New $\Delta_{q}^{v}$ -difference operator and topologıcal features

New

v q



_{-difference operator and topological}

features

Abdulkadir KARAKAŞ*_{, Mahir Salih Abdulrahman ASSAFI}

Siirt University Faculty of Arts and Sciences, Department of Mathematics, Kezer Campus, Siirt.

Geliş Tarihi (Received Date): 20.04.2020 Kabul Tarihi (Accepted Date): 14.10.2020 Abstract

We extented v by using difference operator  . We generated the difference sequence v_q space l _p( v_q) and investigated some of their properties. We showed that, if l _p( v_q) is supplied with an proper norm

,

.

q v

p  then it will be a Banach space. We further more showed that, the sequence spaces

(

( ), . _, v

)

q

p _p

v q

l  _ and

(

l_p, . _p

)

are linearly isometric. At

the end of this studies, it was shown that ( _qv) ( , v)

p p q

l  l  . The family of the Orlicz functions is coincides the  − condition. ₂

Keywords: Difference sequence spaces, isometric sequence spaces, sequence spaces.

Yeni



vq

-fark operatörü ve topolojik özellikleri

Öz

v q

 fark operatörünü kullanarak v

’yi genişlettik. ( )v p q

l  fark dizi uzayını oluşturduk ve bazı topolojik özelliklerini inceledik. Eğer ( )v

p q l  uygun bir , . q v p  normu verilirse bunun bir Banach uzayı olacağını gösterdik. Ayrıca

(

( ), . _, v

)

q p _p v q l  _ ve

(

_p, .

)

p l dizi

uzaylarının lineer izometrik olduklarını gösterdik. Çalışmanın sonunda ise

( _qv) ( , v)

p p q

l  l  olduğu gösterildi. Orlicz fonksiyonlarının ailesi ,  − şartı ile ₂ örtüşmektedir.

* _{Abdulkadir KARAKAŞ, kadirkarakas21@hotmail.com, https://orcid.org/0000-0002-0630-8802}

Anahtar kelimeler: Fark dizi uzayları, izometrik dizi uzayları, dizi uzayları.

1. Introduction

Let c l, and c ₀ be the Banach spaces of convergent, bounded and null sequences 1

( k)

u= u  respectively with complex terms, normed by

sup _k

k

u _ = u ,

where k  .

Kızmaz [1] presented the difference sequence spaces,





( ) ( _k) :

U  = u= u  u U

for U =c, and ,l c_{ 0} where

1

( _k) ( _{k k} )

u u u u ₊

 =  = − .

We have the norm for these Banach spaces as:

1

u _ = u + u _.

Çolak and Et [2] have extended the spaces U ( ) to the U ( v) for U=c l, _ and c₀. Let

U be any sequence spaces and defined





( v) ( ) : v k

U  = u= u  u U

where v  and  =  vu

(

( v−1)u_k

)

for all k  and prove that

0

( v), ( v) and ( v)

c  l_  c  are Banach spaces with the norm

0 ( 1) , v v t k k t t v t u u ₊ =        =



− 1 v i v v i u _ u u  = =



+  .

Karakaş et al. [3] have defined the sequence spaces c(_q), (l _q) and (c₀  . He also _q) presented

1

( ) ( )

qu q ku quk uk+

 =  = −

for q  . Karakaş et al. [4] have presented





v v

for U =c l, and c_{ 0}, where q v , . They showed that the spaces ( v) q U  are Banach spaces by: 1 , v q v q i i v u _ u u  = =



+  where 1 1 1 ( ) ( ) v v v v qu quk q q uk q uk − − +  =  =  −  and 0 ( ) ( 1) v v v t v t q q k k t t v q t u u −u ₊ =        =  =



− .

Recently, Peralta [5] has studied l  and investigated the topological features of this _p( v) space. In this work, we choose p [1, ) . By  , we denote the space of all sequences

( )_k

u= u , for u _k and all kN. Taken u, describe

1 1 p p p k k u u  =   =   



 and let



( ) :



p k _p l = u= u u   .

The linear operator q:

v  

 → is presented recursively as the composition

1

q q

v v

q −

 =   for v  and q 2 . It is obvious that for u and v  we have the 1 following Binomial representation

0 ( 1) v v t v t q k k t t v q t u −u ₊ =        =



− for all k  .

Let v  and define the sequence space l _p( v_q) by





( ) ( ) : p k v v q q p l  = u= u  u l .

The sequence spaces are Banach spaces normed by

1 1 , p v q v _p p v i _{q p} i p u _ u =   =_ + _ 



  (1.1)

For Euler difference sequence spaces and sequence spaces generated by a sequence of Orlicz functions, the reader can consult Altay and Polat [6], Altay and Başar [7] and Qamaruddin and Saifi [8], respectively.

2. Main results

Theorem 2.1. The sequence space (l _p v_q) is a Banach space with the norm

, vq p   . Proof: Let ( ( ))

( )

( ( )n ) k n

u = u is a Cauchy sequence in l _p( v_q). Thus, for   we may 0 find a positive integer N such that

, ( ) ( ) v q n r p u u   − 

whenever ,n r . In other words, we have N

( ) ( ) ( ) ( ) 1 1 p _p v _p v v q q n r n r i i _p i u u u u  =   +         − − 



, for n r, N. Since ( ) ( ) ( ) ( ) , vq r n i i _p n r u u u u  −  − for i=1, 2,3,...,v and ( ) ( ) ( ) ( ) , vq v v q q _p n r n r p u u u u   −  − . Therefore, ( ) (uin )and ( ) )

(v_qun are Cauchy sequences in ℂ and lp, respectively. The

completeness of the spaces ℂ and lp show the existence of elements y i ,

1, 2,3,...,

i= v, and z=(z_k)l_p such that

( ) lim _{i i} 0 n n u −y = (2.1) for i=1, 2,3,...,v and ( ) lim v n 0. p q n  u −z = (2.2) Since

( ) ( ) v v q q n p n k zk z u u  −   − we get ( ) 0 n k k v qu z  − →

as n →  for all k  by equation (2.2). We obtain a recursive formula for lim ( ), 1,

n n v i u+ i as n → . We have 1 1 ( ) ( ) ( ) 1 1 0 ( 1) ( 1) t v v n v n t v t n v q t v q t u u u + − − + =       − =  −



− and so 1 1 1 1 0 ( ) : lim ( 1) ( 1) v v t v t v v n t n v i v q t w u z y − + + − + =         = = − _ − − _ 





Assume that w_v+1,...,w_{v k+ −1},1  have been established. Where k v,

( ) :lim n , 1, 2,..., 1. i n v v i w₊ u ₊ i= k−

Using these, we acquire, for 1 k  v

( 0 1 1 ) ( 1) : lim ( 1) ( 1) v k t v t k t k t v v k _k n v k t k t v t v k t t n v k v q t q z y w u v w − − + = + ₋ − + − + − + = +        _{− −}      = − _{ }   − −      _{ }    =





On the other side, for k  we get v 1 ( ) ( ) ( ) 0 ( 1) ( 1) t k v v n v n t v t n v k q k t v q t u u u₊ − − + =       − =  −



− . So that ) 1 ( 0 : lim ( 1) ( 1) v v t v t v k v k t t n k k n v q t w₊ u ₊ z w₊ − − =         = = − _ − − _ 



.

Let w=

(

y1,...,y wv, v+1,wv+2,... .

)

We assert that ( ) v p q

wl  , that is,   . First, show vqw lp

1 1 1 1 0 1 1 1 1 1 0 0 1 ( ) ( 1) ( 1) ( 1) ( 1) v v t v t v q t v t v v t v t t v t t t t t v q t v v q q t t w y w y z y z − − + + = − − − − + + = =                    = − + −   = − +_ − − _   =







Also, for k=2,3,...,v. We get

1 0 1 ( ) ( 1) ( 1) ( 1) v k v v t v t t v t v q k t k t k v k t t v k k v v q q t t w y w w z − − − − + + + = = − +              = − + − + − =





Similarly, for k we acquire v 1 0 ( ) ( 1) ( 1) . v v t v t v q k t k v k t k v q t w w w z − − + + =        = − + − =



Thus we have presented that v_qw= z l_p. It remains to prove that

( ) , v 0 q p n u w  − → as n →  Then, we obtain , ( ) ( ) ( ) 1 ( ) ( ) 1 lim lim lim lim 0. v q p v p _n p _{v n v} k k q q n n k v _{p p} n v n k k q _p n n n k u y u w u y u z u w  ₌ =   = _ − +  −  _   = − − − +  =





This is proof of the theorem.

Theorem 2.2. The sequence spaces

(

( ), . _, v

)

q

p p

v q

l  _ and

(

l_p, . _p

)

are linearly isometric. Proof: Take in to consideration the map : ( v) _p

q p l l T  → given by Ty u= , where ) ( _k) _p( v_q y= y  l and u=( )u_k with , if 1 ; , if . k k v q k v y k v u y ₋ k v    = _{ } 

The linearity of the difference operator Δ refers the linearity of T . If ( v) p q yl  and Ty=u, then 1 1 1 1 , vq . v _p p p p _v k q k v p p k k v v _p p _v k q k k k p p Ty u y y y y y  − = = +  = =  = = +  = +  =  









This demonstrates that T is well-defined and it is also norm preserving. We presented that T is one-to-one and onto. Assume that Ty =0.

Then, we obtain 0 v qyk  = for all k 1, (2.3) 1 2 ... v 0. y = y = = y = (2.4)

We show that the difference equation (2.3) with initial conditions (2.4) refers that 0

k

y = for all k 1,, that is, y =

(

0, 0,...

)

. Therefore, T is one-to-one.

Assume that u=(uk)lp. Describe the sequence y=(yk) as follows. Let yk = for uk

, 1, 2,..., .

v

k v q k

u ₊ =  u k = v

The succeeding terms of the sequence y is then showed recursively by

1 1 1 1 0 ( 1) ( 1) v v t v t v v t t v q t y u u − − + + + =         = − _ − − _ 



 0 1 1 ( 1) ( 1) , 1 ( 1) v k t v t v k t k t v v k _{k v k t} k t t v t t v q t v q v k t u u y k v y − − + + = + _{− − +} − + =          _{− +}     _{− −}      = −     − −    





and 1 0 ( 1) ( 1) , . v v t v t v k v k t k t v q t y u y k v − − + + + =         = − _ − − _  





Utilizing a similar argument as in the proof of the previous theorem, we prove that

v

qyk uk v+

 =

Thus, we obtain 1 1 . p p v v q _p q k k p k v k p p y y u u  =  + =  =  = =  





So that yl_p( . Since T is onto, ( )_qv) l _p v_q and l_p are linearly isometric.

Definition 2.3. An Orlicz function is a continuous, convex function and nondecreasing



)



)

: 0, 0,

M  →  such that M z =( ) 0, if and only if z = , 0 M u ( ) 0, and

( ) as

M u →  u→ . M is said to fulfil  − condition if there exists a positive ₂ constant K such that M(2 )z KM z( ) for all z  . Let 0 =(M_k) be a sequence of Orlicz functions meeting the  − condition [9]. An Orlicz function M has been defined ₂ in [10] also see [11] for a more general representation in thise direction in the following from:

0

( ) ( )

u

M u =



p t dt

where p , know as the kernel of M , is right-differentable for t0, ( )p t 0, (0)p =0

for t0, p is nondecreasing, and t→ , ( )p t → .

Lindenstrauss and Tzafriri [12] have utilized the view of Orlicz function to find the sequence space,

(

)

1 ) ( ) : ,for some 0 , ( _k p p k k k u u M u l    =   =_ =    _ 





which is a Banach Spaces with respect to the norm

(

)

1 ( )_k inf 0 : _{k k} 1 . k u  M u   =   = _   _ 





The space l( )is closely related to space l_p, which is an Orlicz sequence space with ( ) p, for 1 .

M u = u    p

Describe the sequence spaces as:

(

)

1 ) ( ) : ,for some 0 , ( _k p p k k k u u M u l    =   =_ =    _ 









) ( ) : )

( , v_q v_{q p}(

p u uk

l  = =  ul .

Theorem 2.4. Let =(M_k)be a sequence of Orlicz functions fulfil the  − condition. ₂ If

(

)

1 p k k k M u   =  



(2.5)

for all t,0 then l_p( v_q) l_p( ,v_q).

Proof: Assume that condition (2.5) exists and let u=(u_k)  . Then, we get l_p( v_q)

1 . p v q k k u  =   



(2.6) The convergence of 1 p v q k k u  =   



implies that lim v_{q k} 0. k  u =

Thus, we can find n such that v_qu_k  for all k N1  . Let



1 1



max v ,... v ,1 .

q q N

K =  u  u ₋

Then v_qu_k  for all k  . For K  , utilizing the monotonicity of 0 M , we get _k

(

v

)

(

)

k q k k

M  u  M K  for all k  . This inequality shows that

(

)

(

)

1 1 . p _p v k q k k k k M u  M K    = =  





This estimate proves that  v_qu l_p( ) that is, ul_p( , By equation (2.5) v_q). Therefore, the inclusion (l_p  _qv) l_p( , holds. v_q)

3. Results and discussion

Peralta [5] studied l _p( v_q) and checked the topological properties of this space. Later Karakaş et al. [4] defined difference operator v

q

 . We used Peralta' s [5] studies and extented it by used the generalized difference operator  . We generated the difference v_q sequence space l _p( v_q) and

,

.

q v

p  , and investigated some of their properties. We showed

that, if l _p( v_q) is equipped with an appropriate norm

,

.

q v

p  is a Banach space. We

further more showed that, the sequence spaces

(

( ), . _, v

)

q p p v q l  _ and

(

_p, .

)

p l are linearly

isometric. It is shown that l_p( _qv) l_p( , . Where v_q) a family of Orlicz functions, is coincides the  − condition₂ .

References

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[2] Çolak, R. and Et, M., On some generalized difference sequence spaces and related matrix transformations, Hokkaido Mathematical Journal, 26, 3, 483-492, (1997).

[3] Karakaş, A., Altın, Y. Et, M., On some topological properties of a new type difference sequence spaces, Advancements In Mathematical Sciences, Proceedings of the International Conference on Advancements in Mathematical Sciences (AMS-2015), Fatih University, Antalya, 144, (2015). [4] Karakaş, A., Altın,Y. and Çolak, R., On some topological properties of a new

type difference sequence spaces, International Conference on Mathematics and Mathematics Education (ICMME-2016), Firat University, Elaziğ-Turkey, 167-168, (2016).

[5] Peralta, Isometry of a sequence space generated by a difference operator, International Mathematical Forum, 5, 42, pp. 2077-2083, (2010).

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[7] Altay, B., Başar, F., The matrix domain and the fine spectrum of the difference operator  on the sequence space l_p, 0  Communications in p 1 Mathematics and Applications, 2, 2, 1-11, (2007).

[8] Qamaruddin and Saifi, A. H., Generalized difference sequence spaces defined by a sequence of Orlicz functions, Southeast Asian Bull. Mathematics, 29, 1125-1130, (2005).

[9] Kamthan, P. K., Gupta, M., Sequence Spaces and Series, Marcel Dekker Inc. Newyork, (1981).

[10] Krasnoselskii, M. A., and Rutickii, Y. B., Convex Functions and Orlicz Spaces, Groningen, Netherlands, (1961).

[11] Kamthan, P. K., Convex functions and their applications, Journal of Istanbul University Faculty of Science. A Series, 28, 71-78, (1963).