THE DIFFERENCE ORLICZ SPACE OF ENTIRE SEQUENCE OF

Commun. Korean Math. Soc. 25 (2010), No. 2, pp. 193–206 DOI 10.4134/CKMS.2010.25.2.193

THE DIFFERENCE ORLICZ SPACE OF ENTIRE SEQUENCE OF FUZZY NUMBERS Nagarajan Subramanian and Ayhan Esi Abstract. In this paper we define and study the difference Orlicz space of entire sequence of fuzzy numbers. We study their different properties and statistical convergence in these spaces.

1. Introduction The concept of fuzzy sets and fuzzy set operations were first introduced Zadeh [45] 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. In this paper we introduce and examine the concepts of Orlicz space of entire sequence of fuzzy numbers. £ ¤ Let D be the set of all bounded intervals A = A, A on the real line R. For A, © B ∈ D, define ª A ≤ B if and only if A ≤ B and A ≤ B, d (A, B) = max A − B, A − B . Then it can be easily see that d defines a metric D (cf. [8]) and (D, d) is a complete metric space. A fuzzy number is a fuzzy subset of the real line R which is bounded, convex and normal. Let L (R) denote the set of all fuzzy numbers which are upper semi continuous and have compact support, i.e., if X ∈ L (R), then for any α ∈ [0, 1] , X α is compact, where ½ t : X (t) ≥ α, if 0 < α ≤ 1, α X = t : X (t) > 0, if α = 0. For each 0 < α ≤ 1, the α-level set X α is a nonempty compact subset of R. The linear structure of L (R) includes addition X + Y and scalar multiplication α α α α λX (λ : scalar) in terms of α-level sets, by [X + Y ] = [X] +[Y ] and [λX] = α λ [X] for each 0 ≤ α ≤ 1. Define a map d : L (R) × L (R) → R by d (X, Y ) = sup0≤α≤1 d (X α , Y α ). For X, Y ∈ L (R) define X ≤ Y if and only if X α ≤ Y α Received April 30, 2009. 2000 Mathematics Subject Classification. 40A05, 40C05, 40D05. Key words and phrases. fuzzy numbers, Orlicz space, entire sequence, analytic sequence, difference sequence. c °2010 The Korean Mathematical Society

193

194

NAGARAJAN SUBRAMANIAN AND AYHAN ESI

¡ ¢ for any α ∈ [0, 1] . It is shown that L (R) , d is a complete metric space (cf. [26]). A complex sequence, whose k th terms is xk is denoted by {xk } or simply x. Let φ be the set of all finite sequences. Let `∞ , c, c0 be the sequence spaces of bounded, convergent and null sequences x = (xk ) respectively. In sup

respect of `∞ , c, c0 we have kxk = k |xk | , where x = (xk ) ∈ c0 ⊂ c ⊂ `∞ . 1/k A sequence x = {xk } is said to be analytic if supk |xk | < ∞. The vector space of all analytic sequences will be denoted by Λ. A sequence x is called 1/k entire sequence if limk→∞ |xk | = 0. The vector space of all entire sequences will be denoted by Γ. Orlicz [35] used the idea of Orlicz function to construct the space (LM ). Lindenstrauss and Tzafriri [23] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space `M contains a subspace isomorphic to `p (1 ≤ p < ∞). Subsequently different classes of sequence spaces defined by Parashar and Choudhary [36], Mursaleen et al. [29], Bektas and Altin [4], Tripathy et al. [42], Rao and Subramanian [37] and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [20]. Recall ([35], [20]) 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 M is replaced by M (x + y) ≤ M (x) + M (y), then this function is called a modulus function, introduced by Nakano [30] and further discussed by Ruckle [39] and Maddox [25] and many others. An Orlicz function M is said to satisfy ∆2 -condition for all values of u, if there exists a constant K > 0 such that M (2u) ≤ KM (u) (u ≥ 0). The ∆2 condition is equivalent to M (ù) ≤ K`M (u) for all values of u and for ` > 1. Lindenstrauss and Tzafriri [23] used the idea of Orlicz function to construct Orlicz sequence space ( (1)

`M =

x∈w:

∞ X

µ M

k=1

|xk | ρ

)

¶ 0 .

The space `M with the norm ( (2)

kxk = inf

ρ>0:

∞ X k=1

µ M

|xk | ρ

)

¶ ≤1

becomes a Banach space which is called an Orlicz sequence space. For M (t) = tp , 1 ≤ p < ∞, the space `M coincide with the classical sequence space `p . Given a sequence x = {xk }, its nth section is the sequence x(n) = {x1 , x2 , . . . , xn , 0, 0, . . .}, δ (n) = (0, 0, . . . , 1, 0, 0, . . .) , 1 in the nth place and zero’s else where.

THE DIFFERENCE ORLICZ SPACE OF ENTIRE SEQUENCE

195

2. Definitions and prelimiaries ∞

Let w denote the set of all fuzzy complex sequences x = (xk )k=1 , and M be an Orlicz nfucntion, or a modulus ³ ³ function. ´´ consider ´ 1/k ΓM = x ∈ w : limk→∞ M |xkρ| = 0 for some ρ > 0 and n ´ ³ ³ 1/k ´´ ΛM = x ∈ w : supk M |xkρ| < ∞ for some ρ > 0 . The space ΓM and ΛM is a metric space with the metric ( Ã Ã !! ) 1/k |xk − yk | (3) d (x, y) = inf ρ > 0 : sup M ≤1 ρ k for all x = {xk } and y = {yk } in ΓM . In this paper we define Orlicz space of entire sequence of fuzzy numbers by using regular matrices A = (ank ) (nk = 1, 2, 3, . . .) . By the regularity of A we mean that the matrix which transform convergent sequence into a convergent sequence leaving the limit (cf. Maddox [24]); we prove that these spaces are complete paranormed spaces. If E is a linear space over the filed C, then a paranorm on E is a function g : E → R which satisfies the following axioms; for X, Y ∈ E, (P.1) g (θ) = 0, (P.2) g (X) ≥ 0 for all X ∈ E, (P.3) g (−X) = g (X) for all X ∈ E, (P.4) g (X + Y ) ≤ g (X) + g (Y ) for all X, Y ∈ E, (P.5) If (λn ) is a sequence of scalars with λn → λ (n → ∞) and (Xn ) is a sequence of the elements of E with Xn → X imply λn Xn → λX, where λn λ ∈ C and Xn , X ∈ E; In other words |λn − λ| → 0, g (Xn − X) → 0 imply g (λn Xn − λX) → 0 (n → ∞) . A paranormed space is a linear space E with a paranorm g and is written as (E, g) we now give the following new definitions which will be needed in the sequel. Definition 2.1. A sequence X = (Xk ) of fuzzy number is a function X from the set N of natural numbers into L (R) . The fuzzy number Xn denotes the value of the function n ∈ N and is called nth term of the sequence. We denote by W (F ) the set of all sequences X = (Xk ) of fuzzy numbers. Definition 2.2. Let X = (Xk ) be a sequence of fuzzy numbers. Then the sequence X = (Xk ) of fuzzy numbers is said to be³Orlicz of entire sequence ³ space´´ 1/k

of fuzzy numbers convergent to zero, written as M |Xkρ| for some arbitrarily fixed ρ > 0 and is defined by " Ã Ã !! # 1/k |Xk | → 0 as k → ∞ d M ρ

is denoted by ΓM (F ) , with M being a modulus function.

→ 0 as k → ∞,

196


Definition 2.3. Let X = (Xk ) be a sequence of fuzzy numbers. Then the sequence X = (Xk )n³ of fuzzy ³ numbers ´´ is said o to be Orlicz space of analytic 1/k

sequence if the set M |Xkρ| : k ∈ N of fuzzy numbers. We denote ΛM (F ) the set of all analytic sequence of fuzzy number. 3. Orlicz space of entire sequence of fuzzy numbers

In this papern we define the following ´ ípk o 1/k Pn h ³ ³ ΓM [F, p] = X = (Xk ) : n1 k=1 d M |Xkρ| ,0 → 0 as k → ∞ , n ´ ípk o 1/k Pn h ³ ³ ΛM [F, p] = X = (Xk ) : supn n1 k=1 d M |Xkρ| ,0 < ∞ and call them respectively the spaces of sequences of fuzzy numbers which are strongly Orlicz space of entire to zero and strongly analytic. A metric d on L (R) is said to be a translation invariant if d (X + Z, Y + Z) = d (X, Y ) for X, Y, Z ∈ L (R) . In this paper we define and study the difference Orlicz space of entire sequence of fuzzy numbers. The idea of difference sequences for real numbers was first introduced by Kizmaz [19]. 4. Difference Orlicz space of entire sequence of fuzzy numbers Let X = (Xk ) be a sequence of fuzzy numbers. Write ∆X = (Xk − Xk+1 ) (k = 1, 2, 3, . . .) . We define the following spaces of difference sequences of fuzzy numbers: ΓM [F, A, p, ∆] = {X = (Xk ) ∈ w (F ) : ∆X ∈ ΓM (F, A, p)} , ΛM [F, A, p, ∆] = {X = (Xk ) ∈ w (F ) : ∆X ∈ ΛM (F, A, p)} . Note that ΓM (F, A, p) ⊂ ΓM [F, A, p, ∆] and ΛM (F, A, p) ⊂ ΛM [F, A, p, ∆] . We prove the following results: Proposition 4.1. If d is a translation invariant metric on L (R), then (i) d (∆X + ∆Y, 0) ≤ d (∆X, 0) + d (∆Y, 0), (ii) d (λ∆X, 0) ≤ |λ| d (∆X, 0) , |λ| > 1. Theorem 4.2. ΓM (F, p, ∆) are complete metric spaces with the metric is given by " Ã Ã ! !#p 1/n 1 |∆Xn + ∆Yn | ρ (X, Y ) = sup d M ,0 , n ρ n where X = (∆Xn ) and Y = (∆Yn ) are the difference sequences of fuzzy numbers.


197

Proof. It is easy © to show ª that these are metric spaces. We will show the completeness. Let X (m) be a Cauchy sequence in ΓM (F, p, ∆) . Then  ¯ ¯1/n  ¯ (i) ¯ ∆X ¯ n ¯    M   ρ will be a Cauchy sequence in ΓM (F, p, ∆) . Therefore for each n, then  ¯ ¯1/n  ¯ (i) ¯ ¯∆X n ¯    M   ρ is a Cauchy sequence in L (R) . Since L (R) is complete,   ¯ ¯1/n  ¯ (i) ¯ ∆X ¯ ¯  n 1   (4)  M   → 0 as i → ∞ n ρ µ µ ¶¶ 1/n |∆Xn(i) | put ∆X = (∆Xn ), since M is a Cauchy sequence in ΓM (F, p, ∆) ρ there exists n0 ∈ N such that for all i.   ¯ ¯1/n  p ¯ (j) ¯ (i)  1   ¯∆Xn − ∆Xn ¯   (5)  , 0 → 0 as n → ∞.  d M  n ρ Let j → ∞ we get (5) that   ¯ ¯1/n  p ¯ ¯ (i)  1   ¯∆Xn − ∆Xn ¯   (6)  d M   , 0 → 0 as n → ∞. n ρ µ µ ¶¶ 1/n |∆Xn(i) | Therefore M → 0 as i → ∞. Now we have to show that X ∈ ρ (i)

ΓM (F, p, ∆) . Since ∆X (i) ∈ ΓM (F, p, ∆) , there exists ∆X0 ∈ L (R) such that   ¯ ¯1/n  p ¯ (i) ¯ (i) − ∆X ∆X ¯ n 0 ¯ 1     (7)  d M   , 0 → 0 as n → ∞. n ρ Hence Ã Ã 1 nd

Ã +

1 nd

Ã¯ M Ã M

! ¯ ¯ (j) ¯1/n (i) −∆X0 ¯ ¯∆Xn

!!p

1 nd

,0 ≤ !!p ¯1/n !

ρ

Ã¯

¯ (i) ¯ (i) −∆X0 ¯ ¯∆Xn ρ

,0

Ã

Ã Ã +

1 nd

Ã¯

M Ã M

! ¯ ¯ (j) ¯1/n (i) −∆X0 ¯ ¯∆Xn

!!p

,0 !!p ¯1/n !

ρ

Ã¯

¯ (j) ¯ (i) −∆X0 ¯ ¯∆Xn ρ

,0

198


→ 0 as n → ∞ by (5) and (7). Thus

¶ ¶ µ µ 1/n |∆Xn(i) | M , 0 is a Cauchy ρ

sequence in L(R). Since L(R) is complete, there exists ∆X0 ∈ L(R) such that   ¯ ¯1/n  p ¯ ¯ (i)  1   ¯∆X0 − ∆X0 ¯   (8)  d M   , 0 → 0 as n → ∞. n ρ Therefore by (6), (7) and (8) we have ´ ´´p µ µ µ ∆X (i) −∆X 1/n ¶ ¶¶p ³ ³ ³ | n n| |∆Xn −∆X0 |1/n 1 ,0 ≤ n1 d M ,0 + nd M ρ ρ Ã Ã Ã¯ !!p Ã Ã Ã ¯ !!p ¯1/n ! ¯1/n ! 1 nd

M

¯ (i) ¯ (i) −∆X0 ¯ ¯∆Xn ρ

,0

+

1 nd

M

¯ ¯ (i) ¯∆X0 −∆X0 ¯ ρ

,0

→

0 as n → ∞. This implies that X = (Xk ) ∈ ΓM (F, p, ∆) . Therefore ΓM (F, p, ∆) is complete. This completes the proof. ¤ Theorem 4.3. If d is a transition invariant metric and M is a modulus function, then ΓM (F, p, ∆) are linear spaces over the complex numbers C. Proof. It is easy. Therefore omit the proof.

¤

5. Main results Theorem 5.1. ΓM (F, A, p, ∆) and ΛM (F, A, p, ∆) (inf pk > 0) are complete with respect to the topology generated by the paranorm h is defined by (9) ¯p k Ã " Ã Ã ! !#pk ! ¯ 1/k ¯X ¯ X |∆Xk | ¯ ¯ h (X) = sup ank d M ,0 ,¯ ank ¯ → 0 as k → ∞, ¯ ¯ ρ n k

k

where d is a translation invariant and X = (Xk ) be a sequence of fuzzy numbers. ¡ ¢ Proof. Let X (s) be a Cauchy sequence in ΓM (F, A, p, ∆) . Then ! !# " Ã Ã 1/k |∆X s − ∆X t | ,0 → 0 as s, t → ∞, d M ρ that is (10)    ¯ ¯1/k   ¯ (s) (t) ¯ − ∆X k ¯ X    ¯∆Xk   ank d M    , 0 → 0 as s, t → ∞ for all k. ρ k

Hence

 

¯ ¯1/k   ¯ (s) (t) ¯ − ∆X ¯∆X ¯   k k    d M   , 0 → 0 as s, t → ∞ for all k, ρ


" Ã which implies that d M

Ã¯

! ¯ ¯ (s) ¯1/k ¯∆Xk ¯ ρ

199

!# ,0

is a Cauchy sequence C for each " Ã Ã¯ !# ¯1/k !

k and so there exists Y = (∆Yk ) such that d M

¯ (s) ¯ ¯∆Xk ¯ ρ

,0

→ ∆Yk

as s → ∞ for each k. Now, from (10) we have, for ² > 0, there exists natural number N such that    ¯ ¯1/k   ¯ (s) (t) ¯ X    ¯∆Xk − ∆Xk ¯   (11) ank d M   , 0 < ²  ρ k

for s, t > N and for all n. Hence, for any fixed natural numbers m, we have from (11) (12)    ¯ ¯1/k   ¯ (s) (t) ¯ ∆X − ∆Y ¯ X k k ¯       ank d M    , 0 < ² for s, t > N for all n. ρ k≤m

Now fix s > N and let t → ∞. Then from (11) we have    ¯ ¯1/k   ¯ ¯ (s) ∆X − ∆Y ¯ ¯   X k k     ank d M    , 0 < ² for s > N for all n. ρ k≤m

Since this is valid for any natural number m, we have   ¯  ¯1/k   ¯ ¯ (s) X    ¯∆Xk − ∆Yk ¯   ank d M    , 0 < ² for s > N for all n, ρ k≤m

that is

" Ã d M

Ã¯ ¯ ! ¯∆X (s) − ∆Y ¯1/k ρ

!# ,0

→ 0 as s → ∞,

· µ µ ¶ ¶¸ 1/k |∆X (s) | and thus d M ,0 → ∆Y as s → ∞, and thus ρ " Ã d M

Ã¯ ¯ ! ¯∆X (s) − ∆Y ¯1/k ρ

!# ,0

∈ ΓM (F, A, p, ∆) ,

· µ µ ¶ ¶¸ 1/k 1/k |∆Y −∆X (s) | +|∆X (s) | also writing ∆Y = d M , 0 , we have by linρ earity of ΓM (F, A, p, ∆) ∆Y ∈ ΓM (F, A, p, ∆) ; hence ΓM (F, A, p, ∆) is complete. The completeness of ΛM (F, A, p, ∆) can be similarly obtained. ¤

200


Theorem 5.2. Let X = (Xk ) be a sequence of fuzzy numbers and d be a translation invariant. Let A = (ank ) (n, k = 1, 2, 3, . . .) be an infinite matrix with complex entries. Then A ∈ (ΓM (F, A, p) : ΓM (F, A, p, ∆)) if and only if given ² > 0 there exists M = M (²) > 0 such that |ank − an+1,k | < ²n M k (n, k = 1, 2, 3, . . .) . Proof. Let X = (Xk ) ∈ Γ (F, A, p) and let Ã∞ Ã Ã ! !p ! 1/k X |Xk | Yn = ,0 (n = 1, 2, 3, . . .) , ank d M ρ k=1

´ ´p ´ ³ ³ ³ P∞ |Xk |1/k so that ∆Yn = ,0 . Then (∆Yn ) ∈ k=1 (ank − an+1,k ) d M ρ ΓM (F, A, p) if and only if given any ² > 0 there exists M = M (²) > 0 such that |ank − an+1,k | < ²n M k by using Theorem 4 of [38]. Now (∆Yn ) ∈ ΓM (F, A, p) if and only if (Yn ) ∈ ΓM (F, A, p, ∆) . Thus A ∈ (ΓM (F, A, p) : ΓM (F, A, p, ∆)) if and only if the condition holds. This completes the proof. ¤ Theorem 5.3. Let X = (Xk ) be a sequence of fuzzy numbers and d be a translation invariant. Let A = (ank ) transform ΓM (F, A, p) into ΓM (F, A, p, ∆). Then limn→∞ (ank − an+1,k ) q n = 0 for all integers q > 0 and each fixed k = 1, 2, 3, . . .. ´ ´p ´ ³P ³ ³ ∞ |Xk |1/k Proof. Let Yn = ,0 (n = 1, 2, 3, . . .) formally. k=1 ank d M ρ Let (Xk ) ∈ ΓM (F, A, p) and (Yn ) ∈ ΓM (F, A, p, ∆) . But then (∆Yn ) ∈ ΓM (F, A, p) , Ã∞ Ã Ã ! !p ! 1/k X |Xk | (ank − an+1,k ) d M ,0 (n = 1, 2, 3, . . .) . ∆Yn = ρ k=1

Take (Xk ) = δ k = (0, 0, 0, . . . , 1, 0, 0, . . .) , 1 in the k th place and zero’s elsewhere. Then (Xk ) ∈ P ΓM (F, A, p) . We have ∆Yn = ank − an+1,k . But (∆Yn ) ∈ ∞ ΓM (F, A, p) . Hence k=1 |ank − an+1,k | q n < ∞ for every positive integer q. In particular limn→∞ (ank − an+1,k ) q n = 0 for all integers q and each fixed k = 1, 2, 3, . . . . This completes the proof. ¤ Theorem 5.4. Let X = (Xk ) be a sequence of fuzzy numbers and d be a translation invariant. Let A = (ank ) transform ΓM (F, A, p, ∆) into ΓM (F, A, p) . Then limn→∞ ank q n = 0 for all positive integers q. Proof. Let

Ã tn =

∞ X k=1

Ã ank d M

Ã

1/k

|Xk | ρ

!

!p ! ,0


201

with (Xk ) ∈ ΓM (F, A, p, ∆), (tn ) ∈ ΓM (F, A, p) and Ã∞ Ã Ã ! !p ! 1/k X |Xk | sn = ank d M ,0 , ρ k=1

(sn ) ∈ ΓM (F, A, p) . Then

Ã

Yn = (tn − sn ) = =

∞ X k=1

Ã ank d M

∞ X

Ã

1/k

|∆Xk | ρ

1/k

|Xk − Xk+1 | ρ !p

ank d M

k=1

Ã

Ã !

!

!p ! ,0

,0

and (∆Xk ) ∈ ΓM (F, A, p) and (Yn ) ∈ ΓM (F, A, p) . Hence (ank ) q n → 0 as n → ∞ for all k, by [13]. This completes the proof. ¤ Theorem 5.5. Let X = (Xk ) be a sequence of fuzzy numbers and d be a translation invariant. If A = (ank ) transforms ΓM (F, A, p, ∆) into ΓM (F, A, p, ∆), then ank q n → 0 and an+1,k q n → 0 as n → ∞. Proof. From Theorems 5.2 and 5.3 we have ank q n → 0 and (ank − an+1,k ) q n as n → ∞ for all positive integers q and for all k ⇒ ank q n → 0 and (ank q n − an+1,k q n ) → 0 ⇒ (an+1,k ) q n → 0 and (ank ) q n → 0 as n → ∞ for all k. This completes the proof. ¤ 6. ∆-statistical convergence The idea of statistical convergence of fuzzy numbers was introduced by Nuray and Savas P [34]. The generalized de la Valée-Pousin mean is defined by tn (X) = λ1n k∈In Xk , where λ = (λn ) is a non-decreasing sequence of positive numbers such that λn+1 ≤ λn + 1, λ1 = 1, λn → ∞ as n → ∞ and In = [n − λn + 1, n] . In this section we define this concept for the sequences. Definition 6.1. A sequence X = (Xk ) of fuzzy numbers is said to be ∆statistial convergent to fuzzy number zero if ¯( Ã Ã ! ! )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ lim , 0 ≥ ² ¯ = 0, ¯ k≤n:d M n→∞ n ¯ ¯ ρ ³ ³ ³ ´ ´´ 1/k i.e., d M |∆Xρk | , 0 < ². In this case we write Ã Ã !! 1/k |Xk | St (∆) − lim M = 0. k→∞ ρ Definition 6.2. Let X = (Xk ) be a sequence of fuzzy numbers and p = (pk ) be a sequence of strictly positive real numbers. Then the sequence X = (Xk )

202


is said to be strongly is a fuzzy number zero such that h ³ ∆-convergent ³ ´ ifíthere pk P |∆Xk |1/k 1 limn→∞ n k≤n d M ,0 = 0. ρ Definition 6.3. Let X = (Xk ) be a sequence of fuzzy numbers. Then the sequence n X =³(Xk ) of fuzzy numbers is said to be ∆-Orlicz space of analytic ó |∆xk |1/k if the set M :k∈N of fuzzy numbers is Orlicz space of analytic. ρ By ΛM (∆) we shall denote the set of all ∆-Orlicz space of analaytic sequences of fuzzy numbers. Theorem 6.4. If ³ (X³ St´´´ (∆) and c ∈ L (R), then k ) , (Y ³ k ) ∈1/k ³ ³ ´´ 1/k |Xk | (i) St (∆) − lim c M = cSt (∆) − lim M |Xkρ| , ρ ³ ³ ´´ ³ ³ ´´ |Xk +Yk |1/k |Xk |1/k (ii) St (∆)−lim M = St (∆)−lim M +St (∆)− ρ ρ ³ ³ ´´ 1/k lim M |Ykρ| , where d is a translation invariant. ³ ³ ´´ 1/k Proof. (i) Let St (∆) − lim M |Xkρ| c ∈ L (R) and ² > 0 be given. Then the proof follows from the following inequality ¯( Ã Ã ! ! )¯ 1/k ¯ 1 ¯¯ |c∆Xk | ¯ ,0 ≥ ² ¯ ¯ k≤n:d M ¯ ¯ n ρ ¯( ! ! )¯ Ã Ã 1/k ² ¯¯ |∆Xk | 1 ¯¯ ,0 ≥ ≤ ¯ k≤n:d M ¯. n¯ ρ |c| ¯ (ii) Suppose that Ã Ã St (∆) − lim M

|Xk | ρ

1/k

!!

Ã = 0 and St (∆) − lim M

Ã

1/k

|Yk | ρ

By Minkowski’s inequality we get Ã Ã ! ! 1/k |∆Xk + ∆Yk | ,0 d M ρ Ã Ã ! ! Ã Ã ! ! 1/k 1/k |∆Xk | |∆Yk | =d M ,0 + d M ,0 . ρ ρ Therefore given ² > 0 we have ¯( Ã Ã ! ! )¯ 1/k ¯ 1 ¯¯ |∆Xk + ∆Yk | ¯ ,0 ≥ ² ¯ ¯ k≤n:d M ¯ ¯ n ρ ¯( ! ! )¯ Ã Ã 1/k ² ¯¯ |∆Xk | 1 ¯¯ ,0 ≥ ≤ ¯ k≤n:d M ¯ n¯ ρ 2 ¯ ¯( Ã Ã ! ! )¯ 1/k 1 ¯¯ |∆Yk | ² ¯¯ + ¯ k≤n:d M ,0 ≥ ¯. n¯ ρ 2 ¯

!! = 0.


´´ ³ ³ 1/k k| Hence St (∆) − lim M |Xk +Y = 0. This completes the proof. ρ

203

¤

Theorem 6.5. If a sequence X = (Xk ) is ∆-statistically convergent to the fuzzy number zero and lim inf (n) (λn /n) > 0, then it is ∆-statistically convergent to zero. Proof. Given ² > 0 we have ¯( Ã Ã ! ! )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ ,0 ≥ ² ¯ ¯ k≤n:d M ¯ n¯ ρ ¯( ! ! )¯ Ã Ã 1/k ¯ 1 ¯¯ |∆Xk | ¯ ,0 ≥ ² ¯. ⊃ ¯ k ∈ In : d M ¯ n¯ ρ Therefore,

¯( ! ! )¯ Ã Ã 1/k ¯ 1 ¯¯ |∆Xk | ¯ ,0 ≥ ² ¯ ¯ k≤n:d M ¯ n¯ ρ ¯( Ã Ã ! ! )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ ≥ ¯ k ∈ In : d M ,0 ≥ ² ¯ ¯ n¯ ρ ¯( Ã Ã ! ! )¯ 1/k ¯ |∆Xk | λn 1 ¯¯ ¯ ,0 ≥ ² ¯. ≥ ¯ k ∈ In : d M ¯ n λn ¯ ρ

Taking limit as n → ∞ and using lim inf (n) (λn /n) > 0, we get X is ∆statistically convergent to zero. This completes the proof. ¤ n o Theorem 6.6. Let 0 ≤ pk ≤ qk and let pqkk be bounded. Then ΓM (F, q, ∆) ⊂ ΓM (F, p, ∆). Proof. Let (13)

X ∈ ΓM (F, q, ∆)

and given ² > 0 ¯( ! ! qk Ã Ã )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ (14) ,0 ≥ ² ¯ → 0 as n → ∞. ¯ k≤n:d M ¯ n¯ ρ ³ ³ ´ ´ qk 1/k Let tk = d M |∆Xρk | ,0 and λk = pqkk . Since pk ≤ qk , we have 0 ≤ λk ≤ 1. Take 0 < λ < λk . Define uk = tk (tk ≥ 1) ; uk = 0 (tk < 1) and vk = 0 (tk ≥ 1) ; vk = tk (tk < 1) , tk = uk + vk , i.e., tλk k = uλk k + vkλk . Now it follows that (15)

uλk k ≤ uk ≤ tk and vkλk ≤ vkλ .

204


Since tλk k = uλk k + vkλk , then tλk k ≤ tk + vkλ , ¯ ¯ ! !qk !λk Ã Ã Ã ¯ 1/k ¯¯ ¯ 1¯ |∆Xk | k≤n: d M ,0 ≥ ² ¯¯ ¯ n ¯¯ ρ ¯( Ã Ã Ã ! !!qk )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ ,0 ≥² ¯ ≤ ¯ k≤n: d M ¯ n¯ ρ ¯ ¯ Ã Ã Ã ! ! ! p /q q k k ¯ k 1/k ¯¯ ¯ 1¯ |∆Xk | ¯ k ≤ n : ⇒ d M , 0 ≥ ² ¯¯ n ¯¯ ρ ¯( Ã Ã Ã ! !!qk )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ ≤ ¯ k≤n: d M ,0 ≥² ¯ ¯ n¯ ρ ¯( Ã Ã Ã ! !!pk )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ ⇒ ,0 ≥² ¯ ¯ k≤n: d M ¯ n¯ ρ ¯( ! !!qk Ã Ã Ã )¯ 1/k ¯ 1 ¯¯ |∆Xk | ¯ ≤ ¯ k≤n: d M ,0 ≥ ² ¯. ¯ n¯ ρ ¯n ´ ´´qk o¯ ³ ³ ³ 1/k ¯ ¯ But n1 ¯ k ≤ n : d M |∆Xρk | ,0 ≥ ² ¯ → 0 as n → ∞ by (14). There¯n ´ ´´ o¯ ³ ³ ³ pk 1/k ¯ ¯ ,0 ≥ ² ¯ → 0 as n → ∞. Hence fore n1 ¯ k ≤ n : d M |∆Xρk | (16)

X ∈ ΓM (F, p, ∆) .

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