MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES ... - ijpam

International Journal of Pure and Applied Mathematics Volume 78 No. 4 2012, 509-522 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu

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MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES OF FUZZY NUMBERS DEFINED BY A SEQUENCE OF ORLICZ FUNCTIONS Kuldip Raj1 § , Seema Jamwal2 , Sunil K. Sharma3 1,2,3 School

of Mathematics Shri Mata Vaishno Devi University Katra, 182320, J&K, INDIA

Abstract: In this paper we introduce multiplier generalized double sequence spaces of fuzzy numbers defined by a sequence of Orlicz functions M = (Mk,l ) and multiplier function u = (uk,l ). We also make an effort to prove some topological properties and inclusion relation between these spaces. AMS Subject Classification: 40A05, 40D25 Key Words: fuzzy numbers, Musielak-Orlicz function, de La Vallee Poussin means, statistical convergence, multiplier function

1. Introduction The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [16] and subsequently several authors 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 [8] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties. In [9] Nanda studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Savas [13] Received:

April 9, 2012

§ Correspondence

author

c 2012 Academic Publications, Ltd.

url: www.acadpubl.eu

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introduced and discussed double convergent sequences of fuzzy numbers and showed that the space of all double convergent sequences of fuzzy numbers is complete. Recently Basarir and Mursaleen [2] introduced and studied some new sequence spaces of fuzzy numbers generated by nonnegative regular matrix. The study of Orlicz sequence spaces was initiated with a certain specific purpose in Banach space theory. Parashar and Choudhary [11] 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 [1], Tripathy and Mahanta [15] 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 [4] and Schoenberg [14] for real and complex sequences. Further this concept was studied by Fridy [7], Connor [5] and many others. Statistical convergence is closely related to the concept of convergence in probability. The existing literature on statistical convergence appears to have been restricted to real or complex analysis, but at the first time Nurray and Savas [10] extended the idea to apply the sequences of fuzzy numbers.

2. Definitions and Preliminaries 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 → ∞. If convexity of Orlicz function is replaced by M (x + y) ≤ M (x) + M (y), then this function is called the modulus function see, [12]. An orlicz function M is said to satisfy ∆2 -condition for all values of u, if there exists K > 0 such that M (2u) ≤ KM (u), u ≥ 0. A fuzzy number is a fuzzy set on the real axis, i.e., a mapping X : Rn → [0, 1] which satisfies the following four conditions: 1. X is normal, i.e., there exist an x0 ∈ Rn such that X(x0 ) = 1; 2. X is fuzzy convex, i.e., for x, y ∈ Rn and 0 ≤ λ ≤ 1, X(λx + (1 − λ)y) ≥ min[X(x), X(y)]; 3. X is upper semi-continuous; 4. the closure of {x ∈ Rn : X(x) > 0}, denoted by [X]0 , is compact.

MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES...

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Let C(Rn ) = {A ⊂ Rn : A compact and convex }. The spaces C(Rn ) has a linear structure induced by the operations A + B = {a + b : a ∈ A, b ∈ B} and λA = {λa : a ∈ A} C(Rn )

for A, B ∈ and λ ∈ R. The Hausdorff distance between A and B of n C(R ) is defined as δ∞ (A, B) = max{sup inf ka − bk, sup inf ka − bk} a∈A b∈B

b∈B a∈A

where k.k denotes the usual Euclidean norm in Rn . It is well known that (C(Rn ), δ∞ ) is a complete (not separable) metric space. Let λ = (λm,n ) be a non-decreasing sequence of positive real numbers tending to infinity such that λm+1,n ≤ λm,n + 1, λm,n+1 ≤ λm,n + 1, λm,n − λm+1,n ≤ λm,n+1 − λm+1,n+1 , λ1,1 = 1 and Im,n = {(k, l) : m − λm,n + 1 ≤ k ≤ m, n − λm,n + 1 ≤ l ≤ n}. The generalized double de la Vallee-Poussin mean is defined by X 1 (Xk,l ). tm,n = tm,n (Xk,l ) = λm,n (k,l)∈Im,n

A Fuzzy double sequence is a double infinite array of fuzzy numbers. We denote a fuzzy double sequence by (Xk,l ), where Xk,l ’s are fuzzy numbers for each k, l ∈ N. By s′′ (F ) we denote the set of all double sequences of fuzzy numbers. A double sequence X = (Xk,l ) of fuzzy numbers is said to be convergent in the Pringsheim’s sense or P -convergent to a fuzzy number X0 , if for every ǫ > 0 there exists N ∈ N such that d(Xk,l , X0 ) < ǫ for all k, l > N, where N is the set of natural numbers, and we denote it also by P −lim X = X0 . The number X0 is called the Pringsheim limit of (Xk,l ). More exactly we say that a double sequence (Xk,l ) converges to a finite number X0 if Xk,l tend to X0 as both k and l tends to ∞ independently of one another.

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A double sequence X = (Xk,l ) of fuzzy numbers is said to be λ-statistically convergent to X0 provided that for each ǫ > 0 1 |{(j, k); j ≤ m and k ≤ n : d(Xk,l , X0 ) ≥ ǫ}| = 0. m,n m, n

P − lim

We denote the set of all double λ-statistically convergent sequences of fuzzy numbers by s′′ (λ)F . Let M = (Mk,l ) be a sequence of Orlicz functions, p = (pk,l ) be a bounded sequence of positive real numbers and u = (uk,l ) be a sequence of strictly positive real numbers. In the present paper we define the following classes of sequences: w′′ (λ, M, u, p)F h  d(t (X), X ) ipk,l n X 1 mn 0 uk,l Mk,l = X = (Xk,l ) ∈ s′′ (F ) : lim m,n→∞ λm,n ρ k,l∈Im,n

o = 0, uniformly in m, n for some ρ > 0 , w0′′ (λ, M, u, p)F = n

X = (Xk,l ) ∈ s′′ (F ) :

1

lim

m,n→∞

λm,n

X

k,l∈Im,n

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l ρ

o = 0, uniformly in m, n for some ρ > 0 and ′′ (λ, M, u, p)F = w∞ n X = (Xk,l ) ∈ s′′ (F ) : sup m,n

1 λm,n

X

k,l∈Im,n

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l ρ

o < ∞, uniformly in m, n for some ρ > 0 . where ¯ 0(t) =



1, 0,

t = (0, 0, 0, · · · , 0) otherwise.

If X ∈ w′′ (λ, M, u, p)F , we say that X is strongly almost λ-convergent with respect to the sequence of Orlicz function. In this case we write Xk,l → X0 (w′′ (λ, M, u, p)F ). The following sequence spaces are defined by giving particular values to M, u, p and λ;

MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES... (i) For λm,n = 1, we have w′′ (λ, M, u, p)F = w′′ (M, u, p)F , w0′′ (λ, M, u, p)F = w0′′ (M, u, p)F and ′′ ′′ w∞ (λ, M, u, p)F = w∞ (M, u, p)F ,

(ii) If M = Mk,l (x) = x for all k, l we get w′′ (λ, M, u, p)F = w′′ (λ, u, p)F ,

w0′′ (λ, M, u, p)F = w0′′ (λ, u, p)F

and ′′ ′′ w∞ (λ, M, u, p)F = w∞ (λ, u, p)F ,

(iii) If pk,l = 1 for all k, l ∈ N, then w′′ (λ, M, u, p)F = w′′ (λ, M, u)F ,

w0′′ (λ, M, u, p)F = w0′′ (λ, M, u)F

and ′′ ′′ w∞ (λ, M, u, p)F = w∞ (λ, M, u)F ,

(iv) If M = Mk,l (x) = x for all k, l and pk,l = 1 for all k, l ∈ N, then w′′ (λ, M, u, p)F = w′′ (λ, u)F , w0′′ (λ, M, u, p)F = w0′′ (λ, u)F and ′′ ′′ w∞ (λ, M, u, p)F = w∞ (λ, u)F ,

(v) If pk,l = 1 for all k, l ∈ N, and uk,l = 1 for all k, l then w′′ (λ, M, u, p)F = w′′ (λ, M)F ,

w0′′ (λ, M, u, p)F = w0′′ (λ, M)F

and ′′ ′′ w∞ (λ, M, u, p)F = w∞ (λ, M)F ,

(vi) If M = Mk,l (x) = x, pk,l = 1 and uk,l = 1 for all k, l then w′′ (λ, M, u, p)F = w′′ (λ)F , w0′′ (λ, M, u, p)F = w0′′ (λ)F

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and ′′ ′′ w∞ (λ, M, u, p)F = w∞ (λ)F .

The following inequality will be used throughout the paper. Let p = (pk,l ) be a double sequence of positive real numbers with 0 < pk,l ≤ supk,l pk,l = H,  and let D = max 1, 2H−1 . Then, for the factorable sequences ak and bk in the complex plane, we have |ak,l + bk,l |pk,l ≤ D(|ak,l |pk,l + |bk,l |pk,l ).

(2.1)

In the present paper we study some topological properties and inclusion relation between the above defined sequence spaces.

3. Main Results Theorem 3.1. Suppose M = (Mk,l ) be a sequence of Orlicz functions, p = (pk,l ) be a bounded sequence of positive real numbers and u = (uk,l ) be a sequence of positive real numbers then ′′ w0′′ (λ, M, u, p)F ⊂ w′′ (λ, M, u, p)F ⊂ w∞ (λ, M, u, p)F .

Proof. The inclusion w0′′ (λ, M, u, p)F ⊂ w′′ (λ, M, u, p)F is obvious. Let X ∈ w′′ (λ, M, u, p)F . Then we get 1 λm,n

≤ + ≤

D λm,n D λm,n D λm,n

X

k,l∈Im,n

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l 2ρ

X

h  d(t (X), X ) ipk,l mn 0 u M k,l k,l p k,l 2 ρ

X

h  d(X , ¯0) ipk,l 0 u M k,l k,l p k,l 2 ρ

X

h  d(t (X), X ) ipk,l mn 0 uk,l Mk,l ρ

1

k,l∈Im,n

1

k,l∈Im,n

k,l∈Im,n

+ D max

k,l∈Imn

n

n h  d(X , ¯0) iH oo 0 . max 1, sup uk,l Mk,l ρ

MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES...

515

′′ (λ, M, u, p)F . where sup pk,l = H and D = max(1, 2H−1 ). Thus we get X ∈ w∞ k,l

Theorem 3.2. Suppose M = (Mk,l ) be a sequence of Orlicz functions, p = (pk,l ) be a bounded sequence of positive real numbers and u = (uk,l ) be a sequence of strictly positive real numbers then w′′ (λ, M, u, p)F , w0′′ (λ, M, u, p)F , ′′ (λ, M, u, p)F are linear spaces. and w∞ ′′ (λ, M, u, p)F and α, β ∈ C. Then Proof. Let X = (Xk,l ), Y = (Yk,l ) ∈ w∞ there exist positive numbers ρ1 , ρ2 such that

sup m,n

X

k,l∈Im,n

h  d(t (X), ¯ 0) ipk,l mn uk,l Mk,l < ∞, uniformly in m, n. ρ1

X

h  d(t (X), ¯ 0) ipk,l mn uk,l Mk,l < ∞, uniformly in m, n. ρ2

1 λm,n

and sup m,n

1 λm,n

k,l∈Im,n

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

1 λm,n

≤ sup m,n

≤ +

X

k,l∈Im,n

1 λm,n

h  αd(t (X), ¯0) + βd(t (Y ), ¯0) ipk,l mn mn uk,l Mk,l ρ3

X

k,l∈Im,n

1 1 sup 2 m,n λm,n 1 1 sup 2 m,n λm,n

h  αd(t (X), ¯0) βd(t (Y ), ¯0) ipk,l mn mn uk,l Mk,l + ρ3 ρ3

X

k,l∈Im,n

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l ρ1

X

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l ρ2

k,l∈Im,n

< ∞. ′′ (λ, M, u, p)F is a linear space. Similarly, we can prove This proves that w∞ others. r

Theorem 3.3. Let 0 < pk,l ≤ rk,l for all k, l ∈ N and ( pk,l ) be bounded. k,l Then we have ′′ ′′ w∞ (λ, M, u, r)F ⊂ w∞ (λ, M, u, p)F .

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′′ (λ, M, u, r)F . Then Proof. Let X = (Xk,l ) ∈ w∞ h  d(t (X), ¯0) irk,l X 1 mn uk,l Mk,l < ∞, uniformly in m, n. sup ρ m,n λm,n k,l∈Im,n

Let sk,l = sup m,n

1

X

λm,n

k,l∈Im,n

h  d(t (X), ¯0) irk,l mn and λk,l = uk,l Mk,l ρ

pk,l rk,l .

Since

pk,l ≤ rk,l , we have 0 ≤ λk,l ≤ 1. Take 0 < λ < λk,l . Define  sk,l if sk,l ≥ 1 uk,l = 0 if sk,l < 1 and vk,l =



0 sk,l

if sk,l ≥ 1 if sk,l < 1 λ

λ

λ

λ

sk,lk,l = uk,lk,l + vk,lk,l . It follows that uk,lk,l ≤ uk,l ≤ sk,l ,

sk,l = uk,l + vk,l , λ

λ

λ

λ

λ

λ λ . since s k,l = u k,l + v k,l , then s k,l ≤ s vk,lk,l ≤ vk,l k,l + vk,l k,l k,l k,l k,l

1

sup

λm,n

m,n

X

uk,l

k,l∈Im,n

h  d(t (X), ¯0) rk,l iλk,l mn Mk,l ρ 1

≤ sup

λm,n

m,n

=⇒ sup m,n

1 λm,n

X

uk,l

k,l∈Im,n

≤ sup m,n

=⇒ sup m,n

1 λm,n

X

k,l∈Im,n

X

h

k,l∈Im,n

Mk,l

1 λm,n

 d(t

h  d(t (X), ¯0) irk,l mn uk,l Mk,l ρ

0) mn (X), ¯

X

k,l∈Im,n

ρ

rk,l ipk,l /rk,l

h  d(t (X), ¯0) irk,l mn uk,l Mk,l ρ

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l ρ

≤ sup m,n

1 λm,n

X

k,l∈Im,n

h  d(t (X), ¯0) irk,l mn uk,l Mk,l ρ

But sup m,n

1 λm,n

X

k,l∈Im,n

h  d(t (X), ¯0) irk,l mn < ∞, uniformly in m, n. uk,l Mk,l ρ

MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES...

517

Therefore sup m,n

1 λm,n

X

k,l∈Im,n

h  d(t (X), ¯ 0) ipk,l mn < ∞, uniformly in m, n. uk,l Mk,l ρ

′′ (λ, M, u, p)F . Thus we get w ′′ (λ, M, u, r)F ⊂ w ′′ (λ, M, u, p)F . Hence X ∈ w∞ ∞ ∞

Theorem 3.4. Suppose M = (Mk,l ) be a sequence of Orlicz functions, p = (pk,l ) be a bounded sequence of positive real numbers and u = (uk,l ) be a sequence of strictly positive real numbers. If sup(Mk,l (x))pk,l < ∞ for all fixed k,l

x > 0, then ′′ w′′ (λ, M, u, p)F ⊂ w∞ (λ, M, u, p)F .

Proof. Let X ∈ w′′ (λ, M, u, p)F . Then there exists a positive number ρ1 > 0 such that h  d(t (X), X ) ipk,l X 1 mn 0 uk,l Mk,l = 0, uniformly in m, n. lim m,n→∞ λm,n ρ1 k,l∈Im,n

Define ρ = 2ρ1 . Since M = (Mk,l ) is non-decreasing and convex, for each k, l so by using (2.1), we have sup m,n

≤ sup m,n

n

m,n

λm,n

1 λm,n

≤ D sup + sup

1

m,n

X

k,l∈Im,n

X

k,l∈Im,n

1 λm,n

1 λm,n

h  d(t (X), ¯0) ipk,l mn uk,l Mk,l ρ

h  d(t (X), X ) + d(X , ¯0) ipk,l mn 0 0 uk,l Mk,l ρ 1

X

k,l∈Im,n

X

k,l∈Im,n

h  d(t (X), X ) ipk,l mn 0 u M k,l k,l 2pk,l ρ1

h  d(t (X), ¯0) ipk,l o mn u M k,l k,l 2pk,l ρ1 1

< ∞. ′′ (λ, M, u, p). This completes the proof of the theorem. Thus X ∈ w∞

Theorem 3.5. Let 0 < h = inf pk,l ≤ pk,l ≤ sup pk,l = H < ∞. Then for a sequence of Orlicz functions M = (Mk,l ) which satisfies the ∆2 -condition,

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we have w0′′ (λ, u, p)F ⊂ w0′′ (λ, M, u, p)F , w′′ (λ, u, p)F ⊂ w′′ (λ, M, u, p)F and ′′ (λ, u, p)F ⊂ w ′′ (λ, M, u, p)F . w∞ ∞ Proof. Let X ∈ w′′ (λ, u, p)F . Then we have h d(t (X), X ) ipk,l X 1 mn 0 → 0 as m, n → ∞, uniformly in m, n. uk,l λm,n ρ k,l∈Im,n

Let ǫ > 0 and choose δ with 0 < δ < 1 such that Mk,l (t) < ǫ for 0 ≤ t ≤ δ. Then h  d(t (X), X ) ipk,l X 1 mn 0 uk,l Mk,l λm,n ρ k,l∈Im,n

=

1 λm,n

X

k,l∈Im,n ,d(tmn (X),X0 )≤δ

h  d(t (X), X ) ipk,l mn 0 uk,l Mk,l ρ

X

h  d(t (X), X ) ipk,l mn 0 uk,l Mk,l ρ

1

+

λm,n k,l∈Im,n ,d(tmn (X),X0 )>δ X X = + . 1

where X = 1

1 λm,n

2

X

k,l∈Im,n ,d(tmn (X),X0 )≤δ

h  d(t (X), X ) ipk,l mn 0 uk,l Mk,l < max(ǫ, ǫH ) ρ

by using continuity of (Mk,l ). For the second summation, we shall make the following procedure. Thus we have d(tmn (X), X0 ) 0, we have h  d(t (X), X ) ipk,l X 1 mn 0 uk,l Mk,l λm,n ρ k,l∈Im,n

MULTIPLIER GENERALIZED DOUBLE SEQUENCE SPACES...

= + ≤ + ≤

1 λm,n 1 λm,n 1 λm,n

X

k,l∈Im,n ,d(tmn (X),X0 )≥ǫ

521

h  d(t (X), X ) ipk,l mn 0 uk,l Mk,l ρ

h  d(t (X), X ) ipk,l mn 0 uk,l Mk,l ρ k,l∈Im,n ,d(tmn (X),X0 )