Some New Sequence Spaces Defined by Musielak-Orlicz Functions on a Real n-Normed Space

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Some New Sequence Spaces Defined by Musielak-Orlicz Functions on a Real n-

Normed Space

Tanweer Jalal¹ and Mobin Ahmad Sayed²

1Department of Mathematics, National Institute of Technology Hazratbal, Srinagar- 190006, India

E-mail: tjalal@rediffmail.com

2Department of Mathematics, Faculty of Science Jazan University, P.O. Box 2097, Jazan 45142, Saudi Arabia

E-mail: profmobin@yahoo.com (Received: 23-8-14 / Accepted: 14-3-15)

Abstract

The purpose of this paper is to introduce the sequence

space ^Enq

(

^{B, Μ, p,s, .,...,.}

)

defined by using an infinite matrix and Musielak-Orlicz function. We also study some topological properties and prove some inclusion relations involving this space.

Keywords: Paranorm, Infinite matrix, n-norm, Musielak-Orlicz functions, Euler transform.

1 Introduction and Preliminaries

The concept of 2-normed spaces was initially developed by Gähler [1] in the mid- 1960s, while one can see that of n -normed spaces in Misiak [2]. Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4]

and Gunawan and Mashadi [5]. Let n be a non-negative integer and X be a real vector space of dimension d, where d ≥n≥2. A real-valued function

. ..., ,

. on X ⁿ satisfying the following conditions:

(1)

(

x1^,x2^,...,xn

)

= ⁰ if and only if x₁,x₂,...,x_n are linearly dependent.

(2)

(

x₁,x₂,...,xn

)

is invariant under permutation,

(3) α x1^,x2^,...,xn = α

(

x1^,x2^,...,xn

)

^, for any α ∈ R, (4)

(

x₁ + x,x₂,...,xn

)

≤

(

x₁,x₂,...,xn

)

+

(

x,x₂ ,...,xn

)

is called an n norm on − X and the pair

(

^{X , .,...,.}

)

is called an n normed − space.

A trivial example of an n normed space is − X = Rⁿ, equipped with the Euclidean n norm−

(

x₁,x₂,...,xn

)

_E ⁼ volume of the n-dimensional parallelepiped spanned by the vectors x₁,x₂,...,x_nwhich may be given explicitly by the formula

(

x₁,x₂,...,xn

)

E ⁼ det

( )

xij ⁼ abs

(

det

(

^〈xi , xj^〉

) )

(1) where xi =

(

xi₁,xi₂,...,xin

)

∈Rⁿ for each i = 1, 2, 3...,n. Let

(

^{X , .,...,.}

)

^{be an} n normed space of dimension ⁻ d ≥ n ≥ 2 and

{

a₁,a₂,...,an

}

be a linearly independent set in X. Then the function .,...,. _∞ on X ⁿ⁻¹ is defined by

(

^x1,x2,...,xn

)

∞ = ₁^max_≤i≤n

{

^{x1,x2,...,xn−1,ai}

}

(2) defines an

(

n−¹

)

-norm on X with respect to

{

a₁,a₂,...,an

}

and this is known as the derived

(

n−¹

)

-norm.

The standard n -norm on X a real inner product space of dimension d ≥ n is as follows:

(

^x1,x2,...,xn

)

^{s =}

[

^det

(

^{〈 xi , xj〉}

) ]

^12,

where 〈 , 〉denotes the inner product on X If we take . X = Rⁿ then this n - norm is exactly the same as the Euclidean n -norm

(

x₁,x₂,...,xn

)

_E

mentioned earlier. For n = 1 this n -norm is the usual norm

〉

= 〈 1, 1

1 x x

x for further details (see Gunawan [4]).

We first introduce the following definitions:

A sequence

( )

^xk in an n -normed space

(

^{X , .,...,.}

)

is said to be convergent to some L∈X if

, 0 ,

...

, , ,

lim − _{1 2 −1} =

∞

→ k n

k

z z z L

x for every z₁,z₂,...,z_n−1 ∈X. (3)

A sequence

( )

^xk in an n -normed space

(

^{X , .,...,.}

)

is said to be Cauchy if ,

0 ,

...

, , ,

lim − _{1 2 −1} =

∞

→→ ∞

n p

k k

z z z x x

p

for every z₁,z₂,...,z_n−1 ∈ X. (4)

If every Cauchy sequence space in X converges to some L∈X, then X is said to be complete with respect to the n -norm. A complete n -normed space is said to be a n - Banach space.

An Orlicz function is a function M ^:

[

^0,∞ ⁾ →

[

^0, ∞^), which is continuous, non-decreasing and convex with ^M

( )

^{0 = 0, M}

( )

^{x > 0 as x >0 a}

( )

^{x → ∞ ,}

M as x →∞.

Lindenstrauss and Tzafriri [6] studied some Orlicz type sequence spaces defined as follows:

. 0 ,

: ) (

1 









  <∞ >









∈ 

=

∑

^∞

=

ρ ^{for some} ρ x

M w

x

k

k k

ℓM

(5)

The space ℓ_M with the norm











  ≤









> 

=

∑

^∞

=

1 :

0 inf

1 k

xk

M

x ρ ρ

(6)

becomes a Banach space which is called an Orlicz sequence space. The space ℓM is closely related to the space ℓ_p which is an Orlicz sequence space with

( )

^{t t p,}

M = for 1 ≤ p <∞. An Orlicz function M is said to satisfy condition

−

∆2 for all values of u if there exists a constant , K such that

( )

²u ≤ K M

( )

u ^,u ≥ ⁰

M (see [7]).

A sequence space ^Μ=

( )

^Mk of Orlicz functions is called a Musielak-Orlicz function see ([8], [9]). A sequence space ^Ν=

( )

^Nk defined by

^Nk

( )

^{v = sup}

{

^{v u − M}k

( )

^{u :u ≥ 0}

}

^{, k = 1, 2,...} (7) is called the complimentary function of a Musielak-Orlicz function Μ. For a given Musielak-Orlicz function Μ, the Musielak-Orlicz sequence space function

tΜ and its subspace h_Μ are defined as follows

( )

{

^{∈ : <∞ > 0}

}

^,

= _Μ

Μ x w I c x for some c

t (8)

( )

{

^{∈ : <∞ > 0}

}

^,

= _Μ

Μ x w I c x for all c

h

where I_Μ is a convex modular defined by

( ) ( )

^,

( )

^.

1

M k k

k

k x x x t

M x

I =

∑

^∞ = ∈

= Μ

(9) We consider tΜ equipped with the Luxemberg norm







  ≤



 



> 

=inf 0: Μ 1

k I x k

x

(10)

or equipped with the Orlicz norm

( )

(

¹

)

^{: 0 .}

inf 1

0







 + >

= I_Μ kx k

x k

(11)

Let X be a linear metric space. A function p : X → R is called a paranorm, if

(1) p

( )

x ≥ ^0, for all x∈ X; (2) ^p

(

^−x

)

^{= p}

( )

^{x , for all x∈ X;}

(3) ^p

(

^x + ^y

)

≤ ^p

( )

^x + ^p

( )

^{y ,} for all x y ∈X;

(4) If

( )

^σn is a sequence of scalars with σn →σ as n → ∞ and

( )

^{xn is a}

sequence of vectors with ^p

(

^xn−x

)

^{→ 0 as n → ∞, then}

(

^{x − x}

)

^{→ 0}

p σ_{n n} σ as n → ∞.

A paranorm p for which p

( )

x =⁰ implies x= 0 is called total paranorm, and the pair

(

^{X ,p}

)

is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [10],

Theorem 10. 4.2, P-183). For more details about sequence spaces see [11-24] and the references therein.

Let

( )

sk denotes the sequence of partial sums of the infinite series

∑

^∞

=0 k

ak and q be any positive real number. The Euler transform

(

^{E ,q}

)

of the sequence

( )

sn

s = is defined by

( ) ( ) ∑

=

 −



 





= + ⁿ

o v

v v n n

q

n q s

v n s q

E .

1 1

(12) The series

∑

^∞

=0 n

an is said to be summable

(

^{E ,q}

)

to the number s if

( ) ( ) ∑

=

− → → ∞



 





= + ⁿ

o v

v v n n

q

n q s s as n

v n q

s E

1 1

(13)

and is said to be absolutely summable

(

^{E ,q}

)

or summable E ,q if

( )

− 1

( )

< ∞ ^.

∑

−

k

q k q

k s E s

E

(14)

Let x =

( )

xk be a sequence of scalars we write N_n

( )

x = E_n^q

( )

x − E_n^q₋1

( )

x ^, where ^En^q

( )

^x is defined by (12). After applications of Abel’s transformation, we have

( ) ( ) ∑

⁻

( ) ( ) ( )

=

−

−

−

−

− + − +

+ + + +

− +

= ² 0

1 1

1 1

1 1 ,

1 1

1 1

1 ⁿ

o k

n n n n n

n n k n k

n s

q q q

s q

A A s

x q

x N

(15)

where

1 . 1

1 0

−

−

∑

= ^



 







 



 −

 −



 





= ^k + ^{n i}

i

k q

i n i n q A q

(16)

Note that for any sequences x =

( ) ( )

xn ,y= yn and scalar λ , we have

(

^{x y}

)

^N

( )

^{x N}

( )

^y

N_n + = _n + _n and ^Nn

(

λ ^x

)

= λ ^Nn

( )

^{x .}

Let ^Μ=

( )

^Mk be a sequence of Musielak-Orlicz functions, ^p=

( )

^{pk be a}

bounded sequence of positive real numbers, "B =b_nk" be an infinite matrix, and

(

^{X , .,...,.}

)

^{be an} n normed space, we define the sequence space: ⁻

( )

( ) ^{( )}

_.

. . . , , 0

, 0 , ,

...

, , , :

. , ...

, . , , , ,

1 2

1 1

1 2

1

















∈

>

≥

∞

 <























= 

= Μ

−

∞

= −

∑

X z

z z every for and some

for

s z

z x z

M N k

x b x

s p B E

n

p

k

n k

s k k n k

q n

k

ρ

ρ

(17)

If we take p = pk =1 for all k∈N,we have

( )

( ) ^{( )}

. .

. . , , ,

0

, 0 , ,

...

, , , :

. , ...

, . , , ,

1 2

1 1

1 2

1

















∈

>

≥

∞

<























= 

= Μ

−

∞

= −

∑

X z

z z every for and some

for

s z

z x z

M N k x b

x

s B E

n k

n k

s k k n k

q n

ρ

ρ

(18)

If we take s =0 , we have

( )

( ) ^{( )}

. .

. . , , ,

0

, ,

...

, , , :

. , ...

, . , , ,

1 2

1 1

1 2

1

















∈

>

∞

 <























= 

= Μ

−

∞

= −

∑

X z

z z every for and some

for

z z

x z M N

b x

x

p B E

n k

p

n k

k k n k

q n

k

ρ

ρ

(19)

The following well known inequality will be used throughout the article. Let

( )

pk

p= be any sequence of positive real numbers with

{

^{1, 2 1}

}

max ,

sup

0≤ ≤ k = = ^H−

k

k p H D

p then



 

 +

≤

+ ^{k k} k ^pk

p k p

k

k b D a b

a (20)

for all k∈N and a_k,b_k∈ C. Also ^{a pk ≤ max}

{

^{1, a H}

}

for all a∈C (see [25]).

The main object of the paper is to examine some topological properties and inclusion relations between the above defined sequence spaces.

2 Some Properties of the Sequence Space

(

^{B, , p,s, .,...,.}

)

E_n^q Μ

Theorem 2.1: Let ^Μ=

( )

^Mk be a Musielak-Orlicz function and ^p=

( )

^{pk be a}

bounded sequence of positive real numbers, then the space

(

^{B, , p,s, .,...,.}

)

E_n^q Μ is linear over the real field.

Proof: Let ^x,y∈Enq

(

^{B, Μ, p,s, .,...,.}

)

^{and α ,β ∈ ℜ} (the set of real numbers). Then there exists numbers ρ₁ andρ₂ such that

( )

, , ,..., ,

1

1 2

1 1

∞

 <























∑

^∞ 

= −

pk

k

n k

s k k

n N x z z z

k M b

ρ and

( )

, , ,..., .

1

1 2

1 2

∞

 <























∑

^∞ 

= −

pk

k

n k

s k k

n N y z z z

k M b

ρ (21)

Define ^{ρ3 = max}

(

^{2 α ρ1,2 β ρ2}

)

^.

Since ^Μ=

( )

^Mk is non-decreasing, convex and so by using inequality (20), we have

( )

^pk

k

n k

s k k

n N x y z z z

k M

∑

^∞ b

= −























 +

1

1 2

1 3

, ...

, , ρ ,

β

α

( ) ( )

^pk

k

n k

n k

s k k

n N y z z z

z z x z M N

k

∑

^∞ b

= − −





















 +











≤ 

1

1 3

1 2

1 3

, ...

, , , ,

...

, ,

, ρ

β ρ

α

(22)

( )

+























≤

∑

^∞ 

= −

pk

k

n k

s k k

n N x z z z

k M D b

1

1 2

1 3

, ...

, , ρ ,

( )

, , ,..., .

1

1 2

1 3

∞

 <























∑

^∞ 

= −

pk

k

n k

s k k

n N y z z z

k M D b

ρ

Therefore, ^{α x+β y∈Enq}

(

^{B,Μ, p,s, .,...,.}

)

^.

Hence, ^Enq

(

^{B, Μ, p,s, .,...,.}

)

is a linear space.

Theorem 2.2: Let ^Μ=

( )

^Mk be a sequence of Musielak-Orlicz functions,^p=

( )

^pk

be a bounded sequence of positive real numbers. Then the space

(

^{B, , p,s, .,...,.}

)

E_n^q Μ is a paranormed space with the paranorm defined by

( ) ( )

, ,...

3 , 2 , 1 , 1 ,

...

, , , :

inf

1

1 2

1 1

















=

 ≤





































= ^∞  ₋

∑

= ^{M N x z z z n}

k x b

g

p H n

k k k

s k H n

p_n ^k

ρ ρ

(23)

where max 1, sup .



 



=  _k

k

p H

Proof: It is clear that ^g

( )

^x = ^g

( )

−^x and ^g

(

^x + ^y

)

≤ ^g

( )

^x + ^g

( )

^{y .} Since

( )

⁰ = ^{0 ,}

Mk we get ^inf

{

^{ρpn H}

}

^{=0 for x =0.} Finally, we prove that multiplication is continuous. Let λ ≠ 0 be any complex number, then by definition, we have

( ) ( )

. ,...

3 , 2 , 1 , 1 ,

...

, , , :

inf

1

1 2

1 1

















=

 ≤





































= ^∞  ₋

∑

= ^{M N x z z z n}

k x b

g

p H n

k k

k s

k H n

p_n ^k

ρ ρ λ

λ

(24) Thus, we have

( )

^inf

( )

^:

^{( )}

^{, , ,..., 1, 1,2,3,... ,}

1

1 2

1 1

















=

 ≤





































= ^∞  ₋

∑

= ^{z z z n}

s x M N

k s b

x g

p H n

k k k

s k H n

p_n ^k

λ λ

(25)

where .

λ

= ρ

s Since ^{λ pk ≤ max}

(

^{1, λ H}

)

^{, we have}

( ) (

^max

( )

^1,

)

^inf

^{( )}

^:

^{( )}

^{, , ,..., 1, 1,2,3,... ,}

1

1 2

1 1

1

















=

 ≤





































≤ ^∞  ₋

∑

= ^{M N x z z z n}

k s b

x g

p H n k

k k

s k n H

p H H

k n

λ ρ λ

(26) and therefore, ^g

( )

λ^x converges to zero when g(x) converges to zero in

(

^{B, , p,s, .,...,.}

)

^.

E_n^q Μ Now, suppose that λn → 0 ^as n→∞ and x is in

(

^{B, , p,s, .,...,.}

)

^.

E_n^q Μ For arbitraryε > 0, let n be a positive integer ₀ such that

( )

, 2 ...

, , ,

1

1 2

1

0

ε

ρ _^_ ^<



















∑

^∞ 

+

= −

pk

n k

n k

s k k

n N x z z z

k M b

(27)

for some ρ > 0.This implies that

( )

.

, 2 ...

, , ,

1

1

1 2

1

0

ε

ρ  ^≤





































∑

^∞ 

+

= −

H p

n k

n k

s k k n

k

z z x z M N

k b

(28)

Let 0< λ <1,then using convexity of

( )

^{Mk , we get}

( )

^pk

n k

n k

s k k

n N x z z z

k M

∑

^∞ b

+

= −

























1

1 2

1

0

, ...

, , ρ ,

λ

( )

.

, 2 ...

, , ,

1

1 2

1

0

p H

n k

n k

s k k n

k

z z x z M N

k

b 



 



<























<

∑

^∞ 

+

= − ε

λ ρ

(29)

Since

( )

^Mk is continuous everywhere on [0,∞), then

( )

ⁿ

( )

^pk

k

n k

s k k

n tN x z z z

k M t b

h

∑

= −























= ⁰ 

1

1 2

1, ,..., ρ ,

(30)

is continuous at .0 So there is 0<δ <1 such that h

( )

t <ε ² for 0 <t <δ. Let

K be such that λn <δ for n>K, we have

( )

.

, 2 ...

, , ,

1

1

1 2

1

0 ε

ρ

λ <







































∑



= −

p H n

k

n k

n s k

k n

k

z z x z M N

k b

(31)

Thus,

( )

, , ,..., , .

1

1

1 2

1 z z forn k

x z M N

k

b ^{p H}

k

n k

n s k

k n

k

>

 <





































∑

^∞ 

= − ε

ρ

λ

(32) Hence ^g

( )

λ^x → ^{0 as} λ→ ^0. This completes the proof of the theorem.

Theorem 2.3: If Μ ′=

( )

M′k and Μ ′′=

^{( )}

Mk′′ are two sequences of Musielak- Orlicz functions and s,s₁,s₂ are nonnegative real numbers, then

(i) ^Enq

(

^{B, Μ′, p,s, .,...,.}

)

^{∩ Enq}

(

^{B,Μ ′′,p,s, .,...,.}

)

(

^{B, M , p,s, .,...,.}

)

^.

E_n^q Μ′+ ′′

⊆

(ii) If s₁≤ s₂,then E_n^q

(

B^{, Μ′,} p^,s1^{, .,...,.}

)

^⊆ E_n^q

(

B^, Μ′^, p^,s2^{, .,...,.}

)

^.

Proof: It is obvious, so we omit the details.

Theorem 2.4: Suppose that 0< r_k ≤ p_k<∞, for each k∈N . Then ^Enq

(

^{B, M, r,s, .,...,.}

)

^{⊆ Enq}

(

^{B, M, p,s, .,...,.}

)

^.

Proof: Let ^x∈^En^q

(

^B, Μ^{, r,s, .,...,.}

)

^. Then there exists some ρ>0 such that

( )

, , ,..., .

1

1 2

1 1

∞

 <























∑

^∞ 

= −

rk

k

n k

s k k

n N x z z z

k M b

ρ

(33)

this implies that , Mk

(

Nk

( )

x ρ^,z^1,z^2,...,zn−¹

)

^≤1 for sufficiently large value of k say , k ≥k₀, for some fixed k₀∈N.Since

(

Mk

)

is non decreasing, we get

( )

_≤























∑

^∞ 

≥ −

pk

k k

n k

s k k

n N x z z z

k M b

0

1 2

1 1

, ...

, , ρ ,

( )

, , ,..., .

0

1 2

1 1

∞

 <























∑

^∞ 

≥ −

rk

k k

n k

s k k

n N x z z z

k M b

ρ (34)

Hence ^x∈Enq

(

^{B, Μ, p,s, .,...,.}

)

^.

Theorem 2.5:

(i) If 0< pk ≤1for eachk then ,

(

^{B, M, p,s, .,...,.}

)

E_n^q ⊆ ^En^q

(

^{B, M,s, .,...,.}

)

^.

(ii) If p_k ≥ 1for all k , then ^En^q

(

^{B, M,s, .,...,.}

)

^⊆

(

^{B, M, p,s, .,...,.}

)

^.

E_n^q

Proof: It is easy to prove by using Theorem 2.4, so we omit the details.

Acknowledgements:

The Authors expresses his heartfelt gratitude to the anonymous reviewers for their valuable suggestions which have enormously enhanced the quality and improved the presentation of the paper.

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