On Double Statistical Convergence of Fuzzy Numbers

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Journal of Inequalities and Applications Volume 2009, Article ID 423792,8pages doi:10.1155/2009/423792

Research Article

On Double Statistical

P

-Convergence of

Fuzzy Numbers

Ekrem Savas¸

1

_{and Richard F. Patterson}

2

1_{Department of Mathematics, Istanbul Commerce University, 34672 Uskudar, Istanbul, Turkey} 2_{Department of Mathematics and Statistics, University of North Florida, Building 11, Jacksonville,}

FL 32224, USA

Correspondence should be addressed to Ekrem Savas¸,ekremsavas@yahoo.com

Received 1 September 2009; Accepted 2 October 2009

Recommended by Andrei Volodin

Savas and Mursaleen defined the notions of statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of statistical convergence by proving some theorems.

Copyrightq2009 E. Savas¸ and R. F. Patterson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

For sequences of fuzzy numbers, Nanda1 studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Kwon2introduced the definition of stronglyp-Ces`aro summability of sequences of fuzzy numbers. Savas¸3introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete. Savas¸4studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy and he continue to study in5,6. Recently Mursaleen and Bas¸arir7introduced and studied some new sequence spaces of fuzzy numbers generated by nonnegative regular matrix. Quite recently, Savas¸ and Mursaleen8defined statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of double statistical convergence and introduce the definition of double stronglyp-Ces`aro summabilty of sequences of fuzzy numbers.

2. Definitions and Preliminary Results

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Let CRn _{_A _⊂ _Rn _: _A_{compact and convex}_}_{. The spaces} _CRn _{has a linear}

structure induced by the operations

AB{ab, a∈A, b∈B},

λA{λa, λ∈A}

2.1

forA, B∈CRn_and_λ_∈_R_{. The Hausdor}_ﬀ_{distance between}_A_and_B_of_CRn_{is defined as}

δ∞A, B max

sup

a∈A

inf

b∈Ba−b,sup_b_∈_Bainf∈Aa−b

. 2.2

It is well known thatCRn_{, δ}

∞is a completenot separablemetric space. A fuzzy number is a functionXfromRn_to₀_,₁_satisfying

1Xis normal, that is, there exists anx0∈Rnsuch thatXx0 1;

2Xis fuzzy convex, that is, for anyx, y∈Rn_{and 0}_≤_λ_≤_1,

Xλx 1−λy≥minXx, Xy; 2.3

3Xis upper semicontinuous;

4the closure of{x∈Rn_:_Xx_>₀_}_{; denoted by}_X0_{, is compact.}

These properties imply that for each 0< α≤1, theα-level set

Xα_{_x_∈_Rn_:_Xx_≥_α_} _2.4

is a nonempty compact convex, subset ofRn_{, as is the support}_X0_{. Let}_LRn_{denote the set}

of all fuzzy numbers. The linear structure of LRn _{induces the addition}_X_Y _{and scalar}

multiplicationλX,λ∈R, in terms ofα-level sets, by

XYα Xα Yα,

λXα_λXα 2.5

for each 0≤α≤1.

Define for each 1≤q <∞,

dqX, Y

1

0

δ∞Xα, Yαqdα 1/q

, 2.6

and d∞ sup0≤α≤1δ∞Xα, Yα clearly d∞X, Y limq→ ∞dqX, Ywith dq ≤ dr if q ≤ r. Moreoverdqis a complete, separable, and locally compact metric space9.

Throughout this paper,dwill denotedq_{with 1} _≤ _q_{≤ ∞}_{. We will need the following}

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Definition 2.1. A double sequence X Xkl of fuzzy numbers is said to be convergent in Pringsheim’s sense or P-convergent to a fuzzy numberX0, if for everyε > 0 there exists

N∈ Nsuch that

dXkl, X0< fork, l > N, 2.7

and we denote byP−limXX0. The numberX0is called the Pringsheim limit ofXkl. More exactly we say that a double sequenceXklconverges to a finite numberX0if

Xkltend toX0as bothkandltend to∞independently of one another.

Letc2_F_{denote the set of all double convergent sequences of fuzzy numbers.}

Definition 2.2. A double sequence X Xkl of fuzzy numbers is said to be P-Cauchy sequence if for everyε >0,there existsN∈ Nsuch that

dXpq, Xkl< forp≥k≥N, q≥l≥N. 2.8

LetC2_F_{denote the set of all double Cauchy sequences of fuzzy numbers.}

Definition 2.3. A double sequence X Xklof fuzzy numbers is bounded if there exists a positive numberMsuch thatdXkl, X0< Mfor allkandl,

X_∞,2sup k,l

dXkl, X0<∞. _2.9

We will denote the set of all bounded double sequences byl2 ∞F.

Let K ⊆ N × Nbe a two-dimensional set of positive integers and let Km,n be the

numbers ofi, jinKsuch thati≤nandj ≤m. Then the lower asymptotic density ofKis defined as

P−lim inf

m,n Km,n

mn δ2K. 2.10

In the case when the sequenceKm,n/mn∞_m,n,∞₁_,₁has a limit, then we say thatKhas a natural density and is defined as

P−lim

m,n Km,n

mn δ2K. 2.11

For example, letK{i2_{, j}2_:_{i, j}_{∈ N}}_{, where}_N_{is the set of natural numbers. Then}

δ2K P−lim

m,n Km,n

mn ≤P−limm,n

√

m√n

mn 0 2.12

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Definition 2.4. A double sequence X Xkl of fuzzy numbers is said to be statistically convergenttoX0provided that for each >0,

P−lim

m,n

1

nm|{k, l;k≤m, l≤n:dXkl, X0≥}|0. 2.13

In this case we writest2−limk,lXk,lX0and we denote the set of all double statistically convergent sequences of fuzzy numbers byst2F.

Definition 2.5. A double sequenceX Xklof fuzzy numbers is said to be statisticallyP -Cauchy if for everyε >0,there existNNεandMMεsuch that

P−lim

m,n

1

nm|{k, l;k≤m, l≤n:dXkl, XNM≥}|0. 2.14

That is,dXkl, XNM< ε, a.a.k, l.

LetC2_F_{denote the set of all double Cauchy sequences of fuzzy numbers.}

Definition 2.6. A double sequenceX Xklof fuzzy and letpbe a positive real numbers. The sequenceX is said to be strongly doublep-Cesaro summable if there is a fuzzy numberX0 such that

P−lim

nm

1

nm mn

k,l1,1

dXkl, X0p0. 2.15

In which case we say thatXis stronglyp-Cesaro summable toX0.

It is quite clear that if a sequenceX Xklis statisticallyP-convergent, then it is a statisticallyP-Cauchy sequence8. It is also easy to see that if there is a convergent sequence

Y Yklsuch thatXklYkla.a.k, l, thenX Xklis statistically convergent.

3. Main Result

Theorem 3.1. A double sequenceX Xklof fuzzy numbers is statisticallyP-Cauchy then there is aP-convergent double sequenceY Yklsuch thatXklYkla.a.(k, l).

Proof. Let us begin with the assumption thatX Xklis statisticallyP-Cauchy this grant us a closed ballBBXM1,N1,1that containsXkla.a.k, lfor some positive numbersM1and N1. Clearly we can chooseMandNsuch thatBBXM,N,1/2·2containsXK,La.a.k, l. It is also clear thatXk,l∈B1,1B∩Ba.a.k, l; for

k, lk≤m; l≤n:Xk,l/∈B∩B

k, lk≤m; l≤n:Xk,l/∈B∪ {k, lk≤m; l≤n:Xk,l/∈B},

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we have

P−lim

m,n

1

mn

k≤m; l≤n:Xk,l/∈B∩B≤P−lim

m,n

1

mn

k≤m; l≤n:Xk,l/∈B

P−lim

m,n

1

mn|{k≤m; l≤n:Xk,l/∈B}| 0.

3.2

ThusB1,1is a closed ball of diameter less than or equal to 1 that containsXk,la.a.k, l. Now we let us consider the second stage to this end we chooseM2 andN2 such thatxk,l ∈B

BXM2,N2,1/2

2_·₂2_. _{In a manner similar to the first stage we have}_Xk,l _∈ _B

2,2 B1∩B, a.a.k, l. Note the diameter ofB2,2 is less than or equal 21−2 ·21−2. If we now consider the

m, nth general stage we obtain the following. First a sequence{Bm,n}∞_m,n,∞₁_,₁ of closed balls such that for eachm, n,Bm,n⊃ Bm1,n1, the diameter ofBm,nis not greater than 21−m·21−m withXk,l ∈ Bm,n, a.a.k, l. By the nested closed set theorem of a complete metric space we have∞_m,n,∞₁_,₁ Bmn/ ∅. So there exists a fuzzy numberA ∈∞_m,n,∞₁_,₁Bm,n. Using the fact that

Xk,l/∈ Bm,n, a.a.k, l, we can choose an increasing sequenceTm,nof positive integers such that

1

mn|{k≤m; l≤n:Xk,l/∈Bm,n}|<

1

pq ifm, n > Tm,n. 3.3

Now define a double subsequenceZk,lofXk,lconsisting of all termsXk,lsuch thatk, l > T1,1 and if

Tm,n< k, l≤Tm1,n1, thenXk,l/∈Bm,n. 3.4

Next we define the sequenceYk,lby

Yk,l: ⎧ ⎨ ⎩

A, if Xk,l is a term ofZ,

Xk,l, otherwise.

3.5

ThenP−limk,lYk,l Aindeed if > 1/m, n > 0,andk, l > Tm,n,then either Xk,l is a term of Z. Which means Yk,l A orYk,l Xk,l ∈ Bm,n and dYk,l−A ≤ |Bm,n| ≤ diameter of

Bm,n≤21−m·21−n_{. We will now show that}_Xk,l_Yk,l_a.a._{k, l}_{. Note that if}_Tm,n_{< m, n < Tm} 1,n1, then

{k≤m, l≤n:Yk,l/Xk,l} ⊂ {k≤m, l≤n:Xk,l/∈Bm,n}, 3.6

and by3.3

1

mn|{k≤m, l≤n:Yk,l/Xk,l}| ≤

1

mn|{k≤m, l≤n:Xk,l/∈Bm,n}|<

1

mn. 3.7

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Theorem 3.2. If X Xk,l is strongly p-Cesaro summable or statistically P-convergent to X0,

then there is aP-convergent double sequencesY and a statisticallyP-null sequenceZsuch thatP−

limk,lYk,lX0and st2limk,lZk,l0.

Proof. Note that if X Xk,l is strongly p-Cesaro summable toX0,then X is statistically

P-convergent toX0. LetN0 0 andM0 0 and select two increasing index sequences of positive integersN1< N2<· · · andM1< M2<· · · such thatm > Miandn > Nj, we have

1

mn

k≤m, l≤n:dXk,l, X0≥ 1

ij

< 1

ij. 3.8

DefineY andZ as follows: ifN0 < k < N1 andM0 < l < M1, setZk,l 0 andYk,l Xk,l. Suppose thati, j >1 andNi< k < Ni1,Mj < l < Mj1,then

Y : ⎧ ⎪ ⎨ ⎪ ⎩

Xk,l, dXk,l, X0< 1

ij, X0, otherwise,

Yk,l: ⎧ ⎪ ⎨ ⎪ ⎩

0, dXk,l, X0< 1

ij, Xk,l, otherwise.

3.9

We now show thatP−limk,lYk,lx0. Let >0 be given, picki, jbe given, and pickiandj such that >1/ij, thus fork, l > Mi, Nj, sincedYk,l, X0< dXk,l, X0< ifdXk,l, X0<1/ij anddYk,l, X0 0 ifdXk,l, X0>1/ij,we havedYk,l, X0< .

Next we show thatZis a statisticallyP-null double sequence, that is, we need to show thatP−limm,n1/mn|{k≤ml≤n:Zk,l/0}|0. Letδ >0 ifi, j∈NxN such that 1/ij < δ,

then|{k ≤ml≤n :Zk,l/0}| < δfor allm, n > Mi, Nj. From the construction ofMi, Nj, if

Mi < k ≤ Mi1 andNj < l ≤ Nj1,thenZk,l/0 only ifdXk,l, X0 > 1/ij. It follows that if

Mi< k≤Mi1andNj< l≤Nj1,then

{k≤m;l≤n:Zk,l/0} ⊂

k≤m;l≤n:dXk,l, X0< 1

pq

. 3.10

Thus forMi < m≤Mi1andNj< n≤Nj1andp, q > i, j,then

1

mn|{k≤m;l≤n:Zk,l/0}| ≤

1

mn

k≤m;l≤n:dXk,l, X0> 1

p, q

< 1

mn <

1

ij < δ. 3.11

this completes the proof.

Corollary 3.3. IfX Xk,lis a stronglyp-Cesaro summable toX0or statisticallyP-convergent to

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4. Conclusion

In recent years the statistical convergence has been adapted to the sequences of fuzzy numbers. Double statistical convergence of sequences of fuzzy numbers was first deduced in similar form by Savas and Mursaleen as we explain now: a double sequencesX{Xk,l}is said to beP-statistically convergent toX0provided that for each >0,

P−lim

m,n

1

mn{numbers ofk, l:k≤m, l≤n, dXk,l, X0≥}, 4.1

Since the set of real numbers can be embedded in the set of fuzzy numbers, statistical convergence in reals can be considered as a special case of those fuzzy numbers. However, since the set of fuzzy numbers is partially ordered and does not carry a group structure, most of the results known for the sequences of real numbers may not be valid in fuzzy setting. Therefore, this theory should not be considered as a trivial extension of what has been known in real case. In this paper, we continue the study of double statistical convergence and also some important theorems are proved.

References

1 S. Nanda, “On sequences of fuzzy numbers,”Fuzzy Sets and Systems, vol. 33, no. 1, pp. 123–126, 1989. 2 J.-S. Kwon, “On statistical andp-Cesaro convergence of fuzzy numbers,”Journal of Applied Mathematics

and Computing, vol. 7, no. 1, pp. 195–203, 2000.

3 E. Savas¸, “A note on double sequences of fuzzy numbers,”Turkish Journal of Mathematics, vol. 20, no. 2, pp. 175–178, 1996.

4 E. Savas¸, “On statistically convergent sequences of fuzzy numbers,”Information Sciences, vol. 137, no. 1–4, pp. 277–282, 2001.

5 E. Savas¸, “A note on sequence of fuzzy numbers,”Information Sciences, vol. 124, no. 1–4, pp. 297–300, 2000.

6 E. Savas¸, “On lacunary statistically convergent double sequences of fuzzy numbers,” Applied Mathematics Letters, vol. 21, no. 2, pp. 134–141, 2008.

7 M. Mursaleen and M. Bas¸arir, “On some new sequence spaces of fuzzy numbers,”Indian Journal of Pure and Applied Mathematics, vol. 34, no. 9, pp. 1351–1357, 2003.

8 E. Savas¸ and Mursaleen, “On statistically convergent double sequences of fuzzy numbers,”Information Sciences, vol. 162, no. 3-4, pp. 183–192, 2004.