Generalized lacunary strong Zweier convergent sequence spaces

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Generalized lacunary strong Zweier convergent sequence spaces

KuldipRaj and SuruchiPandoh

Abstract. In this paper we introduce generalized double Zweier lacunary convergent sequence spaces via sequence of Orlicz functions over n-normed spaces. We also make an eﬀort to study some topological properties and inclusion relations between these spaces. Furthermore, we study the concept of double lacunary statistical Zweier convergence overn-normed spaces.

1. Introduction

In [10], Hardy introduced the concept of regular convergence for double sequences. Some important work on double sequences is also found by Bromwich [27]. By the convergence of a double sequence we mean the convergence of the Pringsheim sense i.e., a double sequencex = (xij) has Pringsheim limitL (denoted byP −limx=L) provided that givenϵ > 0 there exists n ∈ N such that |xij −L| < ϵ whenever i, j > n [2]. In case L= 0, we say that double sequencex= (xij) is a Pringsheim null sequence.

The double sequencex= (x_ij) is bounded if there exists a positive integer K such that |xij| < K for all i and j. We denote by l_∞² the space of all bounded double sequences.

Deﬁnition 1.1. [8] The double sequence I_r,s = {(k_r, l_s)} is called double

2010 Mathematics Subject Classiﬁcation. Primary: 40C05, 40J05; Secondary:

40A45.

Key words and phrases. Orlicz function, paranorm, Zweier space, Double lacunary sequence, Double statistical convergence.

9

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lacunary if there exist two increasing integers sequences (kr) and (ls) such that

k0 = 0, hr=kr−kr−1 → ∞asr → ∞ and

l0 = 0, hs=ls−ls−1 → ∞ass→ ∞. Let kr,s=krls, hr,s=hrhs, and θr,s is determined by

Ir,s ={(k, l) :kr−1 < k≤krandls−1 < l≤ls}, q_r = kr

k_r₋₁, q_s = ls

l_s₋₁ andq_r,s =q_rq_s.

Deﬁnition 1.2. An Orlicz function M : [0,∞)→ [0,∞) is a continuous, non-decreasing and convex function such that M(0) = 0, M(x) > 0 for x > 0 and M(x) −→ ∞ as x −→ ∞. If convexity of Orlicz function is replaced by M(x+y) ≤ M(x) +M(y), then this function is called a modulus function.

Lindenstrauss and Tzafriri [15] used the idea of Orlicz to deﬁne the sequence space,

ℓM = {

x= (x_k)∈w:

∑∞ k=1

M (|x_k|

ρ )

<∞, for someρ >0 }

is known as an Orlicz sequence space. The spaceℓ_M is a Banach space with the norm

||x||= inf {

ρ >0 :

∑∞ k=1

M (|x_k|

ρ

)≤1 }

.

Also it was shown in [15] that every Orlicz sequence space ℓ_M contains a subspace isomorphic to ℓ_p(p ≥ 1). An Orlicz function M can always be represented in the following integral form

M(x) =

∫ _x

0

η(t)dt,

where η is known as the kernel of M, is a right diﬀerentiable for t ≥ 0, η(0) = 0, η(t)>0, η is non-decreasing and η(t)→ ∞ as t→ ∞.

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The notion of diﬀerence sequence spaces was introduced by Kızmaz [14]

who studied the diﬀerence sequence spaces l_∞(∆), c(∆) and c₀(∆). The notion was further generalized by Et and C¸ olak [23] by introducing the spaces l_∞(∆ⁿ), c(∆ⁿ) and c0(∆ⁿ). Let w denote the set of all real and complex sequences andnbe a non-negative integer, then for Z =c, c₀ and l_∞, we have sequence spaces

Z(∆ⁿ) ={x= (xk)∈w: (∆ⁿxk)∈Z},

where ∆ⁿx = (∆ⁿxk) = (∆ⁿ⁻¹xk−∆ⁿ⁻¹xk+1) and ∆⁰xk = xk for all k∈N, which is equivalent to the following binomial representation

∆ⁿx_k=

∑n v=0

(−1)^v (

n v

) x_k+v.

Taking n= 1, we get the spaces studied by Et and Ç olak [23]. Similarly, we can define difference operators on double sequences as:

∆x_k,l = (x_k,l−x_k,l+1)−(x_k+1,l−x_k+1,l+1)

= xk,l−xk,l+1−xk+1,l+xk+1,l+1, and

∆ⁿxk,l = ∆ⁿ⁻¹xk,l−∆ⁿ⁻¹xk,l+1−∆ⁿ⁻¹xk+1,l+ ∆ⁿ⁻¹xk+1,l+1.

For more details about sequence spaces see ([17], [18], [26]) and references therein.

Deﬁnition 1.3. A sequenceM= (M_k) of Orlicz functions is said to be a Musielak-Orlicz function (see [21, 16]). A sequence N = (N_k) deﬁned by

N_k(v) = sup{|v|u−M_k(u) :u≥0}, k = 1,2,· · ·

is called a complementary function of the Musielak-Orlicz function (M_k).

For a given Musielak-Orlicz functionM, the Musielak-Orlicz sequence space t_M and its subspace h_M are deﬁned as follows

t_M= {

x∈w:I_M(cx)<∞ for some c >0 }

,

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h_M= {

x∈w:I_M(cx)<∞ for all c >0 }

, where I_M is a convex modular deﬁned by

I_M(x) =

∑∞ k=1

Mk(xk), x= (xk)∈t_M. We consider t_M equipped with the Luxemburg norm

||x||= inf {

k >0 :I_M (x

k )≤1

}

or equipped with the Orlicz norm

||x||⁰ = inf {1

k (

1 +I_M(kx) )

:k >0 }

.

A Musielak-Orlicz function M = (M_k) is said to satisfy ∆₂-condition if there exist constants a, K > 0 and a sequence c = (c_k)^∞_k=1 ∈ l₊¹ (the positive cone ofl¹) such that the inequality

M_k(2u)≤KM_k(u) +c_k holds for allk∈N and u∈R⁺,whenever Mk(u)≤a.

Deﬁnition 1.4. Let X be a linear metric space. A functionp : X →R is called a paranorm, if

1. p(x)≥0 for allx∈X;

2. p(−x) =p(x) for all x∈X;

3. p(x+y)≤p(x) +p(y) for allx, y∈X;

4. if (λn) is a sequence of scalars with λn → λ as n→ ∞ and (xn) is a sequence of vectors with p(x_n−x) → 0as n→ ∞, then p(λ_nx_n− λx)→0 asn→ ∞.

A paranorm p for which p(x) = 0 implies x= 0 is called a 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 [5] Theorem 10.4.2, pp. 183).

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Deﬁnition 1.5. A sequence space E is said to be solid (or normal) if (α_kx_k) ∈ E whenever (x_k) ∈ E and for all sequence (α_k) of scalars with

|αk|<1 for allk∈N.

Deﬁnition 1.6. A sequence space E is said to be symmetric if (x_k) ∈ E implies (x_π(k))∈E,where π is a permutation of N.

Deﬁnition 1.7. A sequence space E is said to be a sequence algebra if (x_ky_k)∈E whenever (x_k),(y_k)∈E.

Deﬁnition 1.8. A sequence spaceE is said to be convergence free if(y_k)∈ E whenever (x_k)∈E and x_k = 0 implies y_k= 0.

Deﬁnition 1.9. Let K = {k₁ < k₂ < · · ·} ⊂ N and let E be a sequence space. A K-step space of E is a sequence space λ^E_K ={(x_k_n)∈w : (x_k)∈ E}.

Definition 1.10. A canonical preimage of a sequence (x_k_n) ∈ λÊ_K is a sequence(y_k)∈w defined by

y_k= {

x_k, ifk∈K 0 , otherwise.

A canonical preimage of a step spaceλÊ_K is a set of canonical preimages of all the elements inλÊ_K, that is, y is in the canonical preimage of λÊ_K if and only ify is a canonical preimage of some x∈λÊ_K.

Deﬁnition 1.11. A sequence spaceE is said to be monotone if it contains the canonical preimages of its step spaces.

The concept of 2-normed spaces was initially developed by G¨ahler [25] in the mid of 1960’s, while that for n-normed spaces one can see in Misiak [1]. Since then, many others have studied this concept and obtained various results, see Gunawan ([11], [12]) and Gunawan and Mashadi [13]. Letn∈N andXbe a real linear space of dimensiond, whered≥n≥2. A real valued function||·, . . . ,·||onXⁿ satisfying the following four conditions:

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1. ||x1, x2, . . . , xn|| = 0 if and only if x1, x2, . . . , xn are linearly depen- dent in X,

2. ||x1, x2, . . . , xn||is invariant under permutation,

3. ||αx₁, x₂, . . . , x_n||=|α| ||x₁, x₂, . . . , x_n|| for anyα∈R, and 4. ||x+x^′, x2, . . . , xn|| ≤ ||x, x2, . . . , xn||+||x^′, x2, . . . , xn||

is called ann-norm onX, and the pair (X,||·, . . . ,·||) is called ann-normed space over the ﬁeldR.

For example, we may take X = Rⁿ being equipped with the n-norm

||x1, x2, . . . , xn||E= the volume of then-dimensional parallelopiped spanned by the vectorsx₁, x₂, . . . , x_n which may be given explicitly by the formula

||x₁, x₂, . . . , x_n||E =|det(x_ij)|,

wherex_i = (x_i1, x_i2, . . . , x_in)∈Rⁿfor eachi= 1,2, . . . , n. Let (X,||·, . . . ,·||) be ann-normed space of dimensiond≥n≥2 and {a1, a2, . . . , an} be linearly independent set in X. Then the following function ||·, . . . ,·||_∞ on Xⁿ⁻¹ deﬁned by

||x1, x2, . . . , xn−1||∞= max{||x1, x2, . . . , xn−1, ai||:i= 1,2, . . . , n} deﬁnes an (n−1)-norm on X with respect to{a1, a2, . . . , an}.

A sequence (x_k) in ann-normed space (X,||·, . . . ,·||) is said to converge to someL∈X if

klim→∞||xk−L, z1, . . . , zn−1||= 0 for every z1, . . . , zn−1 ∈X.

A sequence (x_k) in ann-normed space (X,||·, . . . ,·||) is said to be a Cauchy sequence if

k,plim→∞||x_k−x_p, z₁, . . . , z_n₋₁||= 0 for every z₁, . . . , z_n₋₁∈X.

If every Cauchy sequence in X converges to some L ∈X, then X is said to be complete with respect to then-norm. A completen-normed space is said to be an n-Banach space. For more details on n-normed spaces, see [19], [20] and references therein.

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2. Lacunary strongly Zweier convergent sequence spaces

Zweier sequence spaces for single sequences were deﬁned and studied by S

¸engönül [22], Esi and Sapsızo˘glu [4], Khan et. al [28], [29]. Esi and Acikgoz [3] defined the double Zweier sequence spaces [W², Z], [N_θ_r,s, Z]₀, [N_θ_r,s, Z]

and [N_θ_r,s, Z]_∞ as the set of all double sequences such that Z−transforms of them are in [W²], [N_θ_r,s]₀, [N_θ_r,s] and [N_θ_r,s]_∞ which were introduced by Sava¸s in [7], Sava¸s and Patterson in [9].

We deﬁne the double sequences v = (v_ij) and w = (w_ij) which will be used throughout the paper, as Z-transform of a sequence x = (x_ij) and y= (yij) respectively i.e.,

v_ij = 1

2(x_ij+x_ij₋₁) and w_ij = 1

2(y_ij+y_ij₋₁); (i, j∈N). (2.1) Let (X,||·, . . . ,·||) be ann-normed space and W(n−X) denotes the space ofX-valued sequences. LetM= (Mij) be a Musielak-Orlicz function, p= (p_ij) be a bounded double sequence of positive real numbers andu= (u_ij) be a double sequence of strictly positive real numbers. In the present paper we introduce the new double Zweier sequence spaces as follows:

[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]0

= {

x= (x_ij) :P−lim

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿvij

ρ , z₁, . . . , z_n₋₁°°°)]pij

= 0 for someρ >0

} , [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]

= {

x= (xij) :P−lim

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿv_ij −L

ρ , z1, . . . , zn−1°°°)]_p_ij

= 0 for someLandρ >0

} , [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞

= {

x= (x_ij) : sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]pij

<∞ for someρ >0

} and

[W², Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]

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= {

x= (x_ij) :P−lim

m,n

1 mn

∑m,n i,j=1,1

M_ij [

u_ij(°°°∆ⁿv_ij −L

ρ , z₁, . . . , z_n₋₁°°°)]pij

= 0 for someLandρ >0

} .

Remark 2.1. Let us consider a few special cases of the above sequence spaces:

(i) If M_ij(x) = x, for all i, j ∈ N, then above sequence space reduces to [Nθr,s, Z,∆ⁿ, p, u,∥·, . . . ,·∥]0, [Nθr,s, Z,∆ⁿ, p, u,∥·, . . . ,·∥], [Nθr,s, Z,∆ⁿ, p, u,

∥·, . . . ,·∥]_∞ and [W², Z,∆ⁿ, p, u,∥·, . . . ,·∥].

(ii) By taking (pij) = 1, for all i, j ∈ N, then the above space becomes [N_θ_r,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥]0, [N_θ_r,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥], [N_θ_r,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥]_∞and [W², Z,M,∆ⁿ, u,∥·, . . . ,·∥].

(iii) By taking (uij) = 1, for all i, j ∈ N, then we get the above space as [N_θ_r,s, Z,M,∆ⁿ, p,∥·, . . . ,·∥]₀, [N_θ_r,s, Z,M,∆ⁿ, p,∥·, . . . ,·∥], [N_θ_r,s, Z,M,∆ⁿ, p,∥·, . . . ,·∥]_∞ and [W², Z,M,∆ⁿ, p,∥·, . . . ,·∥].

(iv) If we takeMij(x) =x, (pij) = 1,(uij) = 1,for all i, j ∈N, andn= 0 then the above space reduces to [N_θ_r,s, Z,∥·, . . . ,·∥]₀, [N_θ_r,s, Z,∥·, . . . ,·∥], [Nθr,s, Z,∥·, . . . ,·∥]_∞and [W², Z,∥·, . . . ,·∥].

(v) Also, if we take (p_ij) = 1, (u_ij) = 1, for all i, j ∈ N, and n = 0 then the above space reduces to [N_θ_r,s, Z,M,∥·, . . . ,·∥]₀, [N_θ_r,s, Z,M,∥·, . . . ,·∥], [N_θ_r,s, Z,M,∥·, . . . ,·∥]_∞ and [W², Z,M,∥·, . . . ,·∥].

The following inequality will be used through out the paper. If 0≤pij ≤ supp_ij =G,D= max(1,2^G⁻¹) then

|aij +bij|^pîj ≤D(|aij|^pîj+|bij|^pîj) (2.2)

for alli, j∈N and a_ij, b_ij ∈C. Also|a|^p^ij ≤max(1,|a|^G) for alla∈C.

The main purpose of this paper is to introduce double Zweier lacunary strongly convergent sequence spaces over n-normed spaces and study dif- ferent properties of these spaces like linearity, paranorm, solidity and monotone etc. Some inclusion relations between these spaces are also established.

Finally, we study the concept of the double Zweier lacunary statistical convergence overn-normed spaces.

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3. Main Results

Theorem 3.1. LetM= (Mij)be a sequence of Orlicz functions,p= (pij) be any bounded double sequence of positive real numbers andu= (u_ij) be a double sequence of strictly positive real numbers. Then the double Zweier se- quence spaces[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]0,[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥], [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞and[W², Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]are lin- ear spaces over the ﬁeldR of real numbers .

Proof. Let x = (xij), y = (yij) ∈ [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. Let α, β∈R. Then there exist positive real numbers ρ₁, ρ₂ such that

sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿv_ij ρ1

, z₁, . . . , z_n₋₁°°°)]pij

<∞,

sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿwij

ρ2

, z1, . . . , zn−1°°°)]_p_ij

<∞.

Letρ3= max(2|α|ρ1,2|β|ρ2).SinceMij’s are non-decreasing and convex so by using inequality (2.2), we have

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿ(αvij +βwij)

ρ₃ , z₁, . . . , z_n₋₁°°°)]pij

≤ sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿαv_ij ρ3

, z1, . . . , zn−1°°°) +uij(°°°∆ⁿβwij

ρ3

, z1, . . . , zn−1°°°)]_p_ij

≤ Dsup

r,s

1 hr,s

∑

(i,j)∈Ir,s

1 2^p^ijM_ij

[

, z₁, . . . , z_n₋₁°°°)]pij

+Dsup

r,s

1 hr,s

∑

(i,j)∈Ir,s

1 2^p^ijM_ij

[

u_ij(°°°∆ⁿw_ij ρ2

, z₁, . . . , z_n₋₁°°°)]pij

≤ Dsup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿv_ij

ρ₁ , z₁, . . . , z_n₋₁°°°)]pij

+Dsup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

[

u_ij(°°°∆ⁿwij

ρ₂ , z₁, . . . , z_n₋₁°°°)]pij

< ∞.

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Thus,αx+βy∈[Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. This proves that [Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞ is a linear space. Similarly we can prove that [Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]0, [Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥] and [W², Z,M,∆ⁿ, p, u,∥·, . . . ,·∥] are linear spaces.

Theorem 3.2. LetM= (M_ij)be a sequence of Orlicz functions,p= (p_ij) be any bounded double sequence of positive real numbers and u = (uij) be a double sequence of strictly positive real numbers. Then the double Zweier sequence space[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞is a paranormed space with paranormed deﬁned by

g(x) = inf {

(ρ)^pij^H : sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n−1°°°)]^pij

H≤1,

for someρ >0 }

, where 0< pij ≤suppij =G and H= max(1, G).

Proof. (i) Clearlyg(x)≥0 forx= (x_ij)∈[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. SinceMij(0) = 0,we getg(0) = 0.

(ii)g(−x) =g(x).

(iii) Let x = (x_ij) and y = (y_ij) ∈[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞, then there exist positive numbersρ1 andρ2 such that

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ₁ , z₁, . . . , z_n−1°°°)]pij

≤1

and

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿw_ij

ρ₂ , z₁, . . . , z_n₋₁°°°)]pij

≤1.

Letρ=ρ₁+ρ₂. Then by using Minkowski’s inequality, we have sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿ(v_ij+w_ij)

ρ , z₁, . . . , z_n₋₁°°°)]pij

= sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ₁+ρ₂ , z₁, . . . , z_n₋₁°°°)]pij

≤ sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°° ∆ⁿvij

ρ₁+ρ₂, z₁, . . . , z_n₋₁°°°)

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+uij(°°°∆ⁿwij

ρ1+ρ2

, z1, . . . , zn−1°°°)]_p_ij

≤ (

ρ1

ρ1+ρ2

) sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿvij

ρ1

, z1, . . . , zn−1°°°)]_p_ij

+ (

ρ2

ρ₁+ρ₂ )

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿwij

ρ₂ , z1, . . . , zn−1°°°)]pij

≤ 1 and thus g(x+y)

= inf {

(ρ)^pij^H : sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ₁+ρ₂ , z₁, . . . , z_n₋₁°°°)]^pij

H≤1 }

≤ inf {

(ρ₁)^pij^H : sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

, z₁, . . . , z_n₋₁°°°)]^pij

H ≤1

}

+ inf {

(ρ2)^pij^H : sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿw_ij ρ2

, z1, . . . , zn−1°°°)]^pij

H ≤1

} .

Therefore,g(x+y)≤g(x) +g(y). Finally, we prove that the scalar multi- plication is continuous. Letλbe any complex number. By deﬁnition, g(λx) = inf

{

(ρ)^pij^H : sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

u_ij(°°°∆ⁿλv_ij

ρ , z₁, . . . , z_n₋₁°°°)]^pij

H≤1 }

= inf {

(|λ|t)^pij^H : sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

t , z₁, . . . , z_n₋₁°°°)]^pij

H≤1 }

,

wheret= _|^ρ_λ_| >0. Since |λ|^pîj ≤max(1,|λ|^sup^pîj), we have g(λx)≤max(1,|λ|^sup^pîj)

inf {

t^pij^H : sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿv_ij

t , z1, . . . , zn−1°°°)]^pij

H≤1

} . So, the fact that the scalar multiplication is continuous follows from the above inequality. This completes the proof of the theorem.

Theorem 3.3. If 0 < p_ij < q_ij < ∞ for each i and j, then we have [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞⊂[N_θ_r,s, Z,M,∆ⁿ, q, u,∥·, . . . ,·∥]_∞.

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Proof. Let x = (xij) ∈ [Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. Then there existsρ >0 such that

sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]pij

<∞

This implies that Mij

[

uij(°°°^∆ⁿ_ρ^v^ij, z1, . . . , zn−1°°°)]pij

< 1 for suﬃciently large values ofiand j. SinceM_ij’s are non-decreasing, we get

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]qij

≤ sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿvij

ρ , z1, . . . , zn−1°°°)]pij

< ∞.

Thus, x = (xij) ∈ [N_θ_r,s, Z,M,∆ⁿ, q, u,∥·, . . . ,·∥]_∞. This completes the proof.

Theorem 3.4. Suppose M= (M_ij) be a sequence of Orlicz functions,p= (pij) be a bounded double sequence of positive real numbers and u = (uij) be a double sequence of strictly positive real numbers. Then

(i) If 0<infp_ij < p_ij ≤1, then

[Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞⊂[Nθr,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥]_∞. (ii) If 1≤p_ij ≤supp_ij <∞, then

[Nθr,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥]_∞⊂[Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞.

Proof. (i) Let x = (x_ij) ∈ [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. Since 0 <

infpij ≤1, we obtain the following sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

Mij

[

ρ , z1, . . . , zn−1°°°)]

≤ sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]pij

< ∞,

and hencex= (x_ij)∈[N_θ_r,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥]_∞.

(ii) Let p_ij ≥ 1 for each i and j and supp_ij < ∞. Let x = (x_ij) ∈

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[Nθr,s, Z,M,∆ⁿ, u,∥·, . . . ,·∥]_∞. Then for each 0 < ϵ < 1 there exists a positive integerN such that

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿvij

ρ , z1, . . . , zn−1°°°)]

≤ϵ <1 for allr, s≥N.

This implies that sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]pij

≤ sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]

< ∞.

Therefore, x = (xij) ∈ [Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞ . This completes the proof.

Theorem 3.5. Let M^′ = (M_ij^′ ) and M^′′= (M_ij^′′) be two sequences of Or- licz functions, then we have[N_θ_r,s, Z,M^′,∆ⁿ, p, u,∥·, . . . ,·∥]_∞∩[N_θ_r,s, Z,M^′′,

∆ⁿ, p, u,∥·, . . . ,·∥]_∞⊂[N_θ_r,s, Z,M^′+M^′′,∆ⁿ, p, u,∥·, . . . ,·∥]_∞.

Proof. Letx= (xij)∈[Nθr,s, Z,M^′,∆ⁿ, p, u,∥·, . . . ,·∥]_∞∩[Nθr,s, Z,M^′′,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. Then

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij^′ [

uij(°°°∆ⁿvij

ρ , z1, . . . , zn−1°°°)]_p_ij

, for someρ1 >0 and

sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij^′′

[

ρ , z₁, . . . , z_n₋₁°°°)]pij

<∞, for someρ₂ >0.

Letρ= max{ρ₁, ρ₂}. The result follows from the inequality sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

(M_ij^′ +M_ij^′′) [

ρ , z₁, . . . , z_n₋₁°°°)]pij

= sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij^′ [

ρ , z₁, . . . , z_n₋₁°°°)]pij

+ sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

M_ij^′′

[

ρ , z₁, . . . , z_n₋₁°°°)]pij

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≤ Dsup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij^′ [

uij(°°°∆ⁿvij

ρ , z1, . . . , zn−1°°°)]_p_ij +Dsup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij^′′

[

ρ , z1, . . . , zn−1°°°)]_p_ij

< ∞. Thus, sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

(M_ij^′ +M_ij^′′) [

ρ , z₁, . . . , z_n₋₁°°°)]pij

<∞. There- fore, x = (xij) ∈ [Nθr,s, Z,M^′+M^′′,∆ⁿ, p, u,∥·, . . . ,·∥]_∞. This completes the proof.

Theorem 3.6. For a sequence of Orlicz functions M= (M_ij) , p= (p_ij) be any bounded double sequence of positive real numbers andu= (u_ij) be a double sequence of strictly positive real numbers. Then

(i) [N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]₀ ⊂[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞ (ii)[Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]⊂[Nθr,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞ Proof. The proof is easy so we omit it.

Theorem 3.7. The double Zweier sequence space [N_θ_r,s, Z,M,∆ⁿ, p, u,

∥·, . . . ,·∥]_∞ is solid.

Proof. Suppose x= (xij)∈[N_θ_r,s, Z,M,∆ⁿ, p, u,∥·, . . . ,·∥]_∞ sup

r,s

1 hr,s

∑

(i,j)∈Ir,s

M_ij [

ρ , z₁, . . . , z_n₋₁°°°)]pij

<∞, for someρ >0.

Let (α_ij) be a double sequence of scalars such that|α_ij| ≤1 for alli, j∈N.

Then we get sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿαijvij

ρ , z1, . . . , zn−1°°°)]_p_ij

≤ sup

r,s

1 h_r,s

∑

(i,j)∈Ir,s

Mij

[

uij(°°°∆ⁿvij

ρ , z1, . . . , zn−1°°°)]pij

< ∞.

This completes the proof.