# On the Slowly Decreasing Sequences of Fuzzy Numbers

Authors: Özer Talo; Feyzi Başar
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce the slowly decreasing condition for sequences of fuzzy numbers. We prove that this is a Tauberian condition for the statistical convergence and the Cesáro convergence of a sequence of fuzzy numbers.


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## Body

## 1.  Introduction

The concept of statistical convergence was introduced by Fast [1]. A sequence (xk)k∈ℕ of real numbers is said to be statistically convergent to some number l if for every ε>0 we have
(1)limn→∞1n+1|{k≤n:|xk-l|≥ε}|=0,
where by |S| and ℕ, we denote the number of the elements in the set S and the set of natural numbers, respectively. In this case, we write st-limk→∞xk=l.A sequence(xk) of real numbers is said to be (C,1)-convergent to l if its Cesàro transform {(C1x)n} of order one converges to l as n→∞, where
(2)(C1x)n=1n+1∑k=0n‍xk∀n∈ℕ.
In this case, we write (C,1)-limk→∞xk=l.We recall that a sequence(xk) of real numbers is said to be slowly decreasing according to Schmidt [2] if
(3)limλ→1+liminfn→∞minn<k≤λn(xk-xn)≥0,
where we denote by λn the integral part of the product λn, in symbol λn:=[λn].It is easy to see that (3) is satisfied if and only if for every ε>0 there exist n0=n0(ε) and λ=λ(ε)>1, as close to 1 as we wish, such that
(4)xk-xn≥-εwhenevern0≤n<k≤λn.Lemma 1 (see [3, Lemma 1]).
Let(xk) be a sequence of real numbers. Condition (3) is equivalent to the following relation:
(5)limλ→1-liminfn→∞minλn<k≤n(xn-xk)≥0.A sequence(xk) of real numbers is said to be slowly increasing if
(6)limλ→1+limsupn→∞maxn<k≤λn(xk-xn)≤0.
Clearly, it is trivial that (xk) is slowly increasing if and only if the sequence (-xk) is slowly decreasing.Furthermore, if a sequence(xk) of real numbers satisfies Landau’s one-sided Tauberian condition (see [4, page 121])
(7)k(xk-xk-1)≥-HforsomeH>0andeveryk,
then (xk) is slowly decreasing.Móricz [3, Lemma 6] proved that if a sequence (xk) is slowly decreasing, then
(8)st-limk→∞xk=l⟹limk→∞xk=l.
Also, Hardy [4, Theorem 68] proved that if a sequence (xk) is slowly decreasing, then
(9)(C,1)-limk→∞xk=l⟹limk→∞xk=l.Maddox [5] defined a slowly decreasing sequence in an ordered linear space and proved implication (9) for slowly decreasing sequences in an ordered linear space.We recall in this section the basic definitions dealing with fuzzy numbers. In 1972, Chang and Zadeh [6] introduced the concept of fuzzy number which is commonly used in fuzzy analysis and in many applications.Afuzzy number is a fuzzy set on the real axis, that is, a mapping u:ℝ→[0,1] which satisfies the following four conditions (i)
uis normal; that is, there exists an x0∈ℝ such that u(x0)=1.(ii)
u is fuzzy convex; that is, 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 semicontinuous.(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 onℝ by E1 and call it the space of fuzzy numbers. α-level set[u]α of u∈E1 is defined by
(10)[u]α:={{t∈ℝ:x(t)≥α},(0<α≤1),{t∈ℝ:x(t)>α}¯,(α=0).
The set [u]α is closed, bounded, and nonempty 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
(11)r¯(x):={1,(x=r),0,(x≠r).Letu,v,w∈E1 and k∈ℝ. Then the operations addition and scalar multiplication are defined on E1 by
(12)u+v=w⟺[w]α=[u]α+[v]α∀α∈[0,1]⟺w-(α)=u-(α)+v-(α),w+(α)=u+(α)+v+(α)∀α∈[0,1],[ku]α=k[u]α∀α∈[0,1]
(cf. Bede and Gal [7]).Lemma 2 (see [7]).
The following statements hold.(i)
0¯∈E1 is neutral element with respect to +, that is, u+0¯=0¯+u=u for all u∈E1.(ii)
With respect to0¯, none of u≠r¯, r∈ℝ has opposite in E1.(iii)
For anya,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 anya∈ℝ and any u,v∈E1, we have a(u+v)=au+av.(v)
For anya,b∈ℝ and any u∈E1, we have a(bu)=(ab)u.Notice thatE1 is not a linear space over ℝ.LetW be the set of all closed bounded intervals A of real numbers with endpoints A_ and A¯; that is, A:=[A_,A¯]. Define the relation d on W by
(13)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 (cf. Nanda [8]). Now, we may define the metric D on E1 by means of the Hausdorff metric d as follows:
(14)D(u,v):=supα∈[0,1]d([u]α,[v]α):=supα∈[0,1]max{|u-(α)-v-(α)|,|u+(α)-v+(α)|}.
One can see that
(15)D(u,0¯)=supα∈[0,1]max{|u-(α)|,|u+(α)|}=max{|u-(0)|,|u+(0)|}.
Now, we may give the following.Proposition 3 (see [7]).
Letu,v,w,z∈E1 and k∈ℝ. Then, the following statements hold. (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:(16)A⪯BiffA_≤B_,A¯≤B¯.
Also, the partial ordering relation on E1 is defined as follows:
(17)u⪯v⟺[u]α⪯[v]α⟺u-(α)≤v-(α),u+(α)≤v+(α)∀α∈[0,1].
We say that u≺v if u⪯v and there exists α0∈[0,1] such that u-(α0)<v-(α0) or u+(α0)<v+(α0) (cf. Aytar et al. [9]).Lemma 4 (see [9, Lemma 6]).
Letu,v∈E1 and ε>0. The following statements are equivalent. (i)
D(u,v)≤ε.(ii)
u-ε¯⪯v⪯u+ε¯.Lemma 5 (see [10, Lemma 5]).
Letμ,ν∈E1. If μ⪯ν+ε¯ for every ε>0, then μ⪯ν.Lemma 6 (see [11, Lemma 3.4]).
Letu,v,w∈E1. Then, the following statements hold. (i)
Ifu⪯v and v⪯w, then u⪯w.(ii)
Ifu≺v and v≺w, then u≺w.Theorem 7 (see [11, Teorem 4.9]).
Letu,v,w,e∈E1. Then, the following statements hold (i)
Ifu⪯w and v⪯e, then u+v⪯w+e.(ii)
Ifu⪰0¯ and v≻w, then uv⪰uw.Following Matloka [12], we give some definitions concerning sequences of fuzzy numbers. Nanda [8] introduced the concept of Cauchy sequence of fuzzy numbers and showed that every convergent sequence of fuzzy numbers is Cauchy.A sequenceu=(uk) of fuzzy numbers is a function u from the set ℕ into the set E1. The fuzzy number uk denotes the value of the function at k∈ℕ and is called the kth term of the sequence. We denote by ω(F), the set of all sequences of fuzzy numbers.A sequence(un)∈ω(F) is called convergent to the limit μ∈E1 if and only if for every ε>0 there exists an n0=n0(ε)∈ℕ such that
(18)D(un,μ)<ε∀n≥n0.
We denote by c(F), the set of all convergent sequences of fuzzy numbers.A sequenceu=(uk) of fuzzy numbers is said to be Cauchy if for every ε>0 there exists a positive integer n0 such that
(19)D(uk,um)<ε∀k,m>n0.
We denote by C(F), the set of all Cauchy sequences of fuzzy numbers.Ifuk⪯uk+1 for every k∈ℕ, then (uk) is said to be a monotone increasing sequence.Statistical convergence of a sequence of fuzzy numbers was introduced by Nuray and Savaş [13]. A sequence (uk) of fuzzy numbers is said to be statistically convergent to some number μ0 if for every ε>0 we have
(20)limn→∞1n+1|{k≤n:D(uk,μ0)≥ε}|=0.
Nuray and Savaş [13] proved that if a sequence (uk) is convergent, then (uk) is statistically convergent. However, the converse is false, in general.Lemma 8 (see [14, Remark 3.7]).
If(uk)∈ω(F) is statistically convergent to some μ, then there exists a sequence (vk) which is convergent (in the ordinary sense) to μ and
(21)limn→∞1n+1|{k≤n:uk≠vk}|=0.Basic results on statistical convergence of sequences of fuzzy numbers can be found in [10, 15–17].The Cesàro convergence of a sequence of fuzzy numbers is defined in [18] as follows. The sequence (uk) is said to be Cesàro convergent (written (C,1)-convergent) to a fuzzy number μ if
(22)limn→∞(C1u)n=μ.
Talo and Çakan [19, Theorem 2.1] have recently proved that if a sequence (uk) of fuzzy numbers is convergent, then (uk) is (C,1)-convergent. However, the converse is false, in general.Definition 9 (see [14]).
A sequence(uk) of fuzzy numbers is said to be slowly oscillating if
(23)infλ>1limsupn→∞maxn<k≤λnD(uk,un)=0.
It is easy to see that (23) is satisfied if and only if for every ε>0 there exist n0=n0(ε) and λ=λ(ε)>1, as close to 1 as wished, such that D(uk,un)≤ε whenever n0≤n<k≤λn.Talo and Çakan [19, Corollary 2.7] proved that if a sequence (uk) of fuzzy numbers is slowly oscillating, then the implication (9) holds.In this paper, we define the slowly decreasing sequence overE1 which is partially ordered and is not a linear space. Also, we prove that if (uk)∈ω(F) is slowly decreasing, then the implications (8) and (9) hold.


## 2. The Main Results

Definition 10.
A sequence(uk) of fuzzy numbers is said to be slowly decreasing if for every ε>0 there exist n0=n0(ε) and λ=λ(ε)>1, as close to 1 as wished, such that for every n>n0(24)uk⪰un-ε¯whenevern<k≤λn.
Similarly, (uk) is said to be slowly increasing if for every ε>0 there exist n0=n0(ε) and λ=λ(ε)>1, as close to 1 as wished, such that for every n>n0(25)uk⪯un+ε¯whenevern<k≤λn.Remark 11.
Each slowly oscillating sequence of fuzzy numbers is slowly decreasing. On the other hand, we define the sequence(un)=(∑k=0n‍vk), where
(26)vk(t)={1-tk+1,(0≤t≤1k+1),0,(otherwise).
Then, for each α∈[0,1], since
(27)un-(α)=0,un+(α)=(1-α)∑k=0n‍1k+1,(un) is increasing. Therefore, (un) is slowly decreasing. However, it is not slowly oscillating because for each n∈ℕ and λ>1 we get for α=0 and k=λn the statements k≤λn and
(28)uk+(0)-un+(0)=∑j=n+1k‍1j+1≥k-nk+1≥λn-1-nλn+1≥n(λ-1)λn+1-1λn+1⟶∞(n⟶∞)
hold.Lemma 12.
Let(un) be a sequence of fuzzy numbers. If (un) is slowly decreasing, then for every ε>0 there exist n0=n0(ε) and λ=λ(ε)<1, as close to 1 as wished, such that for every n>n0(29)un⪰uk-ε¯wheneverλn<k≤n.Proof.
We prove the lemma by an indirect way. Assume that the sequence(un) is slowly decreasing and there exists some ε0>0 such that for all λ<1 and m≥1 there exist integers k and n≥m for which
(30)un⋡uk-ε¯wheneverλn<k≤n.
Therefore, there exists α0∈[0,1] such that
(31)un-(α0)<uk-(α0)-εorun+(α0)<uk+(α0)-ε.
For the sake of definiteness, we only consider the case un-(α0)<uk-(α0)-ε. Clearly, (5) is not satisfied by {un-(α0)}. That is, {un-(α0)} is not slowly decreasing. This contradicts the hypothesis that (un) is slowly decreasing.Theorem 13.
Let(un) be a sequence of fuzzy number. If (un) is statistically convergent to some μ∈E1 and slowly decreasing, then (un) is convergent to μ.Proof.
Let us start by settingn=lm in (21), where 0≤l0<l1<l2<⋯ is a subsequence of those indices k for which uk=vk. Therefore, we have
(32)limm→∞1lm+1|{k≤lm:uk=vk}|=limm→∞m+1lm+1=1.
Consequently, it follows that
(33)limm→∞lm+1lm=limm→∞lm+1m+1×m+1m×mlm=1.
By the definition of the subsequence (lm), we have
(34)limm→∞ulm=limm→∞vlm=μ.
Since (un) is slowly decreasing for every ε>0 there exist n0=n0(ε) and λ=λ(ε)>1, as close to 1 as we wish, such that for every n>n0(35)uk⪰un-ε¯2whenevern<k≤λn.
For every large enough m(36)uk⪰ulm-ε¯2wheneverlm<k≤λlm.
By (33), we have lm+1<λlm for every large enough m, whence it follows that
(37)uk⪰ulm-ε¯2wheneverlm<k<lm+1.
By (34) and Lemma 4, for every large enough m we have
(38)μ-ε¯2≺ulm≺μ+ε¯2.
Combining (37) and (38) we can see that
(39)uk≻μ-ε¯wheneverlm<k<lm+1.
On the other hand, by virtue of Lemma 12, for every ε>0 there exist n0=n0(ε) and λ=λ(ε)<1 such that for every n>n0(40)un⪰uk-ε¯2wheneverλn<k≤n.
For every large enough m(41)ulm+1⪰uk-ε¯2wheneverλlm+1<k≤lm+1.
By (33), we have λlm+1<lm for every large enough m, whence it follows that
(42)ulm+1⪰uk-ε¯2wheneverlm<k<lm+1.
By (34) and Lemma 4, for every large enough m we have
(43)μ-ε¯2≺ulm+1≺μ+ε¯2.
Therefore, (42) and (43) lead us to the consequence that
(44)uk≺μ+ε¯wheneverlm<k<lm+1
which yields with (39) for each ε>0 and Lemma 4 that
(45)D(uk,μ)≤εwheneverlm<k<lm+1.
Therefore, (45) gives together with (34) that the whole sequence (uk) is convergent to μ.Lemma 14.
Letμ,ν,w∈E1. If μ+w⪯ν+w, then μ⪯ν.Proof.
Letμ,ν,w∈E1. If μ+w⪯ν+w, then
(46)μ-(α)+w-(α)≤ν-(α)+w-(α),μ+(α)+w+(α)≤ν+(α)+w+(α)
for all α∈[0,1]. Therefore, we have μ-(α)≤ν-(α) and μ+(α)≤ν+(α) for all α∈[0,1]. This means that μ⪯ν.Theorem 15.
Let(un)∈ω(F). If (un) is (C,1)-convergent to some μ∈E1 and slowly decreasing, then (un) is convergent to μ.Proof.
Assume that(un)∈ω(F) is satisfied (22) and is slowly decreasing. Then for every ε>0 there exist n0=n0(ε) and λ=λ(ε)>1, as close to 1 as we wish, such that for every n>n0(47)uk⪰un-ε¯3whenevern<k≤λn.
If n is large enough in the sense that λn>n, then
(48)λn+1λn-n(C1u)λn+(C1u)n=λn+1λn-n(C1u)n+1λn-n∑k=n+1λn‍uk.
For every large enough n, since
(49)λn+1λn-n≤2λλ-1,
we have
(50)limn→∞D[λn+1λn-n(C1u)λn,λn+1λn-n(C1u)n]=limn→∞λn+1λn-nD[(C1u)λn,(C1u)n]≤limn→∞2λλ-1D[(C1u)λn,(C1u)n]=0.
By Lemma 4, we obtain for large enough n that
(51)λn+1λn-n(C1u)n-ε¯3⪯λn+1λn-n(C1u)λn⪯λn+1λn-n(C1u)n+ε¯3.
By (22), for large enough n we obtain
(52)μ-ε¯3⪯(C1u)n⪯μ+ε¯3.
Since (un) is slowly decreasing, we have
(53)1λn-n∑k=n+1λn‍uk⪰un-ε¯3.
Combining (51), (52), and (53) we obtain by (48) for each ε>0 that
(54)λn+1λn-n(C1u)n+ε¯3+μ+ε¯3⪰λn+1λn-n(C1u)n+un-ε¯3.
By Lemma 14, we have
(55)μ+ε¯⪰un.
On the other hand, by virtue of Lemma 12, for every ε>0 there exist n0=n0(ε) and λ=λ(ε)<1 such that for every n>n0(56)un⪰uk-ε¯3wheneverλn<k≤n.
If n is large enough in the sense that λn<n, then
(57)λn+1n-λn(C1u)λn+1n-λn∑k=λn+1n‍uk=(λn+1n-λn+1)(C1u)n.
For large enough n, since
(58)λn+1n-λn≤2λ1-λ,
we have
(59)limn→∞D[λn+1n-λn(C1u)λn,λn+1n-λn(C1u)n]=0.
Using the similar argument above, we conclude that
(60)un⪰μ-ε¯.
Therefore, combining (55) and (60) for each ε≥0 and large enough n, it is obtained that D(un,μ)≤ε. This completes the proof.Now, we define the Landau’s one-sided Tauberian condition for sequences of fuzzy numbers.Lemma 16.
If a sequence(un)∈ω(F) satisfies the one-sided Tauberian condition
(61)nun⪰nun-1-H¯forsomeH>0andeveryn,
then (un) is slowly decreasing.Proof.
A sequence of fuzzy numbers(uk) satisfies
(62)nun⪰nun-1-H¯
for n∈ℕ, where H>0 is suitably chosen. Therefore, for all α∈[0,1] we have
(63)un-(α)-un-1-(α)≥-Hn,un+(α)-un-1+(α)≥-Hn.
For all n<k and α∈[0,1], we obtain
(64)uk-(α)-un-(α)≥∑j=n+1k‍[uj-(α)-uj-1-(α)]≥∑j=n+1k‍-Hj≥-H(k-nn).
Hence, for each ε>0 and 1<λ≤1+ε/H we get for all n<k≤λn(65)uk-(α)-un-(α)≥-H(kn-1)≥-H(λ-1)≥-ε.
Similarly, for all n<k≤λn and α∈[0,1] we have
(66)uk+(α)-un+(α)≥-ε.
Combining (65) and (66), one can see that uk⪰un-ε¯ which proves that (uk) is slowly decreasing.By Theorems13, 15 and Lemma 16, we derive the following two consequences.Corollary 17.
Let(uk) be a sequence of fuzzy numbers which is statistically convergent to a fuzzy number μ0. If (61) is satisfied, then limk→∞uk=μ0.Corollary 18.
Let(uk) be a sequence of fuzzy numbers which is (C,1)-convergent to a fuzzy number μ0. If (61) is satisfied, then limk→∞uk=μ0.Lemma 19.
If the sequence(un)∈ω(F) satisfies (61), then
(67)n(C1u)n⪰n(C1u)n-1-H¯forsomeH>0andeveryn.Proof.
Assume that the sequence(un)∈ω(F) satisfies (61), then for all α∈[0,1] we have
(68)n[un-(α)-un-1-(α)]≥-H,n[un+(α)-un-1+(α)]≥-H.
By the proof of Theorem  2.3 in [20], we obtain
(69)n[(C1u)n-(α)-(C1u)n-1-(α)]≥-H,n[(C1u)n+(α)-(C1u)n-1+(α)]≥-H.
This means that n(C1u)n⪰n(C1u)n-1-H¯, as desired.Corollary 20.
If the sequence(un)∈ω(F) satisfies (61), then
(70)st-limn→∞(C1u)n=μ0⟹limn→∞un=μ0.Proof.
By Lemma19, n(C1u)n⪰n(C1u)n-1-H¯ which is a Tauberian condition for statistical convergence by Corollary 17. Therefore, st-limn→∞(C1u)n=μ0 implies that limn→∞(C1u)n=μ0. Then, Corollary 18 yields that limn→∞un=μ0.


## 3. Conclusion

In the present paper, we introduce the slowly decreasing condition for a sequence of fuzzy numbers. This is a Tauberian condition fromst-limuk=μ0 to limuk=μ0 and from (C,1)-limuk=μ0 to limuk=μ0.Since we are not able to prove the fact that “(C,1)-statistical convergence can be replaced by(C,1)-convergence as a weaker condition, if it is proved that  {(C1u)n}  is slowly decreasing while  (uk)∈ω(F)  is slowly decreasing,” this problem is still open. So, it is meaningful to solve this problem.Finally, we note that our results can be extended to Riesz means of sequences of fuzzy numbers which are introduced by Tripathy and Baruah in [21].



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