if and only if that n₀N and



0



0

so for

every , ,...., \

 

0

0

2

1 x x X

x _n  that integers

0 0,j

i so be valid





    

  

 

 



X j X i

X j X i X

j X i

X j i

x x

x x x

x x

x

0 0

0 0

0 0

0

0 2 min ,

1 2

2

0



Theorem 12 To every of function Musielak-Orlicz



be valid, if



*₂ then for any

1



a that



1 so

 

1

,

,

t

u

t u

a

a









_

 

_









for

every u_R,tT.

Lemma 13 Be discovered



and



 meet the

conditions



₂.

For every





 

0

,

1

that of function is measurable h T_ : _R with I_

 

h_ 



, number

a

   





0

,

1

and









a

 



  



0

,

1

   

t u t v v

u

t , ,

2 1 2

,









  

  



 

for every

 

t h

u _ and a

u v

 .

Lemma 14 If







₂, that for every





 

0

,

1

there is an undeveloped sequence of measurable

sets of up to

 

B_m , so

\

0

1





















m m

B

T





and

for every mN, there k_m 2 so

 

t

,

2

u

k

m



 

t

,

u



_



for



a.e.,

t



B

_m 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

_





and



t u

, 2



k

_

 

t u

,

,







for



a.e., tT _,and

 

t g u _ .

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,

(b) L_ and X P-convex,

(c)

L

_ reflectif dan X P-convex,

(d) X P-convex,







₂and



₂.

Evidence:

   

a



b

Be discovered

L

_





,

X



P-convex. Because of the space L_ dan

X

embedded isometrically

to L_





,X



and characteristics P-convex as well apply on subspace, then obtained

L

_ and

X

P-convex.

   

b



c

Be discovered L_ and

X

P-convex. According

to theorem 9, obtained L_ reflexive.

   

c



d

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

2







and



*₂.

   

d



a

Be discovered

X

P-convex,







_{2 and}

2

  



. Can be chosen n₀N.

Then, for every tT defined

 

 

 

0 0

1 1

4 4

max

,

n n

f t









h

t

g

t















,

there of function

0

1 4n

h

and

0

1 4n

g

is function

measureble with

0

4 1

n





.

as a result,

 

0

2 1

n f

I_  . Can be chosen

a





. Because







₂, then obtained

       

f a

I_ 1 .

Selected set

B

_ma

0 so fulfilling

 

















a m

B

T

n

d

a

t

f

t

0

\

2

0

1

,





.

Selected lR so l a 2

1

 . There are numbers

2

0



a m

k

so

 

k

 

t v v

a

t a l

m ,

1 ,

0





     

for

0

. .

_ma

a e t

B







,

when

v



a

f

 

t

.

Then, chosen u

a v

 and

 





0

0

0

,

1

m l

a m

m

a

k









. Obtained



t,au



_m

 

t,u

0







 ,

For

h

d

t

B

_ma

0

.

.





Defined









defained

 

following two subsets:

 

Obtained,

 

 

_

_{ }

_

Obtained

 

obtained

 





_









 

   

        



 _

     

 0

21 1

0 0

21 1

1 ~ 0 0

1 1

~ ~

0

0 2 2

1 n

i

E E i n

i

n i

E E i

i I f

n n

r f

I f I n

n     





_











_



















 

0

21 1

1 ~

0

0

1

1

2

n

i

E E i

f

I

n

r

n











_

_

















































n

p

n

n

r

n

1

2

1

1

1

2

0 0

0 0

, with





2

1 r

p  .

So, be found i₁,j₁



1,2,...,n₀



so that



f f



p

I _{i j}  

  



 _

1 2

1

1 1

~

 .

as a result, because







₂, obtained



f_i  f_j



1q

 

p

2 1

1

1 , with

0



q

 

p



1

.

obtained that space L_





,X



P-convexs.

4.

Conclusion

Modular

 







 



T

d

t

f

t

f

I

_



,



generates

the Musielak-Orlicz function space

 



0



: 0 .

L_  f L I_ cf   for some c

Furthermore, for every Musielak - Orlicz function



, we define function

 

t

v



u

v

 

t

u



u

,

sup

,

0

*









 , which is also a Musielak - Orlicz function and it is called the complementary function in the sense of Young.

Modular

 







 



T

X

d

t

f

t

f

I

_



,



~

generates

the Musielak-Orlicz function spaces of Bochner type

L

_





X





f

L



T

X

 

f

L

_



X







,

:

.

,

0 .

Furthermore, L_





,X



is a Banach space. 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

is proved that reflexivity is equivalent to P-convexity.

Acknowledgement

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

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.,