Multiplier sequence spaces of fuzzy numbers defined by

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Mathematics

and Applications

No 35, pp 69-81 (2012)

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

Multiplier sequence spaces of fuzzy numbers defined by

a Musielak-Orlicz function

Kuldip Raj

^and

Amit Gupta

Submitted by: Marian Mat loka

Abstract: In this paper we introduce some multiplier sequence spaces of fuzzy numbers by using a Musielak-Orlicz function M = (M_k) and multiplier function u = (uk) and prove some inclusion relations between the resulting sequence spaces.

AMS Subject Classification: 40A05, 40D25

Key Words and Phrases: fuzzy numbers, Musielak-Orlicz function, De La-Vallee Poussin means, Statistical convergence, Multiplier function

1 Introduction and Preliminaries

Fuzzy set theory, compared to other mathematical theories, is perhaps the most easily adaptable theory to practice. The main reason is that a fuzzy set has the prop- erty of relativity, variability and inexactness in the definition of its elements. Instead of defining an entity in calculus by assuming that its role is exactly known, we can use fuzzy sets to define the same entity by allowing possible deviations and inexactness in its role. This representation suits well the uncertainties encountered in practical life, which make fuzzy sets a valuable mathematical tool. The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [20] and subsequently several au- thors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming. Matloka [12] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties. For more details about sequence spaces see ([1], [2], [14], [17]) and refrences therein.

The study of Orlicz sequence spaces was initiated with a certain specific purpose in Ba- nach space theory. Indeed, Lindberg [9] got interested in Orlicz spaces in connection with finding Banach spaces with symmetric Schauder bases having complementary

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subspaces isomorphic to c0. Parashar and Choudhary [16] have introduced and discussed some properties of the sequence spaces defined by using a Orlicz function M which generalized the well-known Orlicz sequence space lM and strongly summable sequence spaces [C, 1, p], [C, 1, p]0 and [C, 1, p]∞. Later on, Basarir and Mursaleen [2], Tripathy and Mahanta [19] used the idea of an Orlicz function to construct some spaces of complex sequences. The concept of statistical convergence was introduced by Fast [6] and also independently by Buck [3] and Schoenberg [18] for real and complex sequences. Further this concept was studied by Fridy [7, Connor [4]] and many others. Statistical convergence is closely related to the concept of convergence in Probability.

A fuzzy number is a fuzzy set on the real axis, i.e., a mapping u : Rⁿ → [0, 1] which satisfies the following four conditions:

1. u is normal, i.e., there exist an x0∈ Rⁿ such that u(x0) = 1;

2. u is fuzzy convex, i.e., for x, y ∈ Rⁿ and 0 ≤ λ ≤ 1, u(λx + (1 − λ)y) ≥ min[u(x), u(y)];

3. u is upper semi-continuous;

4. the closure of {x ∈ Rⁿ : u(x) > 0}, denoted by [u]⁰, is compact.

Denote L(Rⁿ) = {u : Rⁿ → [0, 1] \ u satisfies (1) − (4) above}. If u ∈ L(Rⁿ), then u is called a fuzzy number and L(Rⁿ) is a fuzzy number space.

Let C(Rⁿ) denote the family of all non empty, compact, convex subsets of Rⁿ. If α, β ∈ R and A, B ∈ C(Rⁿ), then

α(A + B) = αA + αB, (αβ)A = α(βA), 1A = A

and if α, β ≥ 0, then (α + β)A = αA + βA. The distance between A and B is defined by the Housdorff metric

δ_∞(A, B) = max{sup

a∈A

inf

b∈Bka − bk, sup

b∈B

inf

a∈Aka − bk},

where k.k denoted the usual Euclidean norm in Rⁿ. It is well known that (C(Rⁿ), δ_∞) is a complete metric space. For 0 < α ≤ 1, the α-level set [u]^α is defined by [u]^α= {x ∈ Rⁿ : u(x) ≥ α}. Then from (1)-(4), it follows that [u]^α ∈ (C(Rⁿ)). For the addition and scalar multiplication in L(Rⁿ), we have

[u + v]^α= [u]^α+ [v]^α, [ku]^α= k[u]^α, where u, v ∈ L(Rⁿ), k ∈ R. Define, for each 1 ≤ q < ∞,

dq(u, v) =Z 1 0

[δ∞([u]^α, [v]^α)]^q¹_q

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and d_∞(u, v) = sup

0≤α≤1

δ_∞([u]^α, [v]^α), where δ_∞is the Housdorff metric.

The idea of statistical convergence depends on the density of subsets of the set N of natural numbers. A subset E of N is said to have density δ(E) if

δ(E) = lim

n→∞

1 n

n

X

k=1

χ_E(k) exists,

where χ_E is the characteristic function of E. It is clear that any finite subset of N has zero natural density and δ(E^c) = 1 − δ(E).

A sequence x = (x_k) is said to be statistically convergent to the number L if for every

> 0, δ({k ∈ N : |xk− L| ≥ }) = 0. In this case, we write S − lim x_k= L. A sequence X = (X_k) of fuzzy numbers is said to be bounded if the set {X_k : k ∈ N} of fuzzy numbers is bounded and convergent to the fuzzy number X0, written as limkXk = X0, i.e if for every > 0 there exists a positive integer k0 such that d(Xk, X0) < , for k > k0. By w^F, l^F_∞and c^F denote the set of all, bounded and convergent sequences of fuzzy numbers, respectively see [12].

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) → ∞ as x → ∞.

Lindenstrauss and Tzafriri [10] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x = (xk), then

l_M =n x ∈ w :

∞

X

k=1

M|x_k| ρ

< ∞o ,

which is called as an Orlicz sequence space. Also l_M is a Banach space with the norm kxk = infn

ρ > 0 :

∞

X

k=1

M|x_k| ρ

≤ 1o .

Also, it was shown in [10] that every Orlicz sequence space l_M contains a subspace isomorphic to l_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 = (Mk) of Orlicz functions is called a Musielak-Orlicz function see ([13],[14]). A sequence N = (Nk) of Orlicz functions defined by

N_k(v) = sup{|v|u − M_k : 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

tM=n

x ∈ w : IM(cx) < ∞, for some c > 0o ,

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h_M=n

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

IM(x) =

∞

X

k=1

Mk(xk), x = (xk) ∈ tM.

We consider tMequipped with the Luxemburg norm kxk = infn

k > 0 : I_Mx k

≤ 1o , or equipped with the Orlicz norm

kxk⁰= infn1 k

1 + IM(kx)

: k > 0o .

Let λ = (λ_n) be a non-decreasing sequence of positive numbers tending to ∞ and λn+1≤ λn+ 1, λ1= 1. The generalized De la Vallee-Poussin mean is defined by

tn(x) = 1 λn

X

k∈In

xk,

where In= [n − λn+ 1, n].

The space ˆc of all almost convergent sequences was introduced by Maddox [12] has defined x = (xk) to be strongly almost convergent to a number l if

limn

1 n

n

X

k=1

|x_k+m− l| = 0, uniformly in m.

The following inequality will be used throughout this paper. Let p = (pk) be a sequence of positive real numbers with 0 < pk≤ sup pk = H, and let D = max(1, 2^H−1).

Then for ak, bk ∈ C, the set of complex numbers for all k ∈ N, we have

|ak+ bk|^p^k≤ D{|ak|^p^k+ |bk|^p^k} (1)

Let Λ denote the set of all non-decreasing sequences λ = (λn) of positive numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 1, M = (Mk) be a Musielak-Orlicz function and p = (pk) be a bounded sequence of positive real numbers. A sequence X = (Xk) of fuzzy numbers is said to be almost λ-statistically convergent to the fuzzy number X0, with respect to the Musielak-Orlicz function, if for every > 0

n→∞lim 1 λn

n

k ∈ In:h

Mk(d(t_km(X), X₀) ρ )i^pk

≥ o = 0, uniformly in m for some ρ > 0,

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where the vertical bars indicate the number of elements in the enclosed set and

tkm(X) = Xm+ Xm+1+ · · · + Xm+k

k + 1 = 1

k + 1

k

X

i=0

Xm+i.

The set of all almost λ-statistically convergent sequences of fuzzy numbers is denoted by ˆS^F(λ, Mk, u, p). In this case, we write Xk → X0( ˆS^F(λ, Mk, u, p)). In the special cases λn = n for all n ∈ N and M^k(X) = X, pk = 1, uk = 1 for all k ∈ N, we shall write ˆS^F(Mk, u, p) and ˆS^F(λ) instead of ˆS^F(λ, Mk, u, p), respectively. Furthermore, the set of all almost statistically convergent sequences of fuzzy numbers is denoted by ˆS^F

Let λ ∈ Λ, M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers and u = (uk) be a sequence of strictly positive real numbers.

Then we define the following classes of sequences in this paper:

ˆ

w^F(λ, M, u, p) =n

X = (X_k) : lim

n→∞

1 λn

X

k∈In

u_kh

M_kd(t_km(X), X₀) ρ

i^pk

= 0

uniformly in m, for some ρ > 0o , ˆ

w^F₀(λ, M, u, p) =n

X = (X_k) : lim

n→∞

1 λ_n

X

k∈I_n

u_kh

M_kd(tkm(X), ¯0) ρ

ipk

= 0

uniformly in m, for some ρ > 0o and

ˆ

w_∞^F(λ, M, u, p) =n

X = (Xk) : sup

m,n

1 λn

X

k∈In

uk

h Mk

d(t_km(X), ¯0) ρ

i^pk

< ∞

uniformly in m, for some ρ > 0o , where

¯0(t) =

1, t = (0, 0, 0, · · · , 0)

0, otherwise .

If X ∈ ˆw^F(λ, M, u, p), we say that X is strongly almost λ-convergent with respect to the Musielak-Orlicz function M = (M_k). In this case we write X_k → X₀( ˆw^F(λ, M, u, p)). The following sequence spaces are defined by giving particular values to M, u, p.

(i) For λn = n ˆ

w^F(λ, M, u, p) = ˆw^F(M, u, p), ˆw^F₀(λ, M, u, p) = ˆw^F₀(M, u, p), and ˆw^F_∞(λ, M, u, p) = ˆ

w^F_∞(M, u, p),

(ii) If M = Mk(x) = x for all k, we get

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ˆ

w^F(λ, M, u, p) = ˆw^F(u, p, λ), ˆw^F₀(λ, M, u, p) = ˆw₀^F(u, p, λ), and ˆw^F_∞(λ, M, u, p) = ˆ

w^F_∞(u, p, λ),

(iii) If pk= 1 for all k ∈ N, then ˆ

w^F(λ, M, u, p) = ˆw^F(M, u, λ), ˆw₀^F(λ, M, u, p) = ˆw₀^F(M, u, λ), and ˆw^F_∞(λ, M, u, p) = ˆ

w^F_∞(M, u, λ),

(iv) If M = M_k(x) = x for all k, and p_k= 1 for all k ∈ N, then ˆ

w^F(λ, M, u, p) = ˆw^F(u, λ), ˆw₀^F(λ, M, u, p) = ˆw^F₀(u, λ), and ˆw_∞^F(λ, M, u, p) = ˆ

w^F_∞(u, λ),

(v) If pk= 1 for all k ∈ N, and u^k = 1 for all k, then ˆ

w^F(λ, M, u, p) = ˆw^F(M, λ), ˆw^F₀(λ, M, u, p) = ˆw^F₀(M, λ), and ˆw^F_∞(λ, M, u, p) = ˆ

w^F_∞(M, λ),

(vi) If M = Mk(x) = x, pk = 1 and uk = 1 for all k, then ˆ

w^F(λ, M, u, p) = ˆw^F(λ), ˆw₀^F(λ, M, u, p) = ˆw₀^F(λ), and ˆw^F_∞(λ, M, u, p) = ˆw_∞^F(λ).

In this paper we shall prove properties of linearity and some inclusion relations between the classes of sequences ˆw^F(λ, M, u, p), ˆw^F₀(λ, M, u, p), ˆw^F_∞(λ, M, u, p) and Sˆ^F(λ, M, u, p).

2. Main Results

Theorem 2.1. For any Musielak-Orlicz function M = (M_k), p = (p_k) be a bounded sequence of strictly positive real numbers and u = (u_k) be a sequence of positive real numbers, we have

ˆ

w^F₀(λ, M, u, p) ⊂ ˆw^F(λ, M, u, p) ⊂ ˆw^F_∞(λ, M, u, p).

Proof. The inclusion ˆw^F₀(λ, M, u, p) ⊂ ˆw^F(λ, M, u, p) is obvious.

Let X ∈ ˆw^F(λ, M, u, p), then 1

λn

X

k∈In

uk

h Mk

d(tkm(X), ¯0) 2ρ

i^pk

≤ D

λn

X

k∈In

1 2^p^kuk

h Mk

d(tkm(X), X0) ρ

i^pk

+ D

λn

X

k∈In

1 2^p^ku_kh

M_kd(X₀, ¯0) ρ

i^pk

≤ D

λn

X

k∈In

uk

h Mk

d(tkm(X), X0) ρ

i^pk

+ D max

k∈I_n

nmaxn 1, sup

k

u_kh

M_kd(X₀, ¯0) ρ

i^Hoo ,

where sup

k

pk= H and D = max(1, 2^H−1). Thus we get X ∈ ˆw^F_∞(λ, M, u, p).

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Theorem 2.2. If M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers and u = (uk) be a sequence of strictly positive real numbers, then ˆw^F(λ, M, u, p), ˆw^F₀(λ, M, u, p) and ˆw^F_∞(λ, M, u, p) are closed under the operations of addition and scalar multiplication.

Proof. Let X = (X_k), Y = (Y_k) ∈ ˆw^F_∞(λ, M, u, p) and α, β ∈ C. Then there exist positive numbers ρ₁, ρ₂ such that

sup

m,n

1 λn

X

k∈In

uk

h Mk

d(tkm(X), ¯0) ρ₁

i^pk

< ∞, uniformly in m

and

sup

m,n

1 λn

X

k∈I_n

u_kh

M_kd(t_km(Y ), ¯0) ρ₂

ip_k

< ∞, uniformly in m.

Define ρ₃= max(2|α|ρ₁, 2|β|ρ₂). Since M = (M_k) is non-decreasing and convex, we have

sup

m,n

1 λn

X

k∈In

uk

h Mk

αd(t_km(X), ¯0) + βd(t_km(Y ), ¯0) ρ₃

i^pk

≤ 1

2sup

m,n

1 λ_n

X

k∈I_n

u_kh

M_kd(tkm(X), ¯0) ρ₁

ip_k

+ 1

2sup

m,n

1 λ_n

X

k∈I_n

uk

h Mk

d(tkm(X), ¯0) ρ₂

ipk

< ∞.

This proves that ˆw^F_∞(λ, M, u, p) is a linear space. Similarly we can prove for other cases.

Theorem 2.3. If 0 < pk ≤ rk < ∞ for all k ∈ N and (p^r^k_k) be bounded, then we have

ˆ

w^F_∞(λ, M, u, r) ⊆ ˆw^F_∞(λ, M, u, p).

Proof. Let X = (X_k) ∈ ˆw^F_∞(λ, M, u, r). Thus, we have sup

m,n

1 λn

X

k∈In

uk

h Mk

d(tkm(X), ¯0) ρ

i^rk

Let s_k = sup

m,n

1 λn

X

k∈In

u_kh

M_kd(t_km(X), ¯0) ρ

i^rk

and λ_k= ^p_r^k

k. Since p_k ≤ rk, we have 0 ≤ λ_k ≤ 1. Take 0 < λ < λ_k. Now define

uk =







sk if sk≥ 1 0 if sk < 1

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and

vk=

0 if sk≥ 1 sk if sk < 1

sk = uk + vk, s^λ_k^k = u^λ_k^k+ v_k^λ^k. It follows that u^λ_k^k ≤ uk ≤ sk, v_k^λ^k ≤ v^λ_k. since s_k^λ^k = u^λ_k^k+ v_k^λ^k, then s^λ_k^k≤ sk+ v_k^λ

sup

m,n

1 λn

X

k∈In

u_kh

M_kd(t_km(X), ¯0) ρ₁

^rki^λk

≤ sup

m,n

1 λn

X

k∈In

u_kh

M_kd(t_km(X), ¯0) ρ₁

i^rk

=⇒ sup

m,n

1 λ_n

X

k∈I_n

u_kh

rkipk/rk

≤ sup

m,n

1 λ_n

X

k∈I_n

u_kh

irk

=⇒ sup

n,m

1 λ_n

X

k∈I_n

uk

h

Mk

d(tkm(X), ¯0) ρ₁

ipk

≤ sup

m,n

1 λ_n

X

k∈I_n

uk

h Mk

d(tkm(X), ¯0) ρ

irk

But

sup

m,n

1 λn

X

k∈In

uk

h Mk

d(tkm(X), ¯0) ρ

i^rk

Therefore

sup

m,n

1 λn

X

k∈I_n

uk

h Mk

d(tkm(X), ¯0) ρ

ipk

Hence x ∈ ˆw^F_∞(λ, M, u, p). Thus we get ˆw^F_∞(λ, M, u, r) ⊆ ˆw^F_∞(λ, M, u, p).

Theorem 2.4. Suppose M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers and u = (uk) be a sequence of strictly positive real numbers. If sup

k

(Mk(t))^p^k< ∞ for all fixed t > 0, then

ˆ

w^F(λ, M, u, p) ⊂ ˆw^F_∞(λ, M, u, p).

Proof. Let X ∈ ˆw^F(λ, M, u, p), then there exists a positive number ρ₁ > 0 such that

n→∞lim 1 λ_n

X

k∈I_n

u_kh

M_kd(tkm(X), X0) ρ₁

ipk

= 0, uniformly in m.

Define ρ = 2ρ₁. Since M = (Mk) is non-decreasing and convex, for each k. So by using (1), we have

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sup

m,n

1 λn

X

k∈In

uk

h Mk

d(tkm(X), ¯0) ρ

i^pk

≤ sup

m,n

1 λn

X

k∈In

uk

h Mk

d(t_km(X), X₀) + d(X₀, ¯0) ρ

i^pk

≤ Dn sup

m,n

1 λn

X

k∈In

1 2^p^kuk

h Mk

d(tkm(X), X0) ρ₁

i^pk

+ sup

m,n

1 λn

X

k∈In

1 2^p^kuk

h Mk

d(t_km(X), ¯0) ρ₁

i^pko

< ∞.

Thus X ∈ ˆw^F_∞(λ, M, u, p), which completes the proof.

Theorem 2.5. Let 0 < h = inf pk ≤ pk ≤ sup pk = H < ∞. Then for a Musielak- Orlicz function M = (Mk) which satisfies the ∆2-condition, we have ˆw^F₀(λ, u, p) ⊂

ˆ

w^F₀(λ, M, u, p), ˆw^F(λ, u, p) ⊂ ˆw^F(λ, M, u, p) and ˆw_∞^F(λ, u, p) ⊂ ˆw_∞^F(λ, M, u, p).

Proof. Let X ∈ ˆw^F(λ, u, p), then we have 1

λ_n X

k∈I_n

uk

hd(tkm(X), X0) ρ

ipk

→ 0 as n → ∞, uniformly in m.

Let > 0 and choose δ with 0 < δ < 1 such that M_k(t) < for 0 ≤ t ≤ δ. Then 1

λn

X

k∈In

u_kh

i^pk

= 1

λn

X

k∈I_n,d(t_km(X),X₀)≤δ

u_kh

i^pk

+ 1

λn

X

k∈I_n,d(t_km(X),X₀)>δ

u_kh

i^pk

= X

1

+X

2

.

where X

1

= 1 λn

X

k∈I_n,d(t_km(X),X₀)≤δ

uk

h Mk

d(tkm(X), X0) ρ

i^pk

< max(, ^H)

by using continuity of (Mk). For the second summation, we will make the following procedure. Thus we have

d(tkm(X), X0)

ρ < 1 +d(tkm(X), X0)/ρ

δ .

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Since M = (Mk) is non-decreasing and convex, so we have u_kh

M_kd(tkm(X), X0) ρ

i

< u_kh M_kn

1 +d(tkm(X), X0)/ρ δ

oi

≤ 1

2uk

h Mk(2)i

+1 2uk

h Mk

n

2d(tkm(X), X0)/ρ δ

oi . Again, since M = (M_k) satisfies the ∆₂-condition, it follows that

uk

h Mk

d(tkm(X), X0) ρ

i ≤ 1

2Lnd(tkm(X), X0)/ρ δ

o uk

h Mk(2)i

+ 1

2Lnd(t_km(X), X₀)/ρ δ

ou_kh M_k(2)i

= Lnd(tkm(X), X0)/ρ δ

o uk

h Mk(2)i

.

Thus, it follows that X

2

= max

k∈In

n

1,hLuk[Mk(2)]

δ

i^Ho 1 λn

X

k∈In

hd(tkm(X), X0) ρ

i^pk

.

Taking the limit as → 0 and n → ∞, it follows that X ∈ ˆw^F(λ, M, u, p). Similarly, we can prove that ˆw^F₀(λ, u, p) ⊂ ˆw^F₀(λ, M, u, p) and ˆw^F_∞(λ, u, p) ⊂ ˆw^F_∞(λ, M, u, p).

Theorem 2.6. If M = (M_k) be a Musielak-Orlicz function, p = (p_k) be a bounded sequence of positive real numbers and u = (u_k) be a sequence of strictly positive real numbers, then

(i) If 0 < inf p_k≤ p_k ≤ 1 for all k, then ˆw^F(λ, M, u) ⊆ ˆw^F(λ, M, u, p), (ii) If 1 ≤ pk ≤ sup pk= H < ∞ then ˆw^F(λ, M, u, p) ⊆ ˆw^F(λ, M, u).

Proof. (i) Let X ∈ ˆw^F(λ, M, u). Since 0 < inf pk≤ pk ≤ 1, we get 1

λn

X

k∈I_n

u_kh

i^pk

≤ 1 λn

X

k∈I_n

u_kh

i

and hence X ∈ ˆw^F(λ, M, u, p).

(ii) Let X ∈ ˆw^F(λ, M, u, p) and 1 ≤ p_k≤ sup p_k= H < ∞. Then for every 0 < < 1, there exists a positive integer n0such that

1 λn

X

k∈In

uk

h Mk

d(tkm(X), X0) ρ

i^pk

≤ < 1

for all n ≥ n₀. This follows that 1

λn

X

k∈I_n

u_kh

i≤ 1 λn

X

k∈I_n

u_kh

ip_k

.

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and hence X ∈ ˆw^F(λ, M, u).

Theorem 2.7. If M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers, u = (uk) be a sequence of strictly positive real numbers and 0 < h = inf pk≤ pk ≤ sup pk = H < ∞. Then ˆw^F(λ, M, u, p) ⊂ ˆS^F(λ).

Proof. The proof of the theorem follows from the following inequality:

1 λ_n

X

k∈I_n

uk

h Mk

d(tkm(X), X0) ρ

ipk

≥ 1

λn

X

k∈I_n,d(t_km(X),X₀)≥

u_kh

ip_k

≥ 1

λn

X

minn

uk[Mk(1)]^h, uk[Mk(1)]^Ho

≥ 1

λ_n n

k ∈ In : d(tkm(X), X0) ≥ o min

k∈In

n

uk[Mk(1)]^h, uk[Mk(1)]^Ho , where 1= _ρ.

Theorem 2.8. Let M = (M_k) be a Musielak-Orlicz function, X = (X_k) be a bounded sequence of fuzzy numbers and 0 < h = inf p_k ≤ p_k ≤ sup p_k = H < ∞.

Then ˆS^F(λ) ⊂ ˆw^F(λ, M, u, p).

Proof. Suppose that X ∈ l_∞^F and Xk → X0( ˆS^F(λ)). Since X ∈ l^F_∞, there exists a constant K > 0 such that d(tkm(X), X0) ≤ K for all k, m. Given > 0, we have

1 λn

X

k∈In

u_kh

i^pk

= 1

λn

X

uk

h Mk

d(tkm(X), X0) ρ

i^pk

+ 1

λn

X

k∈I_n,d(t_km(X),X₀)<

uk

h Mk

d(tkm(X), X0) ρ

i^pk

≤ 1

λn

X

maxn

uk[Mk(K

ρ)]^h, uk[Mk(K ρ)]^Ho

+ 1

λ_n

X

k∈In,d(tkm(X),X0)<

uk[Mk( ρ)]^p^k

≤ max

k∈I_n

nu_k[M_k(T )]^h, u_k[M_k(T )]^Ho 1 λn

nk ∈ I_n : d(t_km(X), X₀) ≥ o

+ max

k∈I_n

nu_k[M_k(₁)]^h, u_k[M_k(₁)]^Ho . where T =^K_ρ,_ρ = 1. Hence X ∈ ˆw^F(λ, M, u, p).

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DOI: 10.7862/rf.2012.6 Kuldip Raj

email: kuldipraj68@gmail.com Amit Gupta

email: kuldeepraj68@rediffmail.com School of Mathematics

Shri Mata Vaishno Devi University, Katra-182320 J & K, India.

Received 18.08.2011