Volume 2010, Article ID 838741,9pages doi:10.1155/2010/838741
Research Article
On Asymptotically
λ, σ
-Statistical Equivalent
Sequences of Fuzzy Numbers
Ekrem Savas¸,
1H. S¸evli,
1and M. Cancan
21Department of Mathematics, Istanbul Commerce University, ¨Usk¨udar, Istanbul 34672, Turkey 2Department of Mathematics, Y¨uz¨unc¨u Yil University, Van 65088, Turkey
Correspondence should be addressed to Ekrem Savas¸,[email protected]
Received 30 December 2009; Accepted 13 April 2010
Academic Editor: Soo Hak Sung
Copyrightq2010 Ekrem Savas¸ et al. 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.
The goal of this paper is to give the asymptoticallyλ, σ-statistical equivalent which is a natural
combination of the definition for asymptotically equivalent, invariant mean and λ-statistical
convergence of fuzzy numbers.
1. Introduction
The concepts of fuzzy sets and fuzzy set operation were first introduced by Zadeh1and 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 and fuzzy mathematical programming. Matloka 2 introduced bounded and convergent sequences of fuzzy numbers and studied their some properties. For sequences of fuzzy numbers, Nuray and Savas¸3introduced and discussed the concepts of statistically convergent and statistically Cauchy sequences.
Quite recently, Savas¸ 4 introduced the idea of asymptotically λ-statistically equivalent sequences of fuzzy numbers. In this paper we extend his result by using invariant means.
2. Preliminaries
Byl∞andc, we denote the Banach spaces of bounded and convergent sequencesx xknormed byxsupn|xn|, respectively. A linear functionalLonl∞is said to be a Banach
limitsee5if it has the following properties:
1Lx≥0 ifxn≥0 for alln;
2Le 1 wheree 1,1, . . .;
3LDx Lx, where the shift operatorDis defined byDxn {xn1}.
LetBbe the set of all Banach limits onl∞. A sequencex∈l∞is said to be almost convergent
if all Banach limits ofxcoincide. Letcdenote the space of almost convergent sequences. Let σ be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functionalφonl∞is said to be an invariant mean or aσ-mean if and only
if
1φx≥0 when the sequencex xkis such thatxk≥0 for allk,
2φe 1 wheree 1,1,1, . . ., and
3φx φxσkfor allx∈l∞.
Throughout this paper we shall consider the mappingσhas having on finite orbits, that is,σmk/kfor all nonnegative integers withm≥1, whereσmkis themth iterate of σatk. Thusσ-mean extends the limit functional oncin the sense thatφx limxfor all x∈ c. Consequently,c ⊂VσwhereVσis the set of bounded sequences all of whoseσ-mean are equal.
In the case whenσk k1, theσ-mean is often called the Banach limit andVσis the set of almost convergent sequences.
A fuzzy real numberX is a fuzzy set onR, that is, a mappingX : R → I 0,1, associating each real numbertwith its grade of membershipXt.
Theα-cut of fuzzy real numberXis denoted byXα, 0< α≤1, whereXα{t∈R: Xt≥α}. Ifα0,then it is the closure of the strong 0-cut. A fuzzy real numberXis said to be upper semicontinuous if for eachε > 0,X−10, aε, for alla∈ I is open in the usual topology ofR. If there existst∈Rsuch thatXt 1, then the fuzzy real numberXis called normal.
A fuzzy numberX is said to be convex, if Xt ≥ Xs∧Xr minXs, Xr wheres < t < r. The class of all upper semi-continuous, normal, convex fuzzy real numbers is denoted by RI and throughout the article, by a fuzzy real number we mean that the number belongs toRI. LetX, Y∈RIand theα-level sets be
Xα
aα1, aα2, Yα
bα1, bα2, α∈0,1. 2.1
Then the arithmetic operations onRIare defined as follows:
X⊕Yt sup{Xs∧Yt−s}, t∈R,
XYt sup{Xs∧Ys−t}, t∈R,
X⊗Yt sup
Xs∧Y
t s
, t∈R,
X/Yt sup{Xst∧Ys}, t∈R.
The above operations can be defined in terms ofα-level sets as follows:
X⊕Yα
aα1 b1α, aα2b2α,
XYα
aα1−bα2, aα2−bα1
,
X⊗Yα min i,j∈{1,2}a
α
i ·bjα,i,jmax∈{1,2}aαi ·bαj
,
X−1 α
aα2−1,aα1−1, 0/∈{X}.
2.3
The additive identity and multiplicative identity inRIare denoted by 0 and 1, respectively. LetDthe set of all closed and bounded intervalsX XL, XR.Then we writeX ≤Y, if and only ifXL≤YLandXR≤YR, and
ρX, Y maxXL−YL,XR−YR. 2.4
It is obvious thatD, ρis a complete metric space. Now we define the metricd:RI×RI → Rby
dX, Y sup 0≤α≤1
ρXα,Yα 2.5
forX, Y∈RI.
We now give the following definitionssee6for fuzzy real-valued sequences.
Definition 2.1. A fuzzy real-valued sequence X Xkis a function X from the set N of natural numbers into RI. The fuzzy real-valued sequence Xn denotes the value of the function atn ∈ N and is called thenth term of the sequence. We denote by wFthe set of all fuzzy real-valued sequencesX Xk.
Definition 2.2. A fuzzy real-valued sequence X Xk is said to be convergent to a fuzzy numberX0,written as limkXk X0, if for every >0 there exists a positive integerN0such that
dXk, X0< fork > N0. 2.6
LetcFdenote the set of all convergent sequences of fuzzy numbers.
Definition 2.3. A sequenceX Xkof fuzzy numbers is said to be bounded if the set{Xk : k∈N}of fuzzy numbers is bounded. We denote by ∞Fthe set of all bounded sequences
of fuzzy numbers.
It is easy to see that
It was shown thatcFand ∞Fare complete metric spacessee7.
3. Definitions and Notations
Definition 3.1. Two fuzzy real-valued sequences X Xk and Y Yk are said to be asymptotically equivalentif
lim k d
Xk Yk,
1
0 3.1
denoted byX ∼F Y.
Let Λ λn be a nondecreasing sequence of positive reals tending to infinity and λ11 andλn1≤λn1.
In4, Savas¸ introduced the concept ofλ-statistical convergence of fuzzy numbers as follows.
Definition 3.2. A fuzzy real-valued sequencesX Xkis said to beλ-statistically convergent orSλ-convergent toLif for every >0,
lim n
1
λn|{k∈In
:dXk, L≥}|0. 3.2
In this case we writeSλ-limitXLorXk → LSλ, and
Sλ{X:∃L∈RI, Sλ-limitXL}. 3.3
The next definition is natural combination of Definitions 3.1 and 3.2., which was defined in4.
Definition 3.3. Two fuzzy real-valued sequences X Xk and Y Yk are said to be asymptoticallySλ-statistical equivalent of multipleLprovided that for every >0
lim n
1 λn
k∈In:d
Xk
Yk, L
≥0 3.4
denoted byX S
L λ∼F
Definition 3.4. Two fuzzy real-valued sequences X Xk and Y Yk are said to be asymptotically statistical equivalent of multipleLprovided that for every >0,
lim n
1 n
the number ofk < n:d
Xk Yk, L
≥
0 3.5
denoted byX SL∼F
Yand simply asymptotically statistical equivalent ifL1.
It is quite naturel to expect the following definition.
Definition 3.5. Fuzzy real-valued sequences X Xk is said to be λ, σ-statistically convergent toLprovided that for every >0
lim n
1 λn
k∈In:d
Xσkm, L
≥0 3.6
uniformly inm.
In this case we writeSλ,σ-limitXLorXk → LSλ,σ, and
Sλ,σF X:∃L∈RI:Sλ,σ-limit XL. 3.7
Following this result we introduce two new notions asymptotically λ, σ-statistical equivalent of multipleLand strongλ, σ-asymptotically equivalent of multipleL.
Definition 3.6. Two fuzzy real-valued sequences X Xk and Y Yk are said to be asymptoticallyλ, σ-statistical equivalent of multipleLprovided that for every >0
lim n
1 λn
k∈In:d
Xσkm
Yσkm, L
≥
0, 3.8
uniformly in m, denoted by X S
L
λ,σF
∼ Yand simply asymptotically λ, σ-statistical equivalent ifL1.
Definition 3.7. Two fuzzy real-valued sequences X Xk and Y Yk are said to be asymptoticallyσ-statistical equivalent of multipleLprovided that for every >0
lim n
1 n
the number ofk < n:d
Xσkm
Yσkm, L
≥
0 3.9
uniformly inm,denoted byXS
L
σF
∼ Yand simply asymptoticallyσ-statistical equivalent ifL1.
We now define the following.
Definition 3.8. Let p pkbe a sequence of positive real numbers; two fuzzy real-valued sequencesX XkandY Ykare strongly asymptoticallyλ, σ-equivalent of multipleL, provided that
lim n
1 λn
k∈In
d
Xσkm
Yσkm, L
pk
0 3.10
denoted byX V
Lp λ,σF
∼ Yand simply strongly asymptoticallyλ, σ-equivalent ifL1.
If we takepkpfor allk∈Nwe writeX
VLpλ,σF
∼ Yinstead ofXV
Lp λ,σF
∼ Y.
In caseλnnin above definition we get following.
Definition 3.9. Letp pkbe a sequence of positive numbers and let us consider two fuzzy real-valued sequencesX XkandY Yk. Two fuzzy real-valued sequencesX Xk andY Ykare said to be strongly asymptotically Ces´aro equivalent of multipleLprovided that
lim n
1 n
n
k1 d
Xσkm
Yσkm, L
pk
0 3.11
denoted byX V
Lp σ F
∼ Y, and simply strong Ces´aro asymptotically equivalent ifL1.
4. Main Results
Theorem 4.1. Letλn∈Λ. Then
1IfXV
Lp
λ,σF
∼ Y thenX S
L
λ,σF
∼ Y;
2IfXandY ∈l∞FandX
SL
λ,σF
∼ YthenXV
Lp
λ,σF
∼ Y
3XS
L
σF
∼ Y∩l∞F X
VLpλ,σF
Proof. Part1: if >0 andXV
Lp
λ,σ∼F
Y then
k∈In
d
Xσkm
Yσkm, L
p
≥
k∈In&d
Xσkm/Yσkm,L≥ d
Xσkm
Yσkm, L
p
≥p
k∈In:d
Xσkm
Yσkm, L
≥ . 4.1
Therefore X S
L
λ,σF
∼ Y. Part 2: suppose that fuzzy real-valued sequences X Xk and Y Yk are inl∞F and X
SL λ∼F
Y. Then we can assume that dXσkm/Yσkm , L ≤
K for allk andm. Let >0 be given andNbe such that
1 λn
k∈In :d
Xσkm
Yσkm, L
≥
2 1/p
≤ 2Kp 4.2
for alln > Nand let
Lk:
k∈In:d
Xσkm
Yσkm, L
≥
2 1/p
. 4.3
Now for alln > Nwe have
1 λn
k∈In
d
Xσkm
Yσkm, L
p
1
λn
k∈Lk
d
Xσkm
Yσkm, L
p
1
λn
k /∈Lk
d
Xσkm
Yσkm, L
p
≤ 1
λn λn 2KpK
p 1 λnλn
2.
4.4
Hence X V
Lp
λ,σF
∼ Y. This completes the proof.Part 3: this immediately follows from 1 and2.
In the next theorem we prove the following relation.
Theorem 4.2. Let0< hinfkpk≤supkpkH <∞. ThenX
VLλ,σpF
∼ Y impliesXS
L
λ,σF
Proof. LetXV
Lp λ,σF
∼ Y and >0 be given. Then
1 λn
k∈In
d
Xσkm
Yσkm, L
pk
1
λn
k∈In&d
Xσkm/Yσkm,L≥ d
Xσkm
Yσkm, L
pk
1
λn
k∈In&d
Xσkm/Yσkm,L< d
Xσkm
Yσkm, L
pk
≥ 1
λn
k∈In&d
Xσkm/Yσkm,L≥ d
Xσkm
Yσkm, L
pk
≥ 1
λn
k∈In&d
Xσkm/Yσkm,L≥
mininfpk,H
≥ 1
λn
k∈In:d
Xσkm
Yσkm, L
≥
mininfpk,H.
4.5
HenceXS
L
λ,σF
∼ Y.
Theorem 4.3. Let fuzzy real-valued sequencesX Xkand Y Ykbe bounded and0 < h
infkpk≤supkpkH <∞. ThenX SL
λ,σF
∼ Y impliesXV
Lp λ,σF
∼ Y.
Proof. Suppose that fuzzy real-valued sequencesX XkandY Ykbe bounded and >0 is given. SinceX XkandY Ykare bounded there exists an integerKsuch that dXσkm/Yσkm , L≤Kfor allkandm; then
1 λn
k∈In
d
Xσkm
Yσkm, L
pk
1
λn
k∈In&d
Xσkm/Yσkm,L≥ d
Xσkm
Yσkm, L
pk
1
λn
k∈In&d
Xσkm/Yσkm ,L< d
Xσkm
Yσkm, L
pk
≤ 1
λn
k∈In&d
Xσkm/Yσkm,L≥
maxKh, KH
1
λn
k∈In&d
Xσkm/Yσkm,L<
max{}pk
≤maxKh, KH 1 λn
k∈In:d
Xσkm
Yσkm, L
≥
maxh, H.
4.6
HenceXV
Lp λ,σF
Remark 4.4. If we takeσk k1 in our results, all results reduce to the results of almost convergence which have not proved so far.
Acknowledgment
This Work was supported by Grant2008-FED-B162of Y ¨uz ¨unc ¨u Yil university.
References
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3 F. Nuray and E. Savas¸, “Statistical convergence of sequences of fuzzy numbers,”Mathematica Slovaca,
vol. 45, no. 3, pp. 269–273, 1995.
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5 S. Banach,Theorie des Operations Linearies, Subwncji Funduszu Narodowej, Warszawa, Poland, 1932.
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