On Statistical Summability (N,P) of Sequences of Fuzzy Numbers

Filomat 30:3 (2016), 873–884 DOI 10.2298/FIL1603873T

Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat

On Statistical Summability (N, P) of Sequences of Fuzzy Numbers ¨ Ozer Taloa , Canan Balb a Department

of Mathematics, Faculty of Art and Sciences, Celal Bayar University, 45040 Manisa, Turkey Central Vocational and Technical Anatolian High School, Manisa, Turkey

b Private

Abstract. In this paper we introduce the concept of statistical summability (N, p) of sequences of fuzzy numbers. We also present Tauberian conditions under which statistical convergence of a sequence of fuzzy numbers follows from its statistical summability (N, p). Furthermore, we prove a Korovkin-type approximation theorem for fuzzy positive linear operators by using the notion of statistical summability (N, p).

1. Introduction Following the introduction of sequences of fuzzy numbers by Matloka [15], summability of sequences of fuzzy numbers is studied by many researcher and bacame a recent research area in fuzzy set theory. Several summablity methods have been defined for sequence of fuzzy numbers and Tauberian theorems are given for these methods. Ces`aro summability method is defined by Subrahmanyam[23] and various Tauberian conditions for Ces`aro summability method are given by Talo and C ¸ akan [24], Talo and F. Bas¸ar [26], C ¸ anak [14]. Norlund and Riesz mean of sequences of fuzzy numbers are studied by Tripathy and ¨ ¨ Baruah [27], C ¸ anak [13], Onder et al. [20]. Furthermore, various power series methods of summability for sequences and series of fuzzy numbers are studied by Yavuz and Talo [28], Sezer and C ¸ anak [22], Yavuz and C ¸ os¸kun [29]. The concept of statistical convergence of sequences of real numbers was originally introduced by Fast [11], and extended to sequences of fuzzy numbers by Nuray and Savas¸ [19]. Altn et. al. [1] have studied the concept of statistical summability (C; 1) for sequences of fuzzy numbers. Talo and C ¸ akan [25] have recently proved necessary and sufficient Tauberian conditions under which statistical convergence follows from statistically (C, 1)-convergence of sequences of fuzzy numbers. In the present paper our primary interest is to generalize the results in [1, 25] to a large class of summability methods (N, p) by weighted means. We also obtain a Korovkin-type approximation theorem for fuzzy positive linear operator by means of the concept of statistical summability (N, p).

2010 Mathematics Subject Classification. Primary 03E72; Secondary 40A35, 40G15 Keywords. Sequences of fuzzy numbers, summability by weigted mean, Tauberian conditions, statistical convergence, fuzzy Korovkin theory Received: 30 August 2015; Revised 01 December 2015; Accepted: 02 December 2015 Communicated by Ljubiˇsa D.R. Koˇcinac and Ekrem Savas¸ ¨ Email addresses: [email protected], [email protected] (Ozer Talo), [email protected] (Canan Bal)

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2. Definitions and Notation Let K be a subset of natural numbers N and Kn = {k ≤ n : k ∈ K}, The natural density of K is given by δ(K) = lim

n→∞

1 |Kn | n+1

if this limit exists, where |A| denotes the number of elements in A. The concept of statistical convergence was introduced by Fast [11]. A sequence (xk )k∈N of real (or complex) numbers is said to be statistically convergent to some number l if for every ε > 0 we have lim

n→∞

1 |{k ≤ n : |xk − l| ≥ ε}| = 0. n+1

In this case, we write st− limk→∞ xk = l. Basic results on statistical convergence may be found in [9, 10, 16] We recall the basic definitions dealing with fuzzy numbers. A fuzzy number is a fuzzy set on the real axis, i.e. a mapping u : R → [0, 1] which satisfies the following four conditions: (i) u is normal, i.e. there exists an x0 ∈ R such that u(x0 ) = 1; (ii) u is fuzzy convex, i.e. u[λx + (1 − λ)y] ≥ min{u(x), u(y)} for all x, y ∈ R and for all λ ∈ [0, 1]; (iii) u is upper semi-continuous; (iv) The set [u]0 := {x ∈ R : u(x) > 0} is compact, where {x ∈ R : u(x) > 0} denotes the closure of the set {x ∈ R : u(x) > 0} in the usual topology of R. We denote the set of all fuzzy numbers on R by E1 and called it as the space of fuzzy numbers. α-level set [u]α of u ∈ E1 is defined by    {x ∈ R : u(x) ≥ α} , if 0 < α ≤ 1, [u]α :=   {t ∈ R : u(x) > α} , if α = 0. The set [u]α is closed, bounded and non-empty interval for each α ∈ [0, 1] which is defined by [u]α := [u− (α), u+ (α)]. R can be embedded in E1 , since each r ∈ R can be regarded as a fuzzy number r defined by ( 1 , if x = r, r(x) := 0 , if x , r. Let u, v, w ∈ E1 and k ∈ R. Then the operations addition and scalar multiplication are defined on E1 by u + v = w ⇐⇒ [w]α = [u]α + [v]α for all α ∈ [0, 1] ⇐⇒ w− (α) = u− (α) + v− (α) and w+ (α) = u+ (α) + v+ (α) for all α ∈ [0, 1], [ku]α = k[u]α for all α ∈ [0, 1]. The operations addition and scalar multiplication on fuzzy numbers have the following properties. Lemma 2.1. ([8]) (i) If 0 ∈ E1 is neutral element with respect to +,i.e u + 0 = 0 + u = u, for all u ∈ E1 ; (ii) With respect to 0, none of u , r, r ∈ R has opposite in E1 ; (iii) For any a, b ∈ R with a, b ≥ 0 or a, b ≤ 0, and any u ∈ E1 , we have (a + b)u = au + bu. For general a, b ∈ R, the above property does not hold; (iv) For any a ∈ R and any u, v ∈ E1 , we have a(u + v) = au + av; (v) For any a, b ∈ R and any u ∈ E1 , we have a(bu) = (ab)u.

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Properties (ii) and (iii) show us that (E1 , +, ·) is not a linear space over R. Let W be the set of all closed bounded intervals A of real numbers with endpoints A and A, i.e. A := [A, A]. Define the relation d on W by d(A, B) := max{|A − B|, |A − B|}. Then it can be easily observed that d is a metric on W and (W, d) is a complete metric space, (see Nanda [18]). Now, we may define the metric D on E1 by means of the Hausdorff metric d as follows D(u, v) := sup d([u]α , [v]α ) := sup max{|u− (α) − v− (α)|, |u+ (α) − v+ (α)|}. α∈[0,1]

α∈[0,1]

One can see that D(u, 0) = sup max{|u− (α)|, |u+ (α)|} = max{|u− (0)|, |u+ (0)|}.

(1)

α∈[0,1]

Now, we may give: Proposition 2.2. ([8]) Let u, v, w, z ∈ E1 and k ∈ R. Then, (i) (E1 , D) is a complete metric space; (ii) D(ku, kv) = |k|D(u, v); (iii) D(u + v, w + v) = D(u, w); (iv) D(u + v, w + z) ≤ D(u, w) + D(v, z); (v) |D(u, 0) − D(v, 0)| ≤ D(u, v) ≤ D(u, 0) + D(v, 0). One can extend the natural order relation on the real line to intervals as follows: A  B if and only if A ≤ B and A ≤ B. Also, the partial ordering relation on E1 is defined as follows: u  v ⇐⇒ [u]α  [v]α for all α ∈ [0, 1] ⇐⇒ u− (α) ≤ v− (α) and u+ (α) ≤ v+ (α) for all α ∈ [0, 1]. Following Matloka [15], we give some definitions concerning the sequences of fuzzy numbers. A sequence u = (uk ) of fuzzy numbers is said to be convergent to µ ∈ E1 , if for every ε > 0 there exists an n0 = n0 (ε) ∈ N such that D(un , µ) < ε for all n ≥ n0 . A sequence (un ) of fuzzy numbers is said to be bounded if there exists M > 0 such that D(un , 0) ≤ M for all n ∈ N. Statistical convergence of sequences of fuzzy numbers was introduced by Nuray and Savas¸ [19]. A sequence (uk : k = 0, 1, 2, . . .) of fuzzy numbers is said to be statistically convergent to a fuzzy number µ0 if for every ε > 0 we have 1 {k ≤ n : D(uk , µ0 ) ≥ ε} = 0. lim n→∞ n + 1 In this case we write st− lim uk = µ0 . k→∞

For more results on statistical convergence of sequences of fuzzy numbers we refer to [4–7, 12, 21].

(2)

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3. Main Results Let p = (pk : k = 0, 1, 2, . . . ) be a sequence of nonnegative numbers such that p0 > 0 and Pn :=

n X

pk → ∞ (n → ∞)

k=0

and set tn :=

n 1 X pk uk , Pn

n = 0, 1, 2, . . .

k=0

We say that the sequence (uk ) of fuzzy numbers is statistically summable to fuzzy number µ0 by the weighted mean method determined by the sequence p = (pk ), briefly: statistically summable (N, p) if st− lim tn = µ0 .

(3)

n→∞

If pn = 1 for all n in (3), we have n

1X uk = µ0 . n→∞ n

(4)

st− lim

k=0

Then we say that the sequence (un ) of fuzzy numbers is statistically summable (C, 1) to µ0 . Example 3.1. Let (uk ) = (u0 , v0 , u0 , v0 , ...) where   t − 1 , if 1 ≤ t ≤ 2,    −t + 3 , if 2 ≤ t ≤ 3, u0 (t) =     0 , otherwise and v0 (t) =

  t+1    −t +1     0

, if − 1 ≤ t ≤ 0, , if 0 ≤ t ≤ 1, , otherwise.

The sequence (un ) is statistically summable (C, 1) to w0 = (u0 + v0 )/2. But (un ) is not statistically summable (N, 2n ). We claim that if a sequence (uk ) of fuzzy numbers is bounded, then st− lim uk = µ0 . implies k→∞

st− lim tn = µ0 . n→∞

In fact, D tn , µ0




= =

  n  1 X  D  pk uk , µ0  Pn k=0   n n X  1 X  1 D  pk uk , pk µ0  Pn Pn k=0

≤

n 1 X pk D(uk , µ0 ) Pn k=0

k=0

(5)

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Since the real sequence of (D(uk , µ0 )) is bounded and st− lim D(uk , µ0 ) = 0, we have n→∞

n 1 X st− lim pk D(uk , µ0 ) = 0. n→∞ Pn k=0

Thus, we have st− lim D tn , µ0 = 0. This means that st− lim tn = µ0 .


n→∞

n→∞

The converse implication in (5) is not true in general, even in real case (see [17]). Our main goal is to find conditions under which st− lim tn = µ0 n→∞

implies

st− lim uk = µ0 . k→∞

(6)

We give two-sided Tauberian conditions, each of which is necessary and sufficient in order that statistical convergence follow from statistical summability (N, p). Throughout this paper, λn denotes the integral part of the product λn; i.e., λn := [λn]. The concepts of statistical limit inferior and superior of a sequence of real numbers were introduced by Fridy and Orhan [10] and the following lemma was proved by Moricz and Orhan [17]. We use it for the ´ proof of our results. Lemma 3.2. ([17, Lemma1]) If (Pn ) is a nondecreasing sequence of positive numbers, then conditions st− lim inf

Pλn >1 Pn

f or every

λ>1

(7)

st− lim inf

Pn >1 Pλn

f or every

0 < λ < 1.

(8)

n→∞

and n→∞

are equivalent. We need the following lemmas. Lemma 3.3. Let p = (pk ) be a sequence of nonnegative numbers such that p0 > 0 and condition (7) is satisfied, and let (uk ) be a sequence of fuzzy numbers which is statistically summable (N, p) to a fuzzy number µ0 . Then for every λ > 0, st− lim tλn = µ0 .

(9)

n→∞

The proof can be carried out in the same way as in the proof of Lemma 2 in [17]. Lemma 3.4. Let p = (pk ) be a sequence of nonnegative numbers such that p0 > 0 and condition (7) is satisfied, and let (uk ) be a sequence of fuzzy numbers which is statistically summable (N, p) to a fuzzy number µ0 . Then, for every λ > 1, λn X 1 st− lim pk uk = µ0 n→∞ Pλn − Pn

(10)

k=n+1

and for every 0 < λ < 1, n X 1 st− lim pk uk = µ0 . n→∞ Pn − Pλn k=λn +1

(11)

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Proof. Case λ > 1. If Pλn > Pn , then   λn X   1 D  pk uk , µ0  = Pλn − Pn

878

  λn X   1 pk uk + tn , µ0 + tn  D  Pλn − Pn k=n+1   λn X   1 D  pk uk , tn  + D(tn , µ0 ) Pλn − Pn

k=n+1

≤

k=n+1

and   λn X   1  pk uk , tn  D  Pλn − Pn

=

k=n+1

= = = =

  λn n X   1 X 1  pk uk  pk uk , D  Pλn − Pn Pn k=0 k=n+1   λ n n n n X X X   1 1 1 X 1  D  pk uk + pk uk , pk uk  pk uk + Pλn − Pn Pλn − Pn Pn Pλn − Pn k=0 k=0 k=0 k=n+1   λ n n X   Pλn 1 X 1 pk uk  pk uk , D  Pλn − Pn Pλn − Pn Pn k=0 k=0   λ n n  Pλn  Pλn 1 X 1 X  pk uk , D  pk uk  Pλn − Pn Pλn Pλn − Pn Pn k=0 k=0   λ n n  1 X  Pλn 1 X pk uk , D  pk uk  Pλ − Pn Pλ Pn n

=

n

k=0

k=0

Pλn
D tλn , tn . Pλn − Pn

So, we have the following inequality   λn X   Pλn
1 pk uk , µ0  ≤ D  D tλn , tn + D(tn , µ0 ). Pλn − Pn Pλn − Pn

(12)

k=n+1

By (7) we have st− lim sup n→∞

( )−1 Pλn Pλ = 1 + −1 + st− lim inf n < ∞. n→∞ Pn Pλn − Pn

Therefore (10) follows from (12), (13), Lemma 3.3 and the statistical convergence of (tn ). Case 0 < λ < 1. This time, we make use of the following inequality:   n X   Pλn
1  D  pk uk , µ0  ≤ D tλn , tn + D(tn , µ0 ) Pn − Pλn Pn − Pλn

(13)

(14)

k=λn +1

provided Pn > Pλn . By Lemma 3.2 we obtain ( )−1 Pλn Pn = −1 + st− lim inf st− lim sup < ∞. n→∞ Pλn Pn − Pλn n→∞ Therefore (11) follows from (14), (15), Lemma 3.3 and the statistical convergence of (tn ). Now we are ready to give our main results.

(15)

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Theorem 3.5. Let p = (pk ) be a sequence of nonnegative numbers such that p0 > 0 and condition (7) is satisfied, and let (uk ) be a sequence of fuzzy numbers which is statistically summable (N, p) to a fuzzy number µ0 .Then (uk ) is statistically convergent to µ0 if and only if one of the following two conditions holds: for every ε > 0,     λn  X    1  1     ≥ ε  =0 inf lim sup n ≤ N : D p u , u (16)  n   k k   Pλ − Pn    λ>1 N→∞ N + 1  n k=n+1 or     n  X     1  1      = 0. n ≤ N : D  pk uk , un  ≥ ε inf lim sup    0 0 we have    )  λn  X   ε    Pλn
ε 1     D tλn , tn ≥ ∪ n ≤ N : D  pk uk , un  ≥  . {n ≤ N : D(tn , un ) ≥ ε} ⊆ n ≤ N :    Pλn − Pn 2 Pλn − Pn 2 (

k=n+1

Given any δ > 0 , by (16) there exists some λ > 1 such that     λn  X    ε 1  1      ≥    ≤ δ. lim sup p u , u n ≤ N : D n   k k    2  Pλ − Pn  N→∞ N + 1  n k=n+1

(18)

On the other hand, by Lemma 3.3, we have ( ) Pλn
ε 1 lim sup D t λn , t n ≥ n≤N: = 0. Pλn − Pn 2 N→∞ N + 1

(19)

Combining (18) with (19) we get that lim sup N→∞

1 |{n ≤ N : D (un , tn ) ≥ ε}| ≤ δ. N+1

Since δ > 0 is arbitrary, we conclude that for every ε > 0, 1 |{n ≤ N : D (un , tn ) ≥ ε}| = 0. N+1 Secondly, we consider the case 0 < λ < 1. Since   n X   Pλn
1 D(tn , un ) ≤ D  pk uk , un  + D tλn , tn Pn − Pλn Pn − Pλn lim

N→∞

k=λn +1

for any ε > 0, we have    )  n  X   ε    Pλn
ε 1   {n ≤ N : D(tn , un ) ≥ ε} ⊆ n ≤ N : D tλn , tn ≥ ∪ n ≤ N : D  pk uk , un  ≥  .    Pn − Pλn 2 Pn − Pλn 2 (

k=λn +1

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Given any δ > 0 , by (17) there exist some 0 < λ < 1 such that     n X   ε   1  1     ≥   lim sup n ≤ N : D p u , u n   k k   ≤ δ.  Pn − Pλ  2  N→∞ N + 1  n k=λn +1 Using a similar argument as in the case λ > 1, by Lemma 3.3 and condition (11), we conclude that 1 |{n ≤ N : D (un , tn ) ≥ ε}| = 0. N+1 This completes the proof. lim

N→∞

Following Talo and C ¸ akan [25], a sequence (uk ) of fuzzy numbers is said to be statistically slowly oscillating if for every ε > 0, ( ) 1 (20) inf lim sup n ≤ N : max D(uk , un ) ≥ ε = 0 λ>1 N→∞ N + 1 n 0, that there exists a number δ > 0 such that | fα± (y) − fα± (x) |< ε holds for every y ∈ [a, b] satisfying |y − x| < δ. Then we immediately get, for all y ∈ [a, b], that | fα± (y) − fα± (x)| ≤ ε + 2M

(y − x)2 , δ2

(25)

where M := supx∈[a,b] D( f (x), 0). Now using the linearity and the positivity of the operators e Ln ,we have , for each n ∈ N , that n n n X   1 X 1 X
1 ± ± ± ± pke pke pke Lk fα ; x − fα (x) ≤ Lk fα (y) − fα (x) ; x + M Lk (e0 ; x) − e0 (x) Pn Pn k=0 Pn k=0 k=0 n n 1 X 2M 1 X 2 pke Lk (e0 ; x) − e0 (x) + 2 pke Lk ((y − x) ; x) ≤ ε + (ε + M) δ Pn Pn k=0 k=0 which yields n 1 X ± ± pke Lk ( fα ; x) − fα (x) Pn k=0

≤

! n 2c2 M 1 X e ε+ ε+M+ pk Lk (e0 ; x) − e0 (x) 2 δ Pn k=0 n n 2M 1 X 4cM 1 X e + 2 pk Lk (e1 ; x) − e1 (x) + 2 pke Lk (e2 ; x) − e2 (x) , δ Pn δ Pn k=0

k=0

¨ Talo, C. Bal / Filomat 30:3 (2016), 873–884 O.

n where c := max{|a|, |b|}. Also letting K(ε) := max ε + M + the above inequallity implies that



X n



1 ± ± e p L ( f ; x) − f (x) k k α α

Pn k=0

≤

2c2 M 4cM 2M , δ2 , δ2 δ2

o

882

and taking supremum over x ∈ [a, b],

n 

X

1

ε + K(ε)

pke Lk (e0 ) − e0

Pn

(26)

k=0

X



X

 n n

1



1

e e + pk Lk (e1 ) − e1 + pk Lk (e2 ) − e2

. Pn Pn k=0

It follows from (22) that  n  1 X D∗  pk Lk ( f ), Pn k=0

  f 

= = =

k=0

  n   1 X pk Lk ( f ; x), f (x) sup D  Pn x∈[a,b] k=0   n n  1 X      1 X e − − + + e  , p L ( f ; x) − f (x) p L ( f ; x) − f (x) sup sup max  k k k k α α α α  Pn Pn    x∈[a,b] α∈[0,1] k=0 k=0



 

n n  1 X



1 X

     − − e

sup max  pk Lk ( fα ) − fα , pke Lk ( fα+ ) − fα+

 .    P P

n k=0

n k=0

 α∈[0,1]

Combining the above equality with (26), we have  



X



X

 n n n n 

X  1 X

1

1

1

∗  pk Lk ( f ), f  ≤ ε + K(ε)

pke Lk (e0 ) − e0

+

pke Lk (e1 ) − e1

+

pke Lk (e2 ) − e2

. (27) D  Pn Pn Pn Pn k=0

k=0

k=0

k=0

Now, for a given ε0 > 0, choose ε > 0 such that 0 < ε < ε0 ,and also define the following sets:     n    1 X     0 ∗   pk Lk ( f ), f  ≥ ε  U: =  , n ∈ N : D      Pn k=0



  n 

ε0 − ε 

1 X     pk Lek (e0 ) − e0

≥ U0 : =  n ∈ N :

,   

3K(ε) 

Pn k=0



  n 

1 X

ε0 − ε      U1 : =  n ∈ N :

pk Lek (e1 ) − e1

≥ ,   

Pn k=0

3K(ε) 



  n 

ε0 − ε 

1 X     e U2 : =  n ∈ N :

pk Lk (e2 ) − e2

≥ ,   

3K(ε) 

Pn k=0

Then inequality (27) gives U ⊆ U0 ∪ U1 ∪ U2 and so δ(U) ≤ δ(U0 ) + δ(U1 ) + δ(U2 ). Then using the hypothesis (23), we get (24). The proof is completed. Example 4.3. Consider the sequence of fuzzy Bernstein-type polynomials [2] Bn ( f ; x) =

n X k=0

f

k n

!

! n k x (1 − x)n−k , k

(28)

where f ∈ CF [0, 1], x ∈ [0, 1]. Let pk = 1 for all k. Then (N, p)-mean is reduced to (C, 1)-mean. Define the sequence x = (xk ) by ( 1 , if k is odd, xk = −1 , if k is even.

¨ Talo, C. Bal / Filomat 30:3 (2016), 873–884 O.

Observe now that, st − limn→∞

1 n

Pn k=0

883

xk = 0. We define the following fuzzy positive linear operators

Ln ( f ; x) = (1 + xn )Bn ( f ; x).

(29)

Then, the sequence {Ln } satisfies conditions of Theorem 4.2. Hence, we have  n  1 X  ∗  st − lim D  Lk ( f ), f  = 0. n→∞ n k=0

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