SOME NEW SEQUENCE SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTIONS
So me New Sequence Sp ares Dcfinrd by A Sequence of
Orlicz Jı'unctions
SAU Fen B11im1eri Enstitüsü Dergısi
6.Ci lt, 3 .Sayı (Eylül 2002)
T.Böyük, M.Başarır
SOME NEW SEQUENCE SPACES DEFINED BY A SEQUENCE OF
ORLICZ FUNCTIONS
Tuncay
BÖYÜK,
Metin
BAŞARIR
X
M(x)=
Ozet-Bu çalışmada, regüler bir matris ve bir Orlicz
••
fonksiyon
dizisi
yardımıyla
üç
yeni
dizi
where q, known as the kernel of M is
,
o
uzayı
right-differentiable for t � O, q(O)=O, q(t)>O for t>O, q is
nondecreasing , and q(t)� co as t-> co .The space !M is
closely rela ted to the space IP w h ic h is an Orlicz sequence
space v1ith M(x)=xP; 1 � p< oo.
tanımlayıp bazı özellikleri incelendi.
Anahtar _'f.elimeler- Dizi uzayları,Orlicz fonksiyonu.
Abstract-ln this paper, we introduce and examinc
r
Jq(t)dt
'�e properties of three sequence spaces defined
Recent ly, Parashar and Cho u dha ry[ S ] intioduced and
examined soıne properties of fol1o�·ing four sequence
spaces defıned by Orlicz function M:
by using a regular matrix and a sequence of Orlicz
functions.
Key words-Sequence spaces, Orlicz function.
Let p=(pk) be any sequence of p osi ti ve real nun1bers.
z�.t(p)=
I.INTRODUCTION
XECU:
Let lrr.ıand c denote the Banach spaces of real bounded
and convergent sequences x=( Xn) normed by
ll x ll =supJ Xn; respectively.
The space !M Vlith
llxJj =inf
,
for
sonıe
W o(\'l,p)
p )0:
for
k=l\
p
,
for sonıe
Pk
-I
n
k=l
some
p>O
�o as n�oo ,
-+O as n�co ,
p>O
XEaJ:
for
p >0
Pk
n
1
Wcc(M,p)=
n
sup I:
n
sonı e
k==l
(
M
p >0
1 xkj
�
P
J
\Vhen Pk=l, for all k, then /M(P) b ec on1es !M. Iftv1(x)=x then
the family of sequences defined above become l(p), [c,l,p],
[c,l,p]o and [c,l,p]oo respectively. We denote W(M,p),
W o(M,p) and W r/J(M,p) as W(M), W0(1\1) and Wo-iM) w·hen
pk= 1, for e ac h k.
<1
I
<oo
==
XECV:
n orrn
C()
k=I
hi
Lfor sofne p>O and f>O
Lindenstrauss and Tzafriri[3] used the idea of Orlicz
function to construct sequence space
<oo
M
XEüJ:
W(M,p)==
An Orlicz function is a functi on M: [O,oo)-�[O,co),
which is continuous, non-decreasing a nd convex with
M(O)=O, M(x)>O for x>O and M(x)�oo as x-4co. If
convexity of Orlicz function M is repla c ed by
M(x+y)sM(x)+i\.1(y) then this hınction is called
modulus function defıned and discussed by Ruc kl e[ 1]
and Maddox[2].
I
Let M=(Mk) be a sequence of Orlicz functions and
that A =(an k) be a regular matrix. We define
becomes a Banach space which is called an Orlicz
sequence space, whe re (J) be the family of real or
complex scquences.
An Orlicz fun ctio n M can always be repre sen ted (see
Krasnoselskii and Rutitsky[4],p.5) in the integral form
Vl 0(A,M,p )=
f
XECO:
I ank
k
for some p >0
T. Böyiik; Kuzuluk �1.Soykan Elementary School, Akyazt-Sakarya
W(A,�1,p)=
M. Başarır; Department of Mathcmatics, Sakarya C n iversitv
;
.
156
J
s uppo s e
Some t\ew Sequence Spaces Dcfıned by A Scquence of
SAU Fen Bilın;eri Enstitüsü Dergisi
6.C1lt, 3.Sayı (Eylül 2002)
xEcv:
Orlicz Functions
T.Böyü� M.Başanr
Pk
'ı
j
.
e
xk
-40
L ank M k l
p
k
)
for s ome p
XEW:
sup
Pk
L ank
k
ll for some p >0
ll
<I:
as n-->-oo
an
k 2Pk
>0 and .f>O
Ww(A,M ,p)=
1
C Lank
k
j
ı
lxk
IYk
+
N! k
Mk
P2
Pı
k
Pk
lv[ k
Pı
Tc
k
!Y
1
Mk
Pı
L ank
k
n___,. oo
<oo
bnear topological
n
C .oudhary.
Proof:
{p
Tlıeorenr 1: Let p=(pk) be bounded. Then W0(A,M,p)
,
vV(i\,�1,p) and Vloo(A,M,p) are linear spaces over the
Clearly G(x)=G(-x).
Pn
/
'
IH }=O for x
set of complex numbers C.
=
O
.
(
The atlıers
can be treated similarly. Let x,y E W0(A,M,p) and
a,jJE C. In order to prove the result we need to find
and
I ank
k
(
�
Mk
Pk
\
lxk
Pı
Defıne p3=rnax
and convex,
�o' as
for a=f3 1,
Since M(O)=O, we
Conversely, sup p ose
ıx ı
k
1
H
Pk
p
pE(O<pc <E) such that
<1
il-400
P1c
�o, as n---j>oo,
sı.
ı
-
H
<
-
SinceM is non decreasing
(jaxk + PYk j
I a,ık M k
l P3
k
H
Thus
(
(2iajpı, 2!BiP2).
1
Pk
some p1
Pk
ı
Pk
�
H
J
Suppose x
\axkl IPYkl Pk
+
I ank M k
P3
P3
k
11
111
Then
157
p
::f::.
O for soıne m.
---7 ro
Let c�O
it follo\:vs that
get inf
G(x)=O:- then
'
=O.
This implies that for a given & >0, there exists some
P!c
Pk
By us ing Theoreml
l'ı;f k
W c sha11 only prove for W0(A,M,p).
and p2 such that
<1'
==1, 2,3, ...
ıve get G(x+y)<G(x)+G(y).
II.MAlN RESULTS
H
P1c
G(x) =inf
and
xy
, E W0(A,M,p) , therefore there exist
ıs
1
.�-'\=(C, 1) Cesaro matrix,
w e have the sequence spaces W0(M,p ), W(M,p ) and
W,.c(M,p) that are defıned by S. D. Parashar and
Since
W0(A,M,p)
.
space paranornı.ed by
respective 1Y.
some p3 such that
�o as
Theorem2: Let H=max(l,sup Pk ) . Then W0(A,M,p) is a
defıned above becomes [A,p]0 , [A,p] and [A,p ]cm
Proof:
Pk
,
linear.
k
$
)
\Vhere C=max( 1 ,2H·1 ) . This proves that
When Nlk(x)=-x for aU k, then the family of sequences
When Mktx)=x for all
Pk
<1.
So me
SAU Fen Bil imieri Enstitüst Dergisi
6.Cilt, 3.Sa)'l
(Eylül 2002)
m
p
\
ı
\Pm
<
H
which is a
�oo
X
11111
==
O for each nı.
p� � a"k (l M
:
N
f(t)=
Let A. be any number. By definition,
L
k=1
r
ank Mk
is 1 >8>0 such that
k
A
1,
<8 for n>K,
\.
<
1 f( t)/
<
for O<t <5. Let K be such that
c
2
then for n > K,
Pk
ll1 /H
c
<2
Thus
ı
H
Pk
<1 ,
Xr
p
n=l,2,3, . . .
<c
for n>K.
Remark: It can be easny veri fied that when Mk(x)=x, the n
the paTanorn1 defined in W0(A,M,p) and paranann defined
where r=p/A..
in [ A,p ]o are saıne.
1 A.jPk <max( 1, :A-IH)
<
ank
is continuous at O. So there
Lank
k=l
Pk
l!ı.l H
p
1
N
Then
( �;"x)==inf
E_t
t xk
\.
n==1,2,3, ...
Since
2:
k=N+l
S in ce M is continuous every where in [O,oo), then
Finally, we prove scalar nıultiplication is continuous.
H
00
)
contradiction. Therefore
G(A.x) = 1nf p
Orlicz Functions
T.Böyük, :vf.Başarır
x,ı
m
New Sequence Spaces Defined by A Sequence of
H
max(l, lA! )
1/H
therefore
DejilıUion(Krasnoseiskii and Rutitsky[4],25):An Orlicz
function M is said to satisfy �2-condition for all values of u,
if there exists, constant K>O, such that M(2u)<Kı\1(u)
.
(u>O).
Hence
G( A.x) :::; max( 1, llıw IH) l
/H
L\2-condition is equivalent to the satisfaction of
inequality M(lu)<K.IM(u) for all values of u and for!> I.
The
Pk
ı
-
H
Theot·eın3:Let A be a nonnegative regular matrix, and
M=(Yfk) b e a sequence of Orlicz functions which satisfies
�ı-condition for all k. Then
< 1,
n==l,2,3,...
Which converges
W0(A,M,p) w here
\V0(A,M,p)=
to zero as G(x) converges to zero in
Pk
i)
ii)
iii)
Where
-;.O, as n -)co forsonıe p>O
[ A p ]0=
,
For arbitrary
00
that
I
k=N+I
E:
>0, let N be a pozitive integer such
<
ank
implies that
ro
L
k=N+l
ank A1k
ı ı
\Pk
[A,p]ocW 0(A,M,p)
[A,p )cW(A,M,p)
[A,p]c/JcW oo(A ,M,p)
E
2
{
l
[A,p]=
for s ome p>O. This
[A,p]oo=
XEW
Sn=
<
�- ) J �
k
XECV
:
supn
,
I a11k \ xk \Pk
k
as n �a:ı
n -..::,oo
�
< ocı
Let xE [A,p l, then
Lank \xk -I\ Pk -')-O (...truncated)
