ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 7 Issue 4(2015), Pages 17-23.
ROUGH CONVERGENCE OF A SEQUENCE OF FUZZY NUMBERS
(COMMUNICATED BY FEYZI BASAR)
FATMA GECIT AKC¸ AY AND SALIH AYTAR*
Abstract. We define the concept of rough limit set of a sequence of fuzzy numbers and obtain the relation between the set of rough limit and the extreme limit points of a sequence of fuzzy numbers. Finally, we investigate some properties of the rough limit set.
1. Introduction
The set of fuzzy numbers is denotedL(R),andddenotes the supremum metric onL(R). Now let rbe a nonnegative real number. Then we say that a sequence
{Xi}of fuzzy numbers is r−convergent to a fuzzy number X∗ and we write
Xi r
→X∗ as i→ ∞, provided that for everyε >0 there is an integeriε so that
d(Xi, X∗)< r+ε
wheneveri≥iε. The set
LIMrXi:={X∗∈L(R) :Xi r
→X∗ , as i→ ∞}
is called ther−limit set of the sequence{Xi}.
According to this definition, a sequence of fuzzy numbers which is divergent can be convergent with a certain roughness degree. For instance, let us define
Xi(x) :=
η(x) , ifiis odd integer µ(x) , otherwise ,
where
η(x) :=
x , ifx∈[0,1]
−x+ 2 , ifx∈[1,2] 0 , otherwise
2000Mathematics Subject Classification. 40A05, 03E72.
Key words and phrases. Fuzzy numbers; Analysis; Rough convergence. c
2015 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted Jul 14, 2015. Published November 11, 2015.
This paper was supported by grant SDU-BAP-2332-YL-10 from the Suleyman Demirel Uni-versity, Isparta, Turkey.
and
µ(x) :=
x−3 , ifx∈[3,4]
−x+ 5 , ifx∈[4,5] 0 , otherwise
.
Then we have
LIMrXi =
∅ , ifr <3/2
[µ−r1, η+r1] ,otherwise
,
wherer1 is nonnegative real number with [µ−r1, η+r1] := {X ∈L(R) :µ−r1X η+r1}.
The idea of rough convergence of a sequence can be interpreted as follows. Let
{Yi}be a convergent sequence of fuzzy numbers. Assume thatYi cannot be deter-mined exactly for everyi∈N(or somei∈N). That is,Yi cannot be calculated so we can use approximate value ofYifor simplicity of calculation. We only know that Yi belongs to the closed intervals [µi, λi], whered(µi, λi)≤r for everyi∈N.We have to do with an approximated and known sequence{Xi}satisfyingXi∈[µi, λi] for alli. Then the sequence{Xi} may not be convergent, but the inequality
d(Xi, X∗)≤d(Xi, Yi) +d(Yi, X∗)≤r+d(Yi, X∗)
implies that the sequence{Xi}isr−convergent.
Phu [10] and Burgin [4] introduced the notion of rough convergence indepen-dently with different titles. Here we will adopt the definitions and notations in [10]. In [10], Phu showed that the set LIMrx is bounded, closed and convex; and he also investigated the dependence of LIMrx on the roughness degree r. In [11], he extended the results given in [10] to infinite dimensional normed spaces. Recently, Aytar [3] proved that the ordinary core of a sequence x = (xi) of real numbers is equal to its 2r-limit set, where r := inf{r≥0 : LIMrx6=∅}. Later, Aytar [2] defined the concept of rough statistical convergence. Defining the set of rough sta-tistical limit points of a sequence, he obtained two stasta-tistical convergence criteria associated with this set. He also examined the relations between the set of all sta-tistical cluster points and the set of all rough stasta-tistical limit points of a sequence. Pal et al. [9] and D¨undar, C¸ akan [7] independently gave an extension of rough convergence at the same time, by using the notion of an ideal. They also stated some basic results related to the rough ideal limit set.
In this paper, we first define the concept of rough convergence of a sequence of fuzzy numbers. In the second step, we obtain the relation between the set of rough limit and the extreme limit points of a sequence of fuzzy numbers. When a sequence is convergent, we characterize the rough limit set of this sequence. Later, we prove that a necessary and sufficient condition for the rough limit set of a sequence to be nonempty is the boundedness of the sequence. Finally, we show that the rough limit set of a sequence is closed, bounded and convex.
2. Preliminary Concepts
In this section, we briefly recall some of the basic notations in the theory of fuzzy numbers and we refer to [1, 5, 6, 8, 12] for more details.
A fuzzy number X is a fuzzy subset of the real line R, which is normal, fuzzy convex, upper semi-continuous, and the set X0 is bounded where X0 := cl{x ∈
α∈(0,1], theα−level setXα defined by
Xα:={x∈R: X(x)≥α}=
h
Xα, Xαi
is a nonempty compact convex subset ofR.
Thesupremum metricdon the setL(R) is defined by
d(X, Y) := sup α∈[0,1]
max(|Xα−Yα|, X
α
−Yα ).
Now, givenX, Y ∈L(R), we define
X Y ifXα≤Yα andXα≤Yα for each α∈[0,1].
We writeX ≺Y ifX Y and there exists anα0∈[0,1] such thatXα0 < Yα0 or
Xα0 < Yα0.
A subsetEofL(R) is said to bebounded above if there exists a fuzzy numberµ, called anupper bound ofE, such thatX µfor everyX ∈E. µis called theleast upper bound (sup) ofEifµis an upper bound andµµ0for all upper boundsµ0. A lower bound and the greatest lower bound (inf) are defined similarly. E is said to bebounded if it is both bounded above and below.
The notions of ”sup” and ”inf” have been defined only for bounded sets of fuzzy numbers. An important fact, proved by Wu and Wu [12], states that if the set E⊂L(R) is bounded then its supremum and infimum exist (see also [8]).
Thelimit infimum and thelimit supremum of a sequence {Xi} is defined by
lim inf
i→∞ Xi:= infAX lim sup
i→∞
Xi:= supBX ,
where
AX:={µ∈L(R) : The set {i∈N:Xi≺µ} is infinite} BX:={µ∈L(R) : The set {i∈N:Xiµ} is infinite}.
Now, given two fuzzy numbersX, Y ∈L(R),we define their sum as Z =X+Y, whereZα:=Xα+Yα andZα:=Xα+Yα for allα∈[0,1].
To any real numbera∈R,we can assign a fuzzy numbera1 ∈L(R), which is defined by
a1(x) =
1 ifx=a 0 otherwise .
Anorder interval inL(R) is defined as
[X, Y] :={Z∈L(R) :XZ Y},
whereX, Y ∈L(R).
A setE of fuzzy numbers is calledconvex if
λµ1+ (1−λ)µ2∈E
3. Main Results
First, we give the relation between the set of rough limit and the extreme limit points of a sequence of fuzzy numbers.
Theorem 3.1. If LIMrX
i 6=∅, then we have
LIMrXi⊆[(lim supXi)−r1,(lim infXi) +r1].
In order to prove this theorem, we need the following lemma.
Lemma 3.2. IfX∗∈LIMrXi, thend(lim supXi, X∗)≤r andd(lim infXi, X∗)≤ r.
Proof of Lemma 3.2. We assume thatd(lim supXi, X∗)> r. Defineεe:=
d(lim supXi,X∗)−r
2 .
By definition of limit supremum, we have that giveni0 e
ε ∈Nthere exists an i∈N withi≥i0
e
ε such thatd(lim supXi, Xi)<eε.
Also, sinceXi r
→X∗ as i→ ∞,there is an integer i00 e
ε so that
d(Xi, X∗)< r+eε
wheneveri≥i00 e
ε. Letieε:= max
n
i0 e ε, i
00
e ε
o
.There exists i∈Nsuch thati≥iεeand
d(lim supXi, X∗) ≤ d(lim supXi, Xi) +d(Xi, X∗)< r+ 2eε
= r+d(lim supXi, X∗)−r=d(lim supXi, X∗).
This contradiction proves the lemma. Similarly,d(lim infXi, X∗)≤rcan be proved
using defination of limit infumum.
Now, we are ready to give the proof of Theorem 3.1.
Proof of Theorem 3.1. We show thatX∗ ∈[(lim supXi)−r1,(lim infXi) +r1] for
an arbitrary X∗ ∈LIMrXi, i.e., (lim supXi)−r1 X∗ (lim infXi) +r1. First,
assume that (lim supXi)−r1X∗does not hold. Thus, there exists anα0∈[0,1]
such that
(lim supXiα0)−r > X∗α0 or (lim supXi α0
)−r > X∗ α0
holds which are equivalent to the inequalities
(lim supXiα0)−X∗α0 > r or (lim supXi α0
)−X∗ α0
> r.
On the other hand, according to Lemma 3.2, we have
(lim supXiα0)−X∗α0
≤r and
(lim supXi
α0
)−X∗ α0
≤r.
Thus we obtain a contradiction. Therefore, we get (lim supXi)−r1X∗.
The second part of this theorem can be proved using the similar arguments in
the first part.
Note that the converse inclusion in this theorem holds for sequences of real numbers, but it may not hold for sequences of fuzzy numbers, as can be seen in the following example.
Example 3.3. Define
Xi(x) :=
−1
and
X∗(x) :=
1 , ifx∈[0,1] 0 , otherwise .
Then we have X
1
∗−X
1
i
=|1−0|= 1,i.e., d(Xi, X∗)≥1 for alli∈N. Although the sequence{Xi}is not convergent toX∗,lim supXiandlim infXiof this sequence are equal to X∗. Hence we get X∗ ∈ [lim supXi−(1/2)1,lim infXi+ (1/2)1], but
X∗∈/LIM1/2Xi.
Theorem 3.4. If a sequence{Xi}converges to the fuzzy numberX∗,then LIMrXi= Br(X∗) :={µ∈L(R) :d(µ, X∗)≤r}.
Proof. Letε >0.Since the sequence{Xi}is convergent to X∗,there is an integer iε so that
d(Xi, X∗)< ε
wheneveri≥iε. LetY ∈Br(X∗). We have
d(Xi, Y)≤d(Xi, X∗) +d(X∗, Y)< ε+r
for everyi≥iε.Hence we haveY ∈LIMrXi. Now letY ∈LIMrX
i. Hence there is an integeri
0
εso that
d(Xi, Y)< r+ε
wheneveri≥i0ε. Leti00ε := max
n
iε, i
0
ε
o
. For alli > i00ε, we obtain
d(Y, X∗)≤d(Xi, Y) +d(Xi, X∗)< r+ 2ε.
Sinceεis arbitrary, we haved(Y, X∗)≤r. Hence we getY ∈Br(X∗). Thus, if the sequence{Xi}converges toX∗, then LIMrXi=Br(X∗).
Theorem 3.5. A sequence {Xi} in L(R)is r−convergent to X∗ if there exists a
sequence {Yi} inL(R)such that
Yi→X∗ asi→ ∞, and d(Xi, Yi)≤r for every i∈N.
Proof. Assume thatYi →X∗ , asi → ∞, and d(Xi, Yi) ≤r for every i ∈N. Yi→X∗ , as i→ ∞means that for everyε >0 there exists aniε such that
d(Yi, X∗)< ε for all i≥iε.
The inequalityd(Xi, Yi)≤ryields
d(Xi, X∗)≤d(Xi, Yi) +d(Yi, X∗)< r+ε if i≥iε.
Hence the sequence{Xi}isr-convergent to the fuzzy numberX∗.
Theorem 3.6. The diameter of anr−limit set is not greater than 2r.
Proof. We have to show that sup{d(Y, Z) :Y, Z∈LIMrXi} ≤2r. Assume on the contrary that sup{d(Y, Z) :Y, Z ∈LIMrXi}>2r.By this assumption, there exist Y, Z ∈LIMrXi satisfyingλ:=d(Y, Z)>2r. For an arbitrary ε∈(0, λ/2−r), we have
∃i0ε∈N:∀i≥i0ε⇒d(Xi, Y)< r+ε
∃i
00
ε ∈N:∀i≥i
00
ε ⇒d(Xi, Z)< r+ε. Defineiε:= max
n
i0ε, i
00
ε
o
.Thus we get
for alli≥iεwhich contradicts to the fact thatλ=d(Y, Z).
The rest of the paper contains some basic properties of the rough limit set.
Theorem 3.7. A sequence {Xi} is bounded if and only if there exists an r ≥ 0 such that LIMrX
i6=∅.
Proof. (⇒) Let{Xi}be a bounded sequence ands:= sup{d(Xi,01) :i∈N}<∞. Then we have 01∈LIMsXi, i.e., LIMrXi6=∅, where r=s.
(⇐) If LIMrX
i 6=∅for somer≥0,then there exists X∗∈LIMrXi. By defini-tion, for everyε >0 there is an integeriε so that
d(Xi, X∗)< r+ε
wheneveri≥iε. Definet=t(ε) := max{d(X∗,01), d(X1,01), d(X2,01), ..., d(Xiε,01), r+
ε}. Then we have Xi ∈ {µ∈L(R) :d(µ,01)≤t+r+ε} for every i ∈ N, which proves the boundedness of the sequence{Xi}.
The next theorem gives the inclusion relation between the rough limit sets of a sequence and its subsequence. Its proof is straightforward.
Theorem 3.8. If {Xki}is a subsequence of {Xi}, then LIM
rX
i⊂LIMrXki.
Theorem 3.9. For all r ≥ 0, the r−limit set LIMrX
i of an arbitrary sequence
{Xi} is closed.
Proof. Let{Yi} ⊂LIMrXi andYi→Y∗ asi→ ∞. Letε >0. Since the sequence
{Yi} converges toY∗,there is an integer jεso that
d(Yi, Y∗)< ε 2 wheneveri≥jε. Since Yjε ∈LIM
rX
i,there is an integeriε so that
d(Xi, Yjε)< r+
ε 2 wheneveri≥iε. Therefore, we have
d(Xi, Y∗) ≤ d(Xi, Yjε) +d(Yjε, Y∗)
< r+ε/2 +ε/2 =r+ε
for everyi≥iε.Hence Y∗∈LIMrXi implies that the set LIMrXi is closed.
4. Discussion
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Fatma Gecit Akc¸ay
Suleyman Demirel University, Department of Mathematics, Isparta, TURKEY E-mail address:[email protected]
Salih Aytar
