Some Strongly Almost Summable Sequence Spaces

Mathematics

and Applications

JMA No 40, pp 171-183 (2017)

COPYRIGHT c by Publishing House of Rzesz´ow University of Technology P.O. Box 85, 35-959 Rzesz´ow, Poland

Some Strongly Almost Summable Sequence Spaces

Sunil K. Sharma and Ayhan Esi

Abstract: In the present paper we introduce some strongly almost summable sequence spaces using ideal convergence and Musielak-Orlicz function M = (Mk) in n-normed spaces. We examine some topological properties of the resulting sequence spaces.

AMS Subject Classification: 40A05, 46A45, 46B70.

Keywords and Phrases: Paranorm space; I-convergence; Λ-convergent; Orlicz func- tion; Musielak-Orlicz function; n-normed spaces.

1. Introduction and preliminaries

Mursaleen and Noman [18] introduced the notion of λ-convergent and λ-bounded sequences as follows:

Let λ = (λk)^∞_k=1 be a strictly increasing sequence of positive real numbers tending to infinity i.e.

0 < λ0< λ1< · · · and λk→ ∞ as k → ∞

and said that a sequence x = (xk) ∈ w is λ-convergent to the number L, called the λ-limit of x if Λm(x) −→ L as m → ∞, where

λm(x) = 1 λm

m

X

k=1

(λk− λk−1)xk.

The sequence x = (x_k) ∈ w is λ-bounded if sup_m|Λ_m(x)| < ∞. It is well known [18]

that if limmxm= a in the ordinary sense of convergence, then lim

m

 1 λm

 ^m X

k=1

(λ_k− λk−1)|x_k− a|



= 0.

This implies that

limm |Λm(x) − a| = lim

m | 1 λm

m

X

k=1

(λ_k− λk−1)(x_k− a)| = 0,

which yields that lim_mΛ_m(x) = a and hence x = (x_k) ∈ w is λ-convergent to a.

The concept of 2-normed spaces was initially developed by G¨ahler [6] in the mid of 1960’s, while that of n-normed spaces one can see in Misiak [15]. Since then, many others have studied this concept and obtained various results, see Gunawan ([8, 9]) and Gunawan and Mashadi [10] and references therein. Let n ∈ N and X be a linear space over the field K, where K is the field of real or complex numbers of dimension d, where d ≥ n ≥ 2. A real valued function ||·, · · · , ·|| on Xⁿ satisfying the following four conditions:

1. ||x₁, x₂, · · · , x_n|| = 0 if and only if x1, x₂, · · · , x_n are linearly dependent in X;

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

3. ||αx1, x2, · · · , xn|| = |α| ||x1, x2, · · · , xn|| for any α ∈ K, and 4. ||x + x⁰, x₂, · · · , x_n|| ≤ ||x, x2, · · · , x_n|| + ||x⁰, x₂, · · · , x_n||

is called a n-norm on X, and the pair (X, ||·, · · · , ·||) is called a n-normed space over the field K.

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

||x1, x2, · · · , xn||E = the volume of the n-dimensional parallelopiped spanned by the vectors x1, x2, · · · , xn which may be given explicitly by the formula

||x1, x2, · · · , xn||E= | det(xij)|,

where xi = (xi1, xi2, · · · , xin) ∈ Rⁿ for each i = 1, 2, · · · , n. Let (X, ||·, · · · , ·||) be a n-normed space of dimension d ≥ n ≥ 2 and {a1, a2, · · · , an} be linearly independent set in X. Then the following function ||·, · · · , ·||∞on Xⁿ⁻¹defined 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 (xk) in a n-normed space (X, ||·, · · · , ·||) is said to converge to some L ∈ X if

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

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

p→∞k→∞

||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 the n-norm. Any complete n-normed space is said to be n-Banach space.

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 −→ ∞.

Lindenstrauss and Tzafriri [13] used the idea of Orlicz function to define the following sequence space,

`_M =n x ∈ w :

∞

X

k=1

M|xk| ρ

< ∞o ,

which is called as an Orlicz sequence space. Also `M is a Banach space with the norm

||x|| = infn ρ > 0 :

∞

X

k=1

M|xk| ρ

≤ 1o .

Also, it was shown in [13] that every Orlicz sequence space `M contains a subspace isomorphic to `p(p ≥ 1). The ∆2- condition is equivalent to M (Lx) ≤ LM (x), for all L with 0 < L < 1. An Orlicz function M can always be represented in the following integral form

M (x) = Z x

0

η(t)dt,

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

A sequence M = (M_k) of Orlicz function is called a Musielak-Orlicz function see ([14, 23]). A sequence N = (Nk) defined by

Nk(v) = sup{|v|u − Mk(u) : u ≥ 0}, k = 1, 2, . . .

is called the complementary function of the Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence space t_Mand its subspace h_M are defined as follows

t_M=n

x ∈ w : I_M(cx) < ∞ for some c > 0o ,

h_M=n

x ∈ w : I_M(cx) < ∞ for all c > 0o , where IM is a convex modular defined by

I_M(x) =

∞

X

k=1

M_k(x_k), x = (x_k) ∈ t_M.

We consider t_M equipped with the Luxemburg norm

||x|| = infn

k > 0 : I_Mx k

≤ 1o

or equipped with the Orlicz norm

||x||⁰= infn1 k



1 + IM(kx)

: k > 0o .

Let X be a linear metric space. A function p : X → R is called 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 (λn) is a sequence of scalars with λn → λ as n → ∞ and (xn) is a sequence of vectors with p(xn− x) → 0 as n → ∞, then p(λnxn− λx) → 0 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 [30], Theorem 10.4.2, p.

183). For more details about sequence spaces (see [16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29]) and reference therein.

A sequence space E is said to be solid (or normal) if (xk) ∈ E implies (αkxk) ∈ E for all sequences of scalars (αk) with |αk| ≤ 1 and for all k ∈ N.

The notion of ideal convergence was introduced first by P. Kostyrko [11] as a general- ization of statistical convergence which was further studied in topological spaces (see [2]). More applications of ideals can be seen in [2, 3].

A linear functional L on `∞is said to be a Banach limit see [1] if it has the properties:

1. L(x) ≥ 0 if x ≥ 0 (i.e. xn≥ 0 for all n), 2. L(e) = 1, where e = (1, 1, . . .),

3. L(Dx) = L(x),

where the shift operator D is defined by (Dx_n) = (x_n+1).

Let B be the set of all Banach limits on `_∞. A sequence x is said to be almost convergent to a number L if L(x) = L for all L ∈ B. Lorentz [12] has shown that x is almost convergent to L if and only if

tkm= tkm(x) = xm+ xm+1+ · · · + xm+k

k + 1 → L as k → ∞, uniformly in m.

Recently a lot of activities have started to study sumability, sequence spaces and re- lated topics in these non linear spaces see [4, 28]. In particular Sahiner [28] combined these two concepts and investigated ideal sumability in these spaces and introduced certain sequence spaces using 2-norm.

We continue in this direction and by using Musielak-Orlicz function, generalized se- quences and also ideals we introduce I-convergence of generalized sequences with respect to Musielak-Orlicz function in n-normed spaces.

Let (X, ||.||) be a normed space. Recall that a sequence (x_n)_n∈N of elements of X is called statistically convergent to x ∈ X if the set A() =n

n ∈ N : ||xⁿ− x|| ≥ o has natural density zero for each  > 0.

A family I ⊂ 2^Y of subsets of a non empty set Y is said to be an ideal in Y if

1. φ ∈ I;

2. A, B ∈ I imply A ∪ B ∈ I;

3. A ∈ I, B ⊂ A imply B ∈ I, while an admissible ideal I of Y further satisfies {x} ∈ I for each x ∈ Y (see [6]).

Given I ⊂ 2^Nbe a non trivial ideal in N. A sequence (xn)_n∈N in X is said to be I-convergent to x ∈ X, if for each  > 0 the set A() = n

n ∈ N : ||xⁿ − x|| ≥ o belongs to I (see [11]).

Let I be an admissible ideal of N, M = (M^k) be a Musielak-Orlicz function and (X, ||·, · · · , ·||) be a n-normed space. Let p = (pk) be a bounded sequence of positive real numbers and u = (uk) be any sequence of strictly positive real numbers. By S(n − X) we denote the space of all sequences defined over (X, ||·, · · · , ·||). We define the following sequence spaces in this paper:

ˆ

w^I(M, Λ, p, ||·, · · · , ·||) = n

x = (xk) ∈ S(n − X) : ∀  > 0, n n ∈ N : 1

n

n

X

k=1

h Mk

||tkm(Λk(x) − L)

ρ , z1, · · · , zn−1||i^pk

≥ o

∈ I

for some ρ > 0, L ∈ X and z₁, · · · , z_n−1∈ Xo , ˆ

w₀^I(M, Λ, p, ||·, · · · , ·||) = n

x = (xk) ∈ S(n − X) : ∀  > 0, n n ∈ N : 1

n

n

X

k=1

h Mk

||tkm(Λk(x))

ρ , z1, · · · , zn−1||i^pk

≥ o

∈ I

for some ρ > 0, and z₁, · · · , z_n−1∈ Xo , ˆ

w_∞(M, Λ, p, ||·, · · · , ·||) =

nx = (x_k) ∈ S(n − X) : ∃ K > 0 such that

sup

n∈N

1 n

n

X

k=1

h M_k

||tkm(Λk(x))

ρ , z₁, · · · , z_n−1||ipk

≤ K

for some ρ > 0, and z1, · · · , zn−1∈ Xo , ˆ

w_∞^I (M, Λ, p, ||·, · · · , ·||) = n

x = (x_k) ∈ S(n − X) : ∃ K > 0 such that

n

n ∈ N : 1 n

n

X

k=1

h M_k

||tkm(Λk(x))

ρ , z₁, · · · , z_n−1||ipk

≥ Ko

∈ I

for some ρ > 0, and z1, · · · , zn−1∈ Xo .

The following inequality will be used throughout the paper. If 0 ≤ pk ≤ sup pk = H, D = max(1, 2^H−1) then

|ak+ bk|^pk≤ D{|ak|^pk+ |bk|^pk} for all k and ak, bk∈ C. Also |a|^pk≤ max(1, |a|^H) for all a ∈ C.

The main aim of this paper is to study some topological properties and inclusion relations between the above defined sequence spaces.

2. Main results

Theorem 2.1. Let M = (M_k) be a Musielak-Orlicz function, p = (p_k) be a bounded sequence of positive real numbers and I be an admissible ideal of N. Then ˆw^I(M, Λ, p, ||·, · · · , ·||), ˆw^I₀(M, Λ, p, ||·, · · · , ·||), ˆw_∞(M, Λ, p, ||·, · · · , ·||) and

ˆ

w_∞^I (M, Λ, p, ||·, · · · , ·||) are linear spaces.

Proof. Let x, y ∈ ˆw^I(M, Λ, p, ||·, · · · , ·||) and α, β ∈ C. So

nn

n ∈ N : 1 n

n

X

k=1

h Mk

||t_km(Λ_k(x) − L) ρ1

, z1, · · · , zn−1||i^pk

≥ o

∈ I for some ρ1> 0,

L ∈ X and z1, · · · , zn−1∈ Xo and

nn

n ∈ N : 1 n

n

X

k=1

h Mk

||tkm(Λk(y) − L)

ρ₂ , z1, · · · , zn−1||ipk

≥ o

∈ I for some ρ2> 0,

L ∈ X and z1, · · · , zn−1∈ Xo .

Since ||·, · · · , ·|| is a n-norm, M = (M_k) be a Musielak-Orlicz function and so by using inequality (1.1), we have

1 n

n

X

k=1

h M_k

||tkm(Λk(αx + βy) − L)

|α|ρ1+ |β|ρ₂ , z₁, · · · , z_n−1||ipk

≤ D1 n

n

X

k=1

h |α|

(|α|ρ1+ |β|ρ2)M_k

||t_km(Λ_k(x) − L) ρ1

, z₁, · · · , z_n−1||i^pk

+ D1 n

n

X

k=1

h |β|

(|α|ρ1+ |β|ρ2)Mk

||t_km(Λ_k(y) − L) ρ2

, z1, · · · , zn−1||i^pk

≤ DF1 n

n

X

k=1

h Mk

||tkm(Λk(x) − L) ρ1

, z1, · · · , zn−1||i^pk

+ DF1 n

n

X

k=1

h Mk

||tkm(Λk(y) − L) ρ2

, z1, · · · , zn−1||i^pk

,

where F = maxh

1, _|α|

(|α|ρ₁+|β|ρ₂)

H

, _|β|

(|α|ρ₁+|β|ρ₂)

Hi

. From the above inequality, we get

n

n ∈ N : 1 n

n

X

k=1

h M_k

||tkm(Λk(αx + βy) − L)

|α|ρ1+ |β|ρ₂ , z₁, · · · , z_n−1||ipk

≥ o

⊆ n

n ∈ N : DF1 n

n

X

k=1

h Mk

||tkm(Λk(x) − L)

ρ₁ , z1, · · · , zn−1||ipk

≥  2 o

∪ n

n ∈ N : DF1 n

n

X

k=1

h M_k

||tkm(Λk(y) − L)

ρ₂ , z₁, · · · , z_n−1||ipk

≥  2 o

. Two sets on the right hand side belong to I and this completes the proof.

Similarly, we can prove that ˆw^I₀(M, Λ, p, ||·, · · · , ·||), ˆw_∞(M, Λ, p, ||·, · · · , ·||) and ˆ

w_∞^I (M, Λ, p, ||·, · · · , ·||) are linear spaces.

Theorem 2.2. Let M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers. For any fixed n ∈ N, ˆw_∞(M, Λ, p, ||·, · · · , ·||) is a paranormed space with the paranorm defined by

g(x) = infn

ρ^pnH : ρ > 0 is such that sup

k

1 n

n

X

k=1

h Mk

||t_km(Λ_k(x))

ρ , z1, · · · , zn−1||i^pk

≤ 1, ∀z₁, · · · , z_n−1∈ Xo .

Proof. It is clear that g(x) = g(−x). Since Mk(0) = 0, we get inf{ρ^pnH} = 0 for x = 0 therefore, g(0) = 0. Let us take x, y ∈ ˆw_∞(M, Λ, p, ||·, · · · , ·||) . Let

B(x) =n

ρ^pnH : ρ > 0, sup

k

1 n

n

X

k=1

hM_k

||t_km(Λ_k(x))

ρ , z₁, · · · , z_n−1||i^pk

≤ 1,

∀z1, · · · , zn−1∈ Xo ,

B(y) =n

ρ^pnH : ρ > 0, : sup

k

1 n

n

X

k=1

hM_k

||t_km(Λ_k(y))

ρ , z₁, · · · , z_n−1||i^pk

≤ 1,

∀z₁, · · · , z_n−1∈ Xo . Let ρ1∈ B(x) and ρ2∈ B(y). If ρ = ρ1+ ρ2, then we have sup

k

1 n

n

X

k=1

h Mk

||tkm(Λk(x + y))

ρ , z1, · · · , zn−1||i

≤ ρ1

ρ₁+ ρ₂sup

k

1 n

n

X

k=1

h M_k

||tkm(Λk(x))

ρ₁ , z₁, · · · , z_n−1||i

+ ρ₂

ρ1+ ρ2

sup

k

1 n

n

X

k=1

hM_k

||t_km(Λ_k(y)) ρ2

, z₁, · · · , z_n−1||i .

Thus sup

k

1 n

n

X

k=1

h Mk

||t_km(Λ_k(x + y)) ρ1+ ρ2

, z1, · · · , zn−1||^pk

≤ 1 and

g(x + y) ≤ infn

(ρ₁+ ρ₂)^pnH : ρ₁∈ B(x), ρ₂∈ B(y)o

≤ infn

ρ₁^pnH : ρ1∈ B(x)o + infn

ρ₂^pnH : ρ2∈ B(y)o

= g(x) + g(y).

Let σ^m→ σ where σ, σ^m∈ C and let g(x^m− x) → 0 as m → ∞. We have to show that g(σ^mx^m− σx) → 0 as m → ∞. Let

B(x^m) =n ρ

pn

mH : ρm> 0, sup

k

1 n

n

X

k=1

h Mk

||tkm(Λk(x^m)) ρm

, z1, · · · , zn−1||i^pk

≤ 1,

∀z1, · · · , zn−1∈ Xo ,

B(x^m−x) =n ρ⁰

pn

mH : ρ⁰_m> 0, sup

k

1 n

n

X

k=1

h Mk

||tkm(Λk(x^m− x))

ρ⁰_m , z1, · · · , zn−1||i^pk

≤ 1,

∀z1, · · · , zn−1∈ Xo . If ρ_m∈ B(x^m) and ρ⁰_m∈ B(x^m− x) then we observe that

1 n

n

X

k=1

h

M_k||tkm(σ^mΛk(x^m) − σΛk(x))

ρ_m|σ^m− σ| + ρ⁰_m|σ| , z₁, · · · , z_n−1||

≤ 1

n

n

X

k=1

h M_k

||tkm(σ^mΛk(x^m) − σΛk(x^m))

ρ_m|σ^m− σ| + ρ⁰_m|σ| , z₁, · · · , z_n−1||

+ ||tkm(σΛk(x^m) − σΛk(x))

ρ_m|σ^m− σ| + ρ⁰_m|σ| , z1, · · · , zn−1||i

≤ |σ^m− σ|ρm

ρm|σ^m− σ| + ρ⁰_m|σ|

1 n

n

X

k=1

h Mk

||t_km(Λ_k(x^m)) ρm

, z1, · · · , zn−1||i

+ |σ|ρ⁰_m

ρm|σ^m− σ| + ρ⁰_m|σ|

1 n

n

X

k=1

h Mk

||tkm(Λk(x^m) − Λk(x))

ρ⁰_m , z1, · · · , zn−1||i .

From the above inequality, it follows that 1

n

n

X

k=1

h Mk

||tkm(σ^mΛk(x^m) − σΛk(x))

ρm|σ^m− σ| + ρ⁰_m|σ| , z1, · · · , zn−1||i^pk

≤ 1

and consequently,

g(σ^mx^m− σx) ≤ infn

ρm|σ^m− σ| + ρ⁰_m|σ|^pn_H

: ρm∈ B(x^m), ρ⁰_m∈ B(x^m− x)o

≤ (|σ^m− σ|)^pnH infn

ρ^pnH : ρm∈ B(x^m)o + (|σ|)^pnH infn

(ρ⁰_m)^pnH : ρ⁰_m∈ B(x^m− x)o

−→ 0 as m −→ ∞.

This completes the proof.

Theorem 2.3. Let M, M⁰, M⁰⁰ are Musielak-Orlicz functions. Then we have (i) ˆw₀^I(M⁰, Λ, p, ||·, · · · , ·||) ⊆ ˆw₀^I(M ◦ M⁰, Λ, p, ||·, · · · , ·||) provided (p_k) is such that

H₀= inf p_k > 0.

(ii) ˆw^I₀(M⁰, Λ, p, ||·, · · · , ·||)∩ ˆw^I₀(M⁰⁰, Λ, p, ||·, · · · , ·||) ⊆ ˆw₀^I(M⁰+M⁰⁰, Λ, p, ||·, · · · , ·||).

Proof. (i) For given  > 0, first choose 0> 0 such that max{^H₀ , ^H₀⁰} < . Since (Mk) is continuous, choose 0 < δ < 1 such that 0 < t < δ, this implies that Mk(t) < 0. Let x ∈ ˆw^I₀(M, Λ, p, ||·, · · · , ·||). Now from the definition

B(δ) =n

n ∈ N : 1 n

n

X

k=1

h M_k⁰

||tkm(Λk(x))

ρ , z1, · · · , zn−1||i^pk

≥ δ^Ho

∈ I.

Thus if n 6∈ B(δ) then 1 n

n

X

k=1

hM_k⁰

||t_km(Λ_k(x))

ρ , z₁, · · · , z_n−1||i^pk

< δ^H

=⇒

n

X

k=1

h M_k⁰

||tkm(Λk(x))

ρ , z₁, · · · , z_n−1||ipk

< nδ^H

=⇒h M_k⁰

||tkm(Λk(x))

ρ , z1, · · · , zn−1||ipk

< δ^H for all k, m = 1, 2, 3, . . .

=⇒h M_k⁰

||t_km(Λ_k(x))

ρ , z1, · · · , zn−1||i

< δ for all k, m = 1, 2, 3, . . . . Hence from above and using the continuity of M = (Mk), we have

h Mk

 M_k⁰

||tkm(Λk(x))

ρ , z1, · · · , zn−1||i

< 0∀ k, m = 1, 2, 3, . . . , which consequently implies that

n

X

k=1

hM_k M_k⁰

||t_km(Λ_k(x))

ρ , z₁, · · · , z_n−1||i^pk

< max{^H₀, ^H₀⁰} < .

Thus 1 n

n

X

k=1

h M_k

M_k⁰

||t_km(Λ_k(x))

ρ , z₁, · · · , z_n−1||ip_k

< .

This shows that n

n ∈ N : 1 n

n

X

k=1

h Mk

 M_k⁰

||tkm(Λk(x))

ρ , z1, · · · , zn−1||i^pk

≥ o

⊂ B(δ)

and so belongs to I. This proves the result.

(ii) Let (xk) ∈ ˆw^I₀(M⁰, Λ, p, ||·, · · · , ·||) ∩ ˆw₀^I(M⁰⁰, Λ, p, ||·, · · · , ·||). Then the fact 1

n h

(M_k⁰ + M_k⁰⁰)

||tkm(Λk(x))

ρ , z1, · · · , zn−1||ipk

≤

D1 n

hM_k⁰

||t_km(Λ_k(x))

ρ , z₁, · · · , z_n−1||i^pk

+ D1 n

hM_k⁰⁰

||t_km(Λ_k(x))

ρ , z₁, · · · , z_n−1||i^pk

gives the result.

Theorem 2.4. The sequence spaces ˆw^I₀(M, Λ, p, ||·, · · · , ·||) and ˆw_∞^I (M⁰, Λ, p, ||·, · · · , ·||) are solid.

Proof. Let x ∈ ˆw₀^I(M, Λ, p, ||·, · · · , ·||), let (α_k) be a sequence of scalars such that

|α_k| ≤ 1 for all k ∈ N. Then we have nn ∈ N : 1

n

n

X

k=1

hM_k

||t_km(Λ_k(α_kx))

ρ , z₁, · · · , z_n−1||i^pko

⊂

n

n ∈ N : C n

n

X

k=1

h M_k

||tkm(Λk(x))

ρ , z₁, · · · , z_n−1||ipk

≥ o

∈ I,

where C = max{1, |αk|^H}. Hence (αkx) ∈ ˆw^I₀(M, Λ, p, ||·, · · · , ·||) for all sequences of scalars αk with |αk| ≤ 1 for all k ∈ N whenever (xk) ∈ ˆw₀^I(M, Λ, p, ||·, · · · , ·||).

Similarly, we can prove that ˆw^I_∞(M⁰, Λ, p, ||·, · · · , ·||) is also solid.

References

[1] S. Banach, Theorie Operations Linearies, Chelsea Publishing Co., New York 1955.

[2] P. Das, P. Kostyrko, W. Wilczynski and P. Malik, I and I* convergence of double sequences, Math. Slovaca 58 (2008) 605-620.

[3] P. Das, P. Malik, On the statistical and I-variation of double sequences, Real Anal. Exchange 33 (2007-2008) 351-364.

[4] A. Esi, Some new sequence spaces defined by a sequence of moduli, Turk. J. Math.

21 (1997) 61-68.

[5] A. Esi, Strongly [V₂, λ₂, M, p]-summable double sequence spaces defined by Orlicz function, Int. J. Nonlinear Anal. Appl. 2 (2011) 110-115.

[6] S. G¨ahler, Linear 2-normietre Rume, Math. Nachr. 28 (1965), 1-43.

[7] M. Gurdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math. 33 (2009) 257-264.

[8] H. Gunawan, On n-inner product, n-norms, and the Cauchy-Schwartz inequality, Sci. Math. Jpn. 5 (2001) 47-54.

[9] H. Gunawan, The space of p-summable sequence and its natural n-norm, Bull.

Aust. Math. Soc. 64 (2001) 137-147.

[10] H. Gunawan, M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 (2001) 631-639.

[11] P. Kostyrko, T. Salat and W. Wilczy´nski, I-convergence, Real Anal. Exchange 26 (2000) 669-686.

[12] G.G. Lorentz, A contribution to the theory of divergent series, Acta Math. 80 (1948) 167-190.

[13] J. Lindenstrauss, L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971) 345-355.

[14] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, De- partamento de Matem´atica, Universidade Estadmal de Campinas, Campinas SP Brasil 1989.

[15] A. Misiak, n-inner product spaces, Math. Nachr. 140 (1989) 299-319.

[16] M. Mursaleen, On some new invariant matrix methods of summability, Quart. J.

Math. Oxford 34 (1983) 77-86.

[17] M. Mursaleen, A. Alotaibi, On I-convergence in radom 2-normed spaces, Math.

Slovaca 61 (2011) 933-940.

[18] M. Mursaleen, A.K. Noman, On some new sequence spaces of non absolute type related to the spaces lp and l_∞ I, Filomat 25 (2011) 33-51.

[19] M. Mursaleen, A.K. Noman, On some new sequence spaces of non absolute type related to the spaces l_p and l_∞ II, Math. Commun. 16 (2011) 383-398.

[20] M. Mursaleen, S.A. Mohiuddine and O.H.H. Edely, On ideal convergence of dou- ble sequences in intuitioistic fuzzy normed spaces, Comput. Math. Appl. 59 (2010) 603-611.

[21] M. Mursaleen, S.A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Math. Reports 64 (2010) 359-371.

[22] M. Mursaleen, S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca 62 (2012) 49-62.

[23] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics 1034, Springer-Verlag Berlin Heidelberg 1983.

[24] K. Raj, S.K. Sharma, Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function, Acta Univ. Sapientiae Math. 3 (2011) 97-109.

[25] K. Raj, S.K. Sharma, Some generalized difference double sequence spaces defined by a sequence of Orlicz-function, CUBO A Mathematical Journal 14 (2012) 167- 190.

[26] K. Raj, S.K. Sharma, Some multiplier sequence spaces defined by a Musielak- Orlicz function in n-normed spaces, New Zealand J. Math. 42 (2012) 45-56.

[27] W. Raymond, Y. Freese and J. Cho, Geometry of Linear 2-Normed Spaces, N.

Y. Nova Science Publishers, Huntington 2001.

[28] A. Sahiner, M. Gurdal, S. Saltan and H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11 (2007) 1477-1484.

[29] B.C. Tripathy, B. Hazarika, Some I-convergent sequence spaces defined by Orlicz functions, Acta Mathematicae Applicatae Sinica 27 (2011) 149-154.

[30] A. Wilansky, Summability Through Functional Analysis, North-Holland Math.

Stud. 85, Elsevier Science Publishers B.V. 1984.

DOI: 10.7862/rf.2017.12 Sunil K. Sharma

email: sunilksharma42@yahoo.co.in Department of Mathematics

Model Institute of Engineering & Technology Kot Bhalwal, Jammu-181122, J&K

INDIA Ayhan Esi

email: aesi23@hotmail.com Department of Mathematics Adiyaman University 02040, Adiyaman TURKEY

Received 14.03.2017 Accepted 26.07.2017