ON SOME TOPOLOGICAL PROPERTIES OF GENERALIZED DIFFERENCE SEQUENCE SPACES DEFINED

Volume 30 No. 2 2017, 151-161

ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:_{http://dx.doi.org/10.12732/ijam.v30i2.6}

ON SOME TOPOLOGICAL PROPERTIES OF

GENERALIZED DIFFERENCE SEQUENCE SPACES DEFINED

G¨ulcan Atıci Turan Department of Mathematics

Fac. of Arts and Sciences Mu¸s Alparslan University

C block, Diyarbakır Road, 49250 – Mu¸s, TURKEY

Abstract: In this paper, we define new generalized difference sequence spaces

ℓM_λ (∆m

v , u) and ℓλN(∆mv , u), where M= (Mk) and N = (Nk) are sequences of Orlicz functions such that Mk and Nk are mutually complementary for each

k. We also examine some topological properties and establish some inclusion relations between these spaces.

AMS Subject Classification: 46A45, 40A05

Key Words: difference sequence spaces, Orlicz function

1. Introduction

An Orlicz function is a function M : [0,∞)→[0,∞) which is continuous, non-decreasing and convex with M(0) = 0, M(x) >0 for x >0 and M(x) → ∞

asx→ ∞.

Lindenstrauss and Tzafriri [4] used the idea of Orlicz function to define the Orlicz sequence space

ℓM = (

x= (xk) :

∞

X

k=1 M

|xk|

ρ

<∞, for some ρ >0 )

which is a Banach space with the norm

kxk= inf (

ρ >0 :

∞

X

k=1 M

|xk|

ρ

≤1 )

.

It is well known that if M is a convex function and M(0) = 0, then M(λx)≤ λM(x) for allλwith 0≤λ≤1.

Any Orlicz functionMk always has the integral representation

Mk(x) = x Z

0

pk(t)dt,

wherepk, known as the kernel ofMk, is non-decreasing, is right continuous for

t >0,pk(0) = 0, pk(t)>0 for t >0 andpk(t)→ ∞ ast→ ∞. Given an Orlicz functionMk with kernel pk(t), define

qk(s) = sup{t:pk(t)≤s, s≥0}.

Thenqk(s) possesses the same properties aspk(t) and the function Nk, defined as

Nk(x) = x Z

0

qk(s)ds,

is an Orlicz function. The functions Mk and Nk are called mutually comple-mentary Orlicz functions [1].

The difference sequence spaces were introduced by Kızmaz [2] and the con-cept was generalized by Et and C¸ olak [7]. Later, Et and Esi [6] extended the difference sequence spaces to the sequence spaces

X(∆m

v ) ={x= (xk) : (∆mv xk)∈X}

forX=ℓ∞, corc0, wherev= (vk) be any fixed sequence of non-zero complex numbers and (∆m

v xk) = ∆mv−1xk−∆mv−1xk+1

, ∆m

v x= m P

i=0

(−1)i m_i

vk+ixk+i for all k∈N_.

The sequence spaces ∆m_v (ℓ∞), ∆m_v (c) and ∆m_v (c0) are Banach spaces normed by

kxk_∆= m X

i=1

|vixi|+k∆mv xk∞.

(i) Solid (or normal), if (αkxk) ∈ λ whenever (xk) ∈ λ for all sequences (αk) of scalars with |αk| ≤1.

(ii) Monotone, if provided λ contains the canonical preimages of all its stepspaces.

(iii) Perfect, ifλ=λαα_{, see [9].}

Proposition 2. λis perfect⇒ λis normal⇒ λis monotone, see [9].

A Banach sequence space (λ, S) is called a BK-space, if the topology S

of λ is finer than the co-ordinatewise convergence topology, or equivalently, the projection maps Pi : λ → K, Pi(x) = xi, i ≥ 1 are continuous, where

K is the scalar field R _{(the set of all reals) or} C _{(the complex plane). For} x = (x1, ..., xn, ...) and n ∈ N (the set of natural numbers), we write the nth section ofx asx(n)_{= (x}

1, ..., xn,0,0, ...). If{x(n)} tends tox in (λ, S) for each

x ∈ λ, we say that (λ, S) is an AK−space. The norm k.k_λ generating the topology S of λis said to be monotone if kxk_λ ≤ kyk_λ forx ={xi}, y ={yi}

∈λwith |xi| ≤ |yi|, for all i≥1, see [8].

Definition 3. Any two Orlicz functions M1 and M2 are said to be

equiv-alent, if there are positive constant α and β, and x0 such that M1(αx) ≤ M2(x)≤M1(β) for allx with 0≤x≤x0, see [9].

2. Main Results

Definition 4. Let Mk and Nk be mutually complementary functions for each k and letλ={λk} be a sequence of strictly positive real numbers. Then we define the following sequence spaces:

ℓM_λ (∆m

v , u) ={x= (xk) : X

k≥1 ukMk

|∆m v xk|

λkρ

<∞, for someρ >0}

and

ℓλ_N(∆mv , u) ={x= (xk) : X

k≥1 ukNk

λk|∆mv xk|

ρ

<∞, for someρ >0}.

Theorem 5. Let M = (Mk) and N = (Nk) be two sequences of Orlicz functions. Then ℓM_λ (∆m

v , u) and ℓλN(∆mv , u) are linear spaces over the field of

complex numbers.

Proof. Let x, y∈ℓM_λ (∆m

v , u) and a, b∈C. Then there exist positive num-bersρ1 and ρ2 such that

X

k≥1 ukMk

|∆m v xk|

λkρ1

<∞

and

X

k≥1 ukMk

|∆m v yk|

λkρ2

<∞.

Define ρ3 = max (2|a|ρ1,2|b|ρ2). Since Mk are non-decreasing and convex functions and ∆m _{is linear, we have}

X

k≥1 ukMk

|∆m

v (axk+byk)|

λkρ3

≤X

k≥1 ukMk

|∆m v xk|

λkρ1

+X

k≥1 ukMk

|∆m v yk|

λkρ2

<∞.

This proves that ℓM_λ (∆m

v , u) is a linear space. The proof for ℓλN(∆mv , u) is similar.

The proofs of the following theorems are easy and thus omitted.

Theorem 6. The sequence spaceℓM_λ (∆m

v , u)is a normed space with norm

kxk∆_λ = m X

i=1

|xi|+ inf{ρ >0 : X

k≥1 ukMk

|∆m v xk|

λkρ

≤1}.

Theorem 7. The sequence space ℓλ

N(∆mv , u)is a normed space with norm

kxkλ_∆= m X

i=1

|xi|+ inf{ρ >0 : X

k≥1 ukNk

λk|∆mvxk|

ρ

≤1}.

Theorem 8. The spaces ℓM_λ (∆m

v , u),k.k∆λ

and ℓλ

N(∆mv , u),k.kλ∆

Theorem 9. The sequence spaces ℓM_λ (∆m

v , u) equipped with the norm

k.k∆_λ andℓλ

N(∆mv , u) equipped with the normk.kλ∆ areBK-spaces. Proof. The space ℓM_λ (∆m

v , u),k.k∆λ

is a Banach space by Theorem 8. Now let

kxn_−xk∆

λ →0

asn→ ∞. Then

|xn

k−xk| →0 asn→ ∞ for each k≤m and

inf{ρ >0 :X k≥1

ukMk

|∆m

v xnk−∆mv xk|

λkρ

≤1} →0

asn→ ∞for all k∈N_{. If ukMk}

|∆m vx

n k−∆

m vxk|

λkkxk∆

λ

≤1, then |∆

m vx

n k−∆m

vxk|

λkkxk∆

λ

≤1

for all k. Therefore we also obtain

|∆m_v xn_k−∆m_v xk| ≤λkkxn−xk∆λ . Since kxn_−xk∆

λ →0, then |∆mv xkn−∆mv xk| →0 and m X i=0

(−1)i

m v

vk+i xnk+i−xk+i →0

asn→ ∞ for all k∈N_{. On the other hand, we may write}

v_k+m xn_k+m−x_k+m ≤ m X i=0

(−1)i

m v

vk+i xnk+i−xk+i + m 0

vk(xnk−xk) +... + m m−1

vk+m−1 xnk+m−1−xk+m−1

.

Then |xn

k−xk| → 0 as n → ∞ for all k ∈ N. Hence

ℓM_λ (∆m

v , u),k.k∆λ

is a

BK-space.

The proof is similar for ℓλ

Theorem 10. Let M = (Mk) be a sequence of Orlicz functions. Then ℓM_λ (∆m−1

v , u) ⊂ ℓMλ (∆mv , u). In general ℓMλ (∆iv, u) ⊂ ℓMλ (∆mv , u), for i = 1,2, ..., m−1.

Proof. Let x∈ℓM_λ (∆m−1

v , u). Then X

k≥1 ukMk

∆m−1

v xk

λk2ρ

!

<∞.

Since Mk are non-decreasing and convex functions,

X

k≥1 ukMk

|∆m v xk|

λk2ρ

=X

k≥1 ukMk

∆m_v−1x_k−∆m_v−1x_k+1

λk2ρ

!

≤X

k≥1 ukMk

∆m−1

v xk

λkρ !

+X

k≥1 ukMk

∆m−1

v xk+1

λkρ !

<∞.

ThusℓM_λ (∆m−1

v , u)⊂ℓMλ (∆mv , u). This completes the proof.

Theorem 11. Let N = (Nk) be a sequence of Orlicz functions. Then

ℓλ

N(∆mv−1, u) ⊂ ℓλN(∆mv , u). In general ℓλN(∆iv, u) ⊂ ℓNλ (∆mv , u), for i = 1,2, ..., m−1.

Proof. Letx∈ℓλ

N(∆mv−1, u). Then X

k≥1 ukNk

∆m_v−1x_k

λkρ !

<∞.

Since Nk are non-decreasing and convex functions

X

k≥1 ukNk

λk|∆mv xk| 2ρ

=X

k≥1 ukNk

λk

∆m_v −1x_k−∆m_v−1x_k+1 2ρ ! ≤X k≥1 ukNk

λk ∆m−1

v xk ρ ! +X k≥1 ukNk

λk ∆m−1

v xk+1

ρ

!

<∞.

Thusℓλ

N(∆mv−1, u)⊂ℓλN(∆mv , u). This completes the proof.

(i)ℓM_λ (∆m

v , u)∩ℓTλ(∆mv , u)⊂ℓM+Tλ (∆mv , u),

(ii)If Mand T are equivalent, then ℓM_λ (∆m

v , u) =ℓTλ(∆mv , u).

Proof. (i)Let x∈ℓM_λ (∆m

v , u)∩ℓTλ(∆mv , u). Then X

k≥1 ukMk

|∆m v xk|

λkρ

<∞

and

X

k≥1 ukTk

|∆m v xk|

λkρ

<∞.

We have

(Mk+Tk)

|∆m v xk|

λkρ

≤

Mk

|∆m v xk|

λkρ

+

Tk

|∆m vxk|

λkρ

.

Applying P k≥1

and multiplyinguk both side of this inequality, we get,

X

k≥1

uk(Mk+Tk)

|∆m v xk|

λkρ

≤X

k≥1 ukMk

|∆m v xk|

λkρ

+X

k≥1 ukTk

|∆m v xk|

λkρ

.

This completes the proof.

(ii) The proof follows from Definition 3.

Theorem 13. LetN = (Nk)and T= (Tk) be any two sequence of Orlicz functions. Then we have:

(i)ℓλ

N(∆mv , u)∩ℓλT(∆mv , u)⊂ℓNλ +T(∆ m v , u),

(ii)If Mand T are equivalent, then ℓλ

N(∆mv , u) =ℓλT(∆mv , u).

Proof. (i)Let x∈ℓM_λ (∆m

v , u)∩ℓTλ(∆mv , u). Then X

k≥1 ukNk

λk|∆mv xk|

ρ

<∞

and

X

k≥1 ukTk

λk|∆mv xk|

ρ

We have

(Nk+Tk)

λk|∆mv xk|

ρ

≤

Nk

λk|∆mv xk|

ρ

+

Tk

λk|∆mv xk|

ρ

.

Applying P k≥1

and multiplyinguk both side of this inequality, we get,

X

k≥1

uk(Nk+Tk)

λk|∆mv xk|

ρ

≤X

k≥1 ukNk

λk|∆mv xk|

ρ

+X

k≥1 ukTk

λk|∆mv xk|

ρ

.

This completes the proof.

(ii) The proof follows from Definition 3.

Theorem 14. Ifµis a normal sequence space containingλ, thenℓM_λ (∆m v , u)

is a proper subspace ofµ. In addition, ifµis equipped with the monotone norm (quasi-norm) k.k_µ, the inclusion map I :ℓM_λ (∆m

v , u)→ µ(∆mv , u) is continuous

withkIk ≤ k{λk}kµ.

Proof. Letx∈ℓM_λ (∆m

v , u). Since P

k≥1 ukMk

_|∆m vxk|

λkρ

<∞ for someρ >0,

then there exists a constant K >0 such that |∆mvxk|

λkρ ≤K for all k∈

N_{. Since} µ is a normal sequence space containing λ, we have (∆m

v xk) ∈ µ and so that

x∈µ(∆m

v ). Hence ℓMλ (∆mv , u)⊂µ(∆mv , u). Further, sinceMk(1) = 1 for all k∈N, then

X

k≥1 ukMk

|∆m v xk|

λkkxk∆_λ !

≤1,

and so that

|∆m_v xk| ≤λkkxk∆λ , for all k∈N. As k.k_µ is monotone, kIxk_µ = k(∆m

v xk)kµ ≤ k{λk}kµkxk

∆

λ and hence kIk ≤

k{λk}kµ.

Theorem 15. If η is a normal sequence space containing {_λ1_k} ≡ λ−1, then ℓλ

N(∆m) is a proper subspace of η. If the norm (quasi-norm) k.kη on η

is monotone, then the inclusion map J :ℓλ

N(∆mv , u) → η(∆mv , u) is continuous

withkJk ≤ {λ−1_k }

The proof is similar to Theorem 10 and therefore we omit it.

3. Interrelationship between the Spaces ℓM_λ (∆mv , u) and ℓλM(∆mv , u) If λk = 1 for all k ∈ N, then the sequence space ℓλM(∆mv , u) reduces to the sequence space

ℓM(∆m_v, u) ={x= (xk) : X

k≥1 ukMk

_| ∆m

v xk|

ρ

<∞, for someρ >0}.

Theorem 16. If λ = {λk} is a bounded sequence such that infλk >0,

thenℓλ

M(∆mv , u) =ℓMλ (∆vm, u) =ℓM(∆mv , u).

Proof. Let x∈ℓM(∆m

v , u). Then P

k≥1 ukMk

|∆m vxk|

ρ

<∞ for someρ >0. Since λ = {λk} is bounded, we can write a ≤ λk ≤ b for some b > a ≥ 0. Define ρ1 =ρb. Since Mk is increasing, it follows that P

k≥1 ukMk

_λk|∆m vxk|

ρ₁

≤

P

k≥1 ukMk

|∆m vxk|

ρ

<∞. HenceℓM(∆m

v , u)⊂ℓλM(∆mv , u). The other inclusion

ℓλ

M(∆mv , u)⊂ℓM(∆mv , u) follows from the inequality X

k≥1 ukMk

|∆m v xk|

ρ/a

≤X

k≥1 ukMk

λk|∆mv xk|

ρ

<∞.

Therefore, ℓλ

M(∆mv , u) =ℓM(∆mv , u). Similarly, one can prove ℓM_λ (∆m

v , u) =ℓM(∆mv , u).

Theorem 17. If {λk} ∈ ℓ∞ with a = supk≥1λk ≥ 1 and {λ−1_k } is

un-bounded, thenℓM_λ (∆m

v , u)is properly contained inℓλM(∆mv, u)and the inclusion

mapT :ℓM_λ (∆mv, u)→ℓλM(∆mv , u) is continuous with kTk ≤a2.

Proof. For any ρ >0 andρ′=ρa2, we have

X

k≥1 ukMk

λk|∆mv xk|

ρ′

≤X

k≥1 ukMk

|∆m v xk|

λkρ

<∞

We now show that the containment ℓM_λ (∆m

v , u) ⊂ ℓλM(∆mv , u) is proper. From the unboundedness of the sequence {λ−1_k }, choose a sequence {kn} of positive integers such thatλ−1_kn ≥n. Define ∆

m

v x={∆mv xk}as follows: ∆m_v xk =

1/n, k=kn, n= 1,2, ...

0, otherwise .

Then x∈ℓλ

M(∆mv , u); but x /∈ℓMλ (∆mv , u).

To prove the continuity of the inclusion mapT, let us first consider the case obtained fora= 1. For x∈ℓM_λ (∆m

v ), we write

AM_λ (∆mv , u) = 





ρ >0 :X k≥1

Mk

|∆m v xk|

λkρ

≤1 





and

B_Mλ (∆m_v, u) = 





ρ >0 :X k≥1

Mk

λk|∆mv xk|

ρ

≤1 



 .

Since Mk are increasing anda= 1, we getAMλ (∆vm, u)⊂BλM(∆mv , u). Hence

kxkλ_∆= infB_Mλ (∆m_v, u)≤infAM_λ (∆m_v , u) =kxk∆_λ , (1)

i.e., kT(x)kλ_∆≤ kxk_λ∆. Thus T is continuous with kTk ≤1 =a2_.

Ifa6= 1, defineβk = λ_ak,k∈N. Thenβk ≤1 and from (1), it follows that

kxkβ_∆≤ kxk∆_β forx∈ℓM_λ (∆m_v , u). (2) Hence from (2)

kT(x)kλ_∆=kxkλ_∆≤a2kxk∆_λ ,

i.e., T is continuous with kTk ≤a2. This completes the proof.

Theorem 18. If {λk} is unbounded with supk≥1λ−1k = d ≥ 1, λk > 0

for allk, thenℓλ

M(∆mv, u)is properly contained inℓMλ (∆mv , u) and the inclusion

mapU :ℓλ

M(∆mv , u)→ℓMλ (∆mv , u) is continuous with kUk ≤d2.

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