ON SOME GENERALIZED DIFFERENCE SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULI Vakeel A. Khan

106

ON SOME GENERALIZED DIFFERENCE SEQUENCE SPACES DEFINED BY A SEQUENCE OF MODULI

Vakeel A. Khan^1,* & Ayaz Ahmad²

1Department of Mathematics, A.M.U. Aligarh (INDIA)

2National Institute of Technology , Patna ,Bihar (INDIA)

*Email : [email protected]

ABSTRACT

In this paper we define some generalized sequenc spaces defined by a sequence of moduli. The results here in proved are analogous to those by ASMA BEKTAS Cigdem (2006)[Journal of Zhejiang University Science A (2006),7(12) 2093-2096] .

Keywords: Difference sequence space , Sequence of moduli, Strongly almost convergent .

1. INTRODUCTION

Let



⁰ be the set of all complex sequences and

l ,

_

c

and

c

₀ be the sets of all bounded, convergent and null sequences

x = ( x

_k

)

with complex terms, respectively , normed by

}.

{1,2,

=

|, sup |

=

||

||

x x

_k

where k I N 

k





The idea of difference sequence space was introduced by Kizmaz (1981). In 1981, Kizmaz defined the sequence spaces : ¹

}, ) ( : ) (

= {

= )

(

⁰ _



 x x   x  l

l

_k



_k

}, ) ( : ) (

= {

= )

( x x

⁰

x c

c 

_k

  

_k



and

c

₀

(  ) = { x = ( x

_k

)  

⁰

: (  x

_k

)  c

₀

},

where

 x = ( x

_k

 x

_k1

)

. These are Banach spaces with the norm

.

||

||

|

=|

||

|| x

_

x

₁

  x

_

After then R. Colak and M. Et (1995) defined the sequence spaces :

}, ) ( : ) (

= {

= )

(

⁰ _



 x x   x  l

l

^m _k



^m _k

}, ) ( : ) (

= {

= )

( x x

⁰

x c

c 

^m _k

  

^m _k



and

c

₀

( 

^m

) = { x = ( x

_k

)  

⁰

: ( 

^m

x

_k

)  c

₀

},

where

m I N

^,

), (

0

= x

k

 x

), (

= 

_1

 x x

_k

x

_k

), (

=



^m

x 

^m1

x

_k

 

^m1

x

_k1

and so that

. 1) (

=

0

=

i k i

m

i k

m i x

m

x _























and show that these are Banach spaces with the norm

.

||

||

|

|

=

||

||

1

=





 ^x   ^x

x

_i ^m

m

i

1 AMS Subject Classification 2000: 40C05, 46A45.

107

Definition 1.1. A function

f : [0,  )  [0,  )

is called a modulus if 1.

f (t ) = 0

if and only if

t = 0,

2.

f ( t  u )  f ( t )  f ( u ) for all t , u  0,

3. f is increasing, and

4. f is continuous from the right of 0.

Let

X

be a sequence space. Then the sequence space

X ( f )

is defined as

}

|)) (|

( : ) (

= {

= )

( f x x

⁰

f x X

X

_k

 

_k



for a modulus f(Maddox 1986 and Ruckle 1973).

Kolk (1993, 1994) gave an extension of

X ( f )

by considering a sequence of moduli

F = ( f

_k

)

i.e.

}.

|)) (|

( : ) (

= {

= )

( F x x

⁰

f x X

X

_k

 

_{k k}



A sequence

x  l

_ is said to be almost convergent (Lorentz, 1984) if all Banach limits of

x

coincide.

Lorentz(1984) proved that

}.

1 , : lim )

(

= { ˆ :=

1

=

0

x uniformly in s x n

x

c

_{k s}

n

n k

k

^{ } 



Maddox (1967; 1978) has defined

x

to be strongly almost convergent to

L

if

0.

>

, 0,

|=

1 | lim

1

=

L some for s in uniformly L

n x

^{k s}

n

n k







Let

p = ( p

_k

)

be a sequence of strictly positive real numbers . Nanda (1984) defined

0},

>

, 0,

=

| 1 |

: lim )

(

= { :=

] ˆ , [

1

=

0

x L uniformly in s for some L x n

x p

c

_{k s} ^pk

n

n k

k

 ^ 

_



}, , 0,

=

| 1 |

: lim )

(

= { :=

] ˆ , [

1

= 0

0

x uniformly in s

x n x p

c

_{k s} ^pk

n

n k

k

^{ } 



}.

, 0,

=

| 1 |

sup : ) (

= { :=

] ˆ , [

1

=

0

x L uniformly in s

x n x p

c

_{k s} ^pk

n

n k

k



_





^ 

2. MAIN RESULTS

Let

F = ( f

_k

)

be a sequence of moduli,

u = ( u

_k

)

be any sequence such that

u

_k

=  0

for all

k

^and

)

(

= p

_k

p

be any sequence space of strictly positive real numbers then we define the following sequence spaces :

s in uniformly L

x u n f

x x p

F

c

_{k k} ^m _{k s} ^pk

n

n k k

m

u

1 [ (| |)] = 0,

: lim )

(

= { :=

) ](

, ˆ , [

1

=

0

 



 ^ 

_

0},

>

, for some L

}, 0,

=

|)]

(|

1 [ : lim )

(

= { :=

) ( ] , ˆ , [

1

= 0

0

f u x uniformly in s

x n x p

F

c

_{k k} ^m _{k s} ^pk

n

n k k

m

u

 



 ^ 

}, ,

<

|)]

(|

1 [ sup : ) (

= { :=

) ( ] , ˆ , [

1 , =

0

f u x uniformly in s

x n x p

F

c

_{k k} ^m _{k s} ^pk

n

n k s k

m

u

  



_



^ 

If

f

_k

( x ) = x

for every

k

, then

[ c ˆ , F , p ]( 

^m_u

) = [ c ˆ , p ]

,

[ c ˆ , F , p ]

₀

( 

^m_u

) = [ c ˆ , p ]

₀

( 

^m_u

)

and

) ( ] ˆ , [

= ) ( ] , ˆ ,

[ c F p

_



_u^m

c p

_



^m_u .We denote

[ c ˆ , F , p ]( 

^m_u

)

^,

[ c ˆ , F , p ]

₀

( 

^m_u

)

^and

[ c ˆ , F , p ]

_

( 

_u^m

)

^by

)

](

ˆ ,

[ c F 

^m_u ^,

[ c ˆ , F ]

₀

( 

^m_u

)

^and

[ c ˆ , F ]

_

( 

_u^m

)

^{, when}

p

_k

= 1

for all

k

^.

108

Theorem 2.1. For a sequence

F = ( f

_k

)

of moduli , the following statements are equivalent:

1.

[ c ˆ , p ]

_

( 

^m_u

)  [ c ˆ , F , p ]

_

( 

^m_u

),

2.

[ c ˆ , p ]

₀

( 

_u^m

)  [ c ˆ , F , p ]

₀

( 

^m_u

),

3.

1 [ ( )] < ( > 0).

sup

1

=

t t

n f

pk k n

n k

 

Proof. (i)



(ii) is clear.

(ii)



(iii): Let

[ c ˆ , p ]

₀

( 

_u^m

)  [ c ˆ , F , p ]

₀

( 

^m_u

)

. Suppose that (iii) is not true. Then for some

t

.

= )]

( 1 [

sup

1

=



^{n k pk}



n k

t n f

and there exists a sequence

( n

_i

)

of positive integers such that

(1) 1,2,

=

>

1 ) 1 (

1

=

 i

for i i

n f

^k

ni

i k



Now we define

x = { x

_k

}

by









 . )

>

( , 0

, 1,2,

= , 1

1 ,

=

i i k

n k

i n k i if

x 

Then

x  [ c ˆ , p ]

₀

( 

^m_u

)

but by Eqn (1),

x  [ c ˆ , F , p ]

_

( 

_u^m

)

which contradicts (ii).

Hence (iii) is true.

(iii)



(i) :

Let (iii) is true and

x  [ c ˆ , p ]

_

( 

_u^m

).

If we suppose that

x  [ c ˆ , F , p ]

_

( 

_u^m

)

^{. Then}

.(2)

=

|)]

(|

1 [ sup

1 , =





_



^{n k k m k s pk}

n k s

x u n f

If we take

t =| u

_k



^m

x

_ks

|

for each

k

and fixed

s

, then by Eqn(2)

,

= )]

( 1 [

sup

1

=



^{n k pk}



n k

t n f

which contradicts (iii). Hence

[ c ˆ , p ]

_

( 

^m_u

)  [ c ˆ , F , p ]

_

( 

^m_u

)

^.

Theorem 2.2. Let

1   sup

_k

< 

k

k

p

p

. For a sequence of moduli

F = ( f

_k

)

the following statements are equivalent:

1.

[ c ˆ , F , p ]

₀

( 

^m_u

)  [ c ˆ , p ]

₀

( 

^m_u

),

2.

[ c ˆ , F , p ]

₀

( 

_u^m

)  [ c ˆ , p ]

_

( 

^m_u

),

3.

1 [ ( )] > 0 ( > 0).

inf

1

=

t t

n f

pk k n

n



k

Proof. (i)



(ii) is clear.

(ii)



(iii): Let

[ c ˆ , F , p ]

₀

( 

_u^m

)  [ c ˆ , p ]

_

( 

^m_u

).

Suppose that (iii) does not hold. Then

(3) 0),

>

( 0

= )]

( 1 [

inf

1

=

t t

n f

pk k n

n



k

and there exists a sequence

( n

_i

)

of positive integers such that

109

. 1,2,

= 1 ,

<

)]

( 1 [

1

=

 i

i for i

n f

pk k ni

i k



Now define the sequence

x = { x

_k

}

by

 

 

  

.

>

, 0

, 1,2,

= , 1

,

=

_i

i

k

k n

i for n k if i x



By Eqn.(3),

x  [ c ˆ , F , p ]

₀

( 

^m_u

)

^but

x  [ c ˆ , p ]

_

( 

^m_u

),

which contradicts (ii).

Hence (iii) is true.

(iii)



(i) :

Let (iii) is true and

x  [ c ˆ , F , p ]

₀

( 

^m_u

)

^i.e.

}.(4) 0,

=

|)]

(|

1 [ lim

1

=

s in uniformly x

u n f

pk s k m k k n

n



k





Suppose that

x  [ c ˆ , p ]

₀

( 

_u^m

)

.Then for some number



₀

> 0

and positive integer

n

₀ we have

|

0

| u

_k



^m

x

_ks

 

for some

s  s 

and

1  k  n

₀. Therefore

pk s k m k k pk

k

f u x

f ( )] [ (| |)]

[ 

₀

 

_

and hence

lim [ (

0

)] = 0

1

=

pk k n

n k

f 



, which contradicts (iii). Thus

[ c ˆ , F , p ]

₀

( 

^m_u

)  [ c ˆ , p ]

₀

( 

^m_u

).

Theorem 2.3.

Let

1   sup

_k

< 

k

k

p

p

. The inclusion

[ c ˆ , F , p ]

_

( 

^m_u

)  [ c ˆ , p ]

₀

( 

_u^m

),

holds if and only if

0.(5)

>

= )]

( 1 [

lim

1

=

t for t

n f

pk k n

n k

 

Proof. Let

[ c ˆ , F , p ]

_

( 

^m_u

)  [ c ˆ , p ]

₀

( 

_u^m

),

. Suppose Eqn(5) does not hold. Then there exists a number

t

₀

> 0

and a sequence

( n

_i

)

of positive integers such that

.(6) 1,2,

= ,

<

)]

( 1 [

0 1

=

 i

M t

n f

pk k ni

i k



 

Noe we define the sequence

x = ( x

_k

)

by

 

 

  

.

>

, 0

, 1,2,

= 1

,

=

0

i

i

k

k n

i for n k if t

x



Thus by Eqn (6),

x  [ c ˆ , F , p ]

_

( 

^m_u

)

^but

x  [ c ˆ , p ]

₀

( 

^m_u

)

.So that Eqn (5) must hold.

Conversely let Eqn (5) hold. If

x  [ c ˆ , F , p ]

_

( 

_u^m

)

, then for each

s

^and

n

.(7)

<

|)]

(|

1 [

1

=





 ^{f u}  ^{x M} n

pk k m k k n

k

Suppose that

x  [ c ˆ , p ]

₀

( 

_u^m

)

. Then for some number



₀

> 0

and positive integer

s

₀ and index

n

₀ we have

|

0

| u

_k



^m

x

_ks

 

for

s  s

₀. Therefore

,

|)]

(|

[ )]

(

[ f

_k



₀ ^pk

 f

_k

u

_k



^m

x

_ks ^pk

and hence for each

k

and

s

we get

110

,

<

)]

( 1 [

0 1

=



 ^f  ^M n

pk k n

k



for some

M > 0

, by Eqn (7) which contradicts Eqn (5). Hence

) ( ] ˆ , [ ) ( ] , ˆ ,

[ c F p

_



_u^m

 c p

₀



_u^m

.

Theorem 2.4.

Let

1   sup

_k

< 

k

k

p

p

. The inclusion

[ c ˆ , p ]

_

( 

^m_u

)  [ c ˆ , F , p ]

₀

( 

_u^m

),

holds if and only if

0.(8)

>

0

= )]

( 1 [

lim

0

1

=

t for t

n f

pk k n

n



k

Proof. Suppose that

[ c ˆ , p ]

_

( 

_u^m

)  [ c ˆ , F , p ]

₀

( 

^m_u

),

but (8) does not hold. Then for some

t

₀

> 0

0.(9)

=

= )]

( 1 [

lim

0

1

=

 ^{f t L}  n

pk k n

n k

Define the sequence

x = ( x

_k

)

by

 







 

















^

 ^{ }



k k m t

x

^m

k k

1 1)

(

=

0

= 0

for

k = 1,2,  .

Then

x  c

₀

( F , p , 

^m_v

)

, by Eqn (6).Hence Eqn (5) must hold.

Conversely let

x  [ c ˆ , p ]

_

( 

^m_u

)

, and suppose that Eqn (8) holds. Then for every

k

and

s

.

<

|

| u

_k



^m

x

_ks

 M 

Therefore

, )]

( [

|)]

(|

[ f

_k

u

_k



^m

x

_ks ^pk

 f

_k

M

^pk

and

(8).

0,

= )]

( 1 [

) lim

|)]

(|

1 [ lim

1

= 1

=

Eqn by M

n f x

u n f

pk k n

n k pk s k m k k n

n



k

^

^

^ 

Hence

x  [ c ˆ , F , p ]

₀

( 

_u^m

)

. 3. REFERENCES [1]. Çi

g 

dem . A. B., 2006. On some difference sequence spaces defined by a sequence of orlicz functions,Journal of Zhejiang University Science A .,7(12) :2093-2096.

[2]. Colak, R., and M. Et,1995. On some generalized difference sequence spaces , Soochow J. Math., 21(4):

377-386.

[3]. Kizmaz,H., 1981. on certain sequence spaces Candian Math. Bull.,24(2): 169-176.

[4]. Kolk,E., 1993. on strong boundedness and summmability with respect to a sequence of moduli, Acta Comment.

Univ. Tartu,960 :41-50

[5]. Kolk,E., 1994. Inclusion theorems for some sequence spaces defined by a sequence of moduli, Acta Comment.

Univ. Tartu,970 :65-72

[6]. Maddox, I.J.,1978. A new type convergence, Math.Proc.Camp.Phil.Soc., 83: 61-64.

[7]. Maddox, I.J.,1986. Sequence spaces defined by a modulus, Math. Camb. Phil. Soc., 100, 161-166.

[8]. Maddox,I.J.,1967. Spaces of strongly summable sequences. Quart. J. Math., 18 : 61 - 64.

[9]. Ruckle,W.H., 1973. FK spaces in which the sequence of coordinate vectors is bounded , Canad.

J.Math.,25,973-975.

[10]. Lorentz, G.G., 1984. A contribution to the theory of divergent sequences. Acta Mathematica, 80 (1) : 167 - 190.

[11]. Nanda , S., 1984. Strongly almost convergent sequences. Bull. Cal Math.Soc., 76 : 236 - 240.