The Sequence Space

ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

The Sequence Space ^{ w F p ( , )} Defined By Means of The Modulus Function Sequences Among the Spaces

with Semi-Norms

¹

Kamil Akbayır and Tunay Bilgin

Department of Education of Mathematics and Natural Sciences, Faculty of Education, University of Yüzüncü Yıl, 65080, Van, TURKEY.

email:[email protected], [email protected].

DOI : http://dx.doi.org/10.29055/jcms/727 (Received on: January 7, 2018)

ABSTRACT

By means of the modulus function sequence F  ( f

_k

) we have generalized the sequence space  w f p ( , ) studied by Başarır

⁵

and the sequence space

( , , , )

w A V  f studied by Bilgin

⁹

to build the sequence space  w F p ( , ) and given some of its properties.

Keywords: Modulus function, Sequence spaces, Summability, Paranormed spaces.

1. INTRODUCTION

The idea of modulus function suggested by Nakano

²⁵

gained a new dimension to the studies of the sequence space. A function

^{f :}



^0,

^  ^ 

^0,

^  satisfying the following properties is called a modulus function (Nakano

²⁵

):

i) f(x)   0

_{x 0}

_,

ii) For every

^{x, y 0}

one has

^f(x

^

^y)

^

^f(x)

^

^f(y)

, iii) f is incresaing,

iv) f is continuous at 0 from right.

A modulus function f might be bounded or unbounded. From (ii)

^f(x)

^

^f(y)

^

^f(x

^

^y)

and (iv) we see that f is everywhere continuous on

[0,)

.

Maddox

²⁴

defined the class of strong Cesàro summable sequences with respect to modulus, a generalization of the definition of the strong Cesàro summability, as ^{w f}   . Connor

¹²

1 This work consists of a part of the Ph.D thesis of Kamil AKBAYIR titled “Modulus Fonksiyon Dizileri Yardımıyla Tanımlanmış Bazı Dizi Uzayları.”

generalized the definition of (Maddox

²⁴

) to ^{w A f}  ^,  summability method as any non- negative regular matrix instead of the Cesàro matrix.

We need to point out that so many spaces have been constructed by means of the modulus function (Banerji and Galiz

²

; Soomer

²⁸

; Esi

¹⁴

; Kolk

19,20,21,22,23

; Esi and Et

¹³

; Pehlivan and Fisher

²⁶

; Gupta and Bhola

¹⁶

; Bilgin

^7,8,10

; Raj and Sharma

²⁷

; Karakaya and Şimşek

¹⁸

; Bhardwaj and Bala

⁶

; Işık

¹⁷

and the others).

In this work by means of the modulus function sequence F  ( f

_k

)

we will generalize the sequence space  w f p ( , ) studied by Başarır

⁵

and the sequence space w A V ( , , , )  f studied by Bilgin

⁹

to build the sequence space  w F p ( , ) and give some of its properties.

Let F    f

k

be a modulus function sequence, and for every k p 

_k

0 ,

sup

_k

k

H  p

and C

,

_k



k

b

a . The following conditions and inequalities will be used in our future discussion:

(M1)

sup

_k

( ) ,

k

f t  

for   t 0 _, (M2) lim

⁰ _k

( ) 0,

t

f t





(uniform for k  1 _{) (Kolk}

¹⁹

_).

(E1) ^a

^k

^{ b}

^{k pk}

^{ C a} 

^{k pk}

^{ b}

^{k pk}

 ^, C  max(1, 2

^H1

)   t 0 (E2) ^

^pk

^{ max 1,}  ^

^H

 _(Maddox

²⁴

₎

2. MAIN RESULT

2.1. The Sequence Space  w F p ( , ) and Some of Its Properties

Given a sequence x  ( ) x

_k

and a modulus function sequence F    f

k

we can define the following spaces:

 

 

1 0

1

1

1

( , ) ( ) : lim 0

( , ) ( ) : sup

k

k

n p

k k k

n k

n p

k k k

n k

w F p x x s n f x

w F p x x s n f x



 







   

         

 

 

   

          

 

 





Here by choosing the sequences F    f

k

_and ( p

_k

) specifically one can obtain different spaces as special cases.

 

1

1

( , ) ( ) : lim

^k

0,

n p

k k k

n k

w F p x x s n

^

f x l for some l

 

   

          

 

  

Definition 2.1.1: Let x  ( ) x

_k

be a sequence for ( k  IN ) . If for    0 ^{we have}

 

lim

1

:

_k

0

n

n

^

k n x s 



    

we say that the sequence x  -statistically converges to the number s and we use the notation

 

S  to express this situation (Başarır

⁵

).

If we choose p

k

 p _for  k ^{we write}  w f p  ^,   w

p

 f ^,   . If we choose f x ( )  x we obtain w

p

 f ^,    w

p

   defined by Başarır

³

.

We will first give the linearity of the spaces  w F p ( , ) and  w F p

0

( , ) _.

Theorem 2.1.2:  w F p ( , ) and  w F p

0

( , ) are linear spaces on C .

Proof: Here we will only show that  w F p

0

( , ) is linear, the linearity of  w F p ( , ) _{can be} seen by the same way. It is sufficient to show that  x   y   w F p

0

 ,  _for

 

,

0

,

x y   w F p and for the scalars   ,  C. From the inequality (E1) and definition of the modulus function we have

 

 

 ^f

^k

^{  x}

^k

^{  y}

^{k pk}

 _  ^f

^k

^{    x}

^k

^{     y}

^k

^{ }

^pk



  ^f

^k

 ^  ^{ x}

^k

  ^{ f}

^k

 ^{   y}

^k

^  

^pk

_ ^{C f} 

^k

^{    x}

^k

^{ }

^pk

^{ f}

^k

^{    y}

^k

^{ }

^pk



 ^{C T} 

^H

^f

^k

 ^{ x}

^k



^pk

^{ K}

^H

^f

^k

 ^{ y}

^k



^pk

 _ CT

^H

f

k

  x

k



^pk

 CK f

^H k

  y

k



^pk

where T and K are positive integers such that T    1     _and K    1     . Thus we have

 

 

1

1

lim

^k

n p

k k k

n k

n

^

f  x  y

 

 

 

¹

   

1

lim

^k

n p

H

k k

n k

CT n

^

f x

 

  _



¹

   

1

lim

^k

n p

H

k k

n k

CK n

^

f y

 

 

and hence we see that  x   y   w F p

0

 , 

. Let us now show that  w F p

0

 , 

is a para-normed linear topological space:

Theorem 2.1.3: The space  w F p

0

 , 

is a linear topological space para-normed by g

defined by

 

1 1

1

( ) sup

^k

n M

p

k k

n k

g x n

^

f x



 

   

  

Proof: It is clear that g ( )   0, g x ( )  g (  x ) _{. Let} x y ,   w F p

0

 , 

. Since

k

1 p M 

and 1

M  , we see from the inequality (E2) and the definition of the modulus function that for each n we have

 

 

1 1

1

k

n p M

k k k

k

n

^

f x y



 

 

 

  

      

1 1

1

k

n p M

k k k k

k

n

^

f x f y



 

  

 

  



 

1 1

1

k

n M

p

k k

k

n

^

f x



 

  

  

  

1 1

1

k

n M

p

k k

k

n

^

f y



 

  

  

.

Now let   _{C. When} ^K    ¹     , from the definition of the modulus function we have

( )

g  x   

1 1

1

sup

^k

n M

p

k k

n k

n

^

f  x



 

  

  

 K

^{H M}

g x ( ) _. Now let   0 for any constant x _with ^{g x  ( ) 0} _{. For}   1

from Definition 2.1.1 we have

 

1

1

k

n p

k k

k

n

^

f  x



 

 ,  _for n  N ( )  (2.1) On the other hand, since f

^k

is continuous for 1  n  N if we choose  small enough we get

 

1

1

k

n p

k k

k

n

^

f  x 



 



. (2.2) When we consider (2.1) and (2.2) together when   0 we get g (  x )  0 . Thus our proof is complete.

We will lastly give some inclusion relations:

Theorem 2.1.4: Let 0  p

_k

 q

_k

and  p q

k k



be bounded, then we have

 ^,   ^, 

w F q w F p

  

.

Proof: Let ^{x   w F q}  ^,  . We can write t

k

 f

k

  x

k

 l 

^qk

and

k k

k

p

  q

such that 0    

_k

 1

. Define

, 1 0, 1 ,

k k

k

k

t t

u t

 

   

^,

0, 1

1

k k

k k

v t

t t

 

    _.

Then we have t

_k

 u

_k

 v

_k

, t

_k^k

 u

_k^k

 v

_k^k

. Thus it is clear that u

_k^k

 u

_k

 t

_k

, v

_k^k

 v

_k^

. Hence we obtain

1 1 1

1 1 1

k

n n n

k k k

k k k

n t n t n v





  

  

 

   

 

  

and this shows that ^{x   w F p}  ^,  . Thus our proof is complete.

Lemma 2.1.5: Let p 

_k

0 _and q 

_k

0 _{. If} ^{lim inf} 

k k

 ⁰

k

p q 

then we have  c q

⁰

( )   c p

⁰

( ) (Başarır and Et

⁴

).

Theorem 2.1.6: Suppose that lim inf

_k

0

k

p 

and (M2) is satisfied. Then if  x

^k

 l

we have

 ^, 

x

k

 l    w F p   _.

Proof: Let  x

^k

 l

. From (M2) we have f

k

  x

k

 l   ⁰

. Since ^{lim inf}  

k

⁰

k

p 

from Lemma 2.1.5 we clearly have f

k

  x

k

 l 

^pk

 ⁰

for q

_k

 1,  k

. We thus see that

 ^, 

x

k

 l    w F p   _.

Theorem 2.1.7: Let 0   h inf p

_k

 sup p

_k

 H  

and suppose that (M1), (M2) are satisfied.Then we have

( ) ( , ), w p w F p

   

₀

w p ( )  

₀

w F p ( , )

and 

_

w p ( )  

_

w F p ( , ) .

Proof: We will only show the first inclusion, the others are obtained by the same way. Let 0   h inf p

_k

 sup p

_k

 H  

and x   w p ( ) then we have

 

1

1

( )

^k

0,

n p

n k

k

s p n

^

x l



     _{( n   )} .

Let (M2) 1    0 from (M2). For 0   t  _{choose a}  _with 0    0 1   t  such that

k

( ) f t  

. Write t

^k

  x

^k

 l

and suppose that

 

1 1 2

k

n p

k k

k

f t



 

  

where the first sum was taken over t

^k

 

and the second sum was taken over t

^k

 

. In this case

1

h

n



  _{and for}

t

k

  _{we have}

 

k k

1

k

t  t    t 

where   ^{t ,} denotes the integer part of t . From Definition 2.1.2 and (M1) for t

^k

 

we get

 

k k

f t   ¹   t

k

   f

k

⁽¹⁾

 2 (1) f

_k

t

_k



.

Thus we have ^

₂

^{ max}   ^{2 }

^1

^f

^k

⁽¹⁾  

^h

^{, 2 }

^1

^f

^k

⁽¹⁾ 

^H

 ^ns

ⁿ

^{( ) p} and so with

1 h

n



 

the first inclusion is satisfied.

3. CONCLUSION

In this paper, by means of the modulus function sequence F  ( f

_k

) _{we have} generalized the sequence space  w f p ( , ) studied by Başarır

⁵

and the sequence space

( , , , )

w A V  f studied by Bilgin

⁹

to build the sequence space  w F p ( , ) and given some of its properties.

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