Gen. Math. Notes, Vol. 8, No. 2, February 2012, pp.49-59
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2012
www.i-csrs.org
Available free online at http://www.geman.in
Lacunary Invariant Statistical
Convergence of Fuzzy Numbers
Mehmet Açıkgöz1 and Ayhan Esi2
1
University of Gaziantep, Faculty of Science and Arts
Department of Mathematics, 27310 Gaziantep, Turkey
E-mail: acikgoz@gantep.edu.tr
2
Adiyaman University, Science and Art Faculty
Department of Mathematics, 02040 Adiyaman-Turkey
E-mail: aesi23@hotmail.com, aesi@adiyaman.edu.tr
(Received: 10-12-11/ Accepted: 15-1-12)
Abstract
In this paper, we introduce the concepts of invariant convergence, lacunary
invariant statistical convergence of sequences of fuzzy numbers and lacunary
strongly invariant convergence of sequences of fuzzy numbers. We give some
relations related to these concepts.
Keywords: Fuzzy numbers, lacunary sequence, almost convergence, statistical convergence, invariant mean.
1
Introduction
Schaefer [17] defined the σ-convergence as follows:
Let σ be a one- toof the set of positive integers into itself
one mapping
k
k−1
such that σ (n) = σ σ
(n) , k = 1, 2, 3, · · · . A continuous linear functional
ϕ on l∞ , the set of all bounded sequences, is said to be an invariant mean or
a σ-mean if and only if
(i) ϕ (x) ≥ 0 when the sequence x = (xn ) has xn ≥ 0 for all n,
(ii) ϕ (e) = 1, where e = (1, 1, 1, · · ·) and
(iii) ϕ ({xσ (n)}) = ϕ ({xn }) all x = (xn ) ∈ l∞ .
For certain kinds of mappings σ, every invariant mean ϕ extends the limit
functional on the space c, the set of all convergent sequences, in the sense
50
Mehmet Açıkgöz et al.
that ϕ (x) = lim x for all x = (xn ) ∈ c. Consequently, c ⊂ vσ , where vσ is
the set of bounded sequences all of whose σ-means are equal. In the case σ
is the translation mapping n → n + 1, a σ-mean is often called a Banach
limit and vσ is the set
of almost convergent sequences. If x = (xk ), write
T x = (T xk ) = xσ(k) . It can be shown that
vσ = x ∈ l∞ : lim tkn (x) = l, unif ormly in n, l = σ − lim x ,
k→∞
where tkn (x) = (k + 1)−1
k
P
i=0
xσi (n) , here σ k (n) denotes the k th iterate of the
mapping σ at n. By a lacunary sequence we mean an increasing integer sequence θ = (kr ) such that ko = 0 and hr = kr − kr−1 → ∞ as r → ∞.
Throughout this paper the intervals determined by θ = (kr ) will be denoted
kr
will be abbreviated as qr . Lacunary
by Ir = (kr−1 , kr ] and the ratio kr−1
sequences have been discussed by [3], [7], [9], [11] and many others.
The notion of statistical convergence was introduced by Fast [6] and Schoenberg [18], independently. Over the years and under different names statistical
convergence has been discussed in the different theories such as the theory of
Fourier analysis, ergodic theory and number theory. Later on, it was further
investigated from the sequence space point of view and linked with summability theory by Fridy [8], Salat [16], Connor [2] and many others. This concept
extended the idea to apply to sequences of fuzzy numbers with Nuray [14],
Kwon et al. [11], Altin et al. [1], Nuray and Savaş [15] and many others.
A sequence x = (xk ) is said to be statistically convergent to l if for every
ε > 0,
lim n−1 |{k ≤ n : |xk − l| ≥ ε}| = 0
n→∞
where the vertical bars denote the cardinality of the set which they enclose, in
which case we write S−lim x = l. The concept of fuzzy sets was first introduced
by Zadeh [19]. Bounded and convergent sequences of fuzzy numbers were
introduced by Matloka [12]. Matloka show that every convergent sequences of
fuzzy numbers is bounded. Later on sequences of fuzzy numbers have been
discussed by Nanda [13], Nuray [14], [15], [10], Esi [5] and many others. Briefly,
we recall some of the basic notations in the theory of fuzzy numbers and we
refer readers to Matloka [12] and Diamond and Kloeden [4] for more details.
Let C (Rn ) = {A ⊂ Rn : A is compact and convex set}. The space C (Rn )
has a linear structure induced by the operations
A + B = {a + b : a ∈ A, b ∈ B} and γA = {γa : a ∈ A}
for A, B ∈ C (Rn ) and γ ∈ R. The Hausdorff distance between A and B in
C (Rn ) is defined by
(
)
δ∞ (A, B) = max sup inf ka − bk , sup inf ka − bk .
a∈A b∈B
b∈B a∈A
51
Lacunary Invariant Statistical...
It is well-known that (C (Rn ) , δ∞ ) is a complete metric space. A fuzzy number
is a function X from Rn to [0, 1] which is normal, fuzzy convex, upper semicontinuous and the closure of {X ∈ Rn : X (x) > 0} is compact. These properties
imply that for each 0 < α ≤ 1, the α-level set X α = {X ∈ Rn : X (x) ≥ α}
is a non-empty compact, convex subset of Rn , with support X 0 . Let L (Rn )
denote the set of all fuzzy numbers. The linear structure of L (Rn ) induces the
addition X + Y and the scalar multiplication λX in terms of α-level sets, by
[X + Y ]α = [X]α + [Y ]α and [λX]α = λ [X]α
for each 0 ≤ α ≤ 1. Consider the Hausdorff metrics dq and d∞ defined by
dq (X, Y ) =
 1q
 1
Z
 δ∞ (X α , Y α ) dα
(1 ≤ q ≤ ∞)
0
and
d∞ (X, Y ) = sup δ∞ (X α , y α ) .
0≤α≤1
Clearly, d∞ (X, Y ) = limq→∞ dq (X, Y ) with dq (X, Y ) ≤ ds (X, Y ) if q ≤ s, [4].
For simplicity in notation, we shall write throughout d instead of dq with
1 ≤ q ≤ ∞. The metric d has the following properties:
d (cX, cY ) = |c| d (X, Y )
(1)
d (X + Y, Z + W ) ≤ d (X, Z) + d (Y, W ) .
(2)
for c ∈ R and
A metric on L (R) is said to be translation invariant d (X + Z, Y + Z) =
d (X, Y ) for all X, Y, Z ∈ L (R). A sequence X = (Xk ) of fuzzy numbers is a
function X from the set N of natural numbers into L (R). The fuzzy number
Xk denotes the value of the function at k ∈ N [12]. We denote by wF the
set of all sequences X = (Xk ) of fuzzy numbers. A sequence X = (Xk ) of
fuzzy numbers is said to be bounded if the set {Xk : k ∈ N } of fuzzy numbers
F
is bounded [2]. We denote by l∞
the set of all bounded sequences X = (Xk )
of fuzzy numbers. A sequence X = (Xk ) of fuzzy numbers is said to be
convergent to a fuzzy number Xo if for every ε > 0 there is a positive integer
ko such that d (Xk , Xo ) < ε for k > ko [12]. We denote by cF the set of all
convergent sequences X = (Xk ) of fuzzy numbers. It is straightforward to see
F
F
that cF ⊂ l∞
⊂ wF . In [13], it was shown that cF and l∞
are complete metric
spaces.
In the present note, we introduce and examine the concepts of invariant convergence of sequences, lacunary invariant statistical convergence of sequences
52
Mehmet Açıkgöz et al.
of fuzzy numbers and lacunary strongly invariant convergence of sequences
of fuzzy numbers. We give some relations related to these concepts. Let
p = (pk ) ∈ l∞ , then the following well-known inequality will be used in the
paper: For sequences (ak ) and (bk ) of complex numbers we have
|ak + bk |pk ≤ K (|ak |pk + |bk |pk ) ,
(3)
where K = max 1, 2H−1 and H = supk pk < ∞.
Definition 1.1 A sequence X = (Xk ) of fuzzy numbers is said to be invariant convergent to fuzzy number Xo if
lim d (tkn (X) , Xo ) = 0
(4)
k→∞
uniformly in n, where tkn (X) = (k + 1)−1
Pk
i=0
Xσi (n) .
This means that for every ε > 0, there exists a ko ∈ N such that d (tkn (X) , Xo )
< ε whenever k ≥ ko and for all n. If the limit in (4) exists, then we write
vσF − lim X = Xo . Let vσF be the space of all invariant convergent sequences of
F
F
denotes the classes of
⊂ vσF , where vσ,0
fuzzy numbers. It is evident that vσ,0
all invariant convergent to zero of fuzzy numbers.
Definition 1.2 Let θ = (kr ) be lacunary sequence. A sequence X = (Xk )
of fuzzy numbers is said to be lacunary invariant statistically convergent to
fuzzy number Xo if, for every ε > 0
lim h−1
r |{k ∈ Ir : d (tkn (X) , Xo ) ≥ ε}| = 0
r→∞
uniformly in n. In this case we write Xk → Xo Sbθσ or Sbθσ − lim Xk = Xo .
The set of all lacunary invariant statistically convergent sequences of fuzzy
numbers is denoted by Sθσ . In special case θ = (2r ), we shall write Sbσ instead
of Sbθσ .
Definition 1.3 Let θ = (kr ) be lacunary sequence and p = (pk ) be any
sequence of strictly positive real numbers. A sequence X = (Xk ) is said to be
lacunary strongly invariant convergent if there is a fuzzy number Xo such that
lim h−1
r
r→∞
X
[d (tkn (X) , Xo )]pk = 0
k∈Ir
cσ , p .
uniformly in n. In this case we write Xk → Xo M
θ
The set of all lacunary
strongly
invariant convergent sequences of fuzzy
c σ , p . In special case θ = (2r ) and p = 1 for all
numbers is denoted by M
k
θ
c σ , p , respectively.
k ∈ N , we shall write Cb σ , p and Cbθσ instead of M
θ
53
Lacunary Invariant Statistical...
2
Main Results
In this section, we prove the results of this paper.
Theorem 2.1 Let X = (Xk ) and Y = (Yk ) be sequences of fuzzy numbers.
(i) If Sbθσ − lim Xk = Xo and a ∈ R, then Sbθσ − lim aXk = aXo .
(ii) Sbθσ − lim Xk = Xo and Sbθσ − lim Yk = Yo , then Sbθσ − lim (Xk + Yk ) =
Xo + Yo .
Proof. (i) Let Xk → Xo Sbθσ and a ∈ R. Then by taking into account
the properties (1) and (2) of the metric d, we have
h−1
r
|{k ∈ Ir : d (tkn (aX) , aX0 ) ≥ ε}| ≤
h−1
r
k
ε ∈ Ir : d (tkn (X) , Xo ) ≥
a which yields that Sbθσ − lim aXk = aXo .
(ii) By combining the Minkowski’s inequality with the property (2) of the
metric d, we derive that
d (tkn (X) + tkn (Y ) , Xo + Yo ) ≤ d (tkn (X) , Xo ) + d (tkn (Y ) , Yo ) .
Therefore given ε > 0 we have,
h−1
r |{k ∈ Ir : d (tkn (X) + tkn (Y ) , Xo + Yo ) ≥ ε}|
≤ h−1
r |{k ∈ Ir : d (tkn (X) , Xo ) + d (tkn (Y ) , Yo ) ≥ ε}|
ε k ∈ Ir : d (tkn (X) , Xo ) ≥
≤ h−1
r 2 ε +h−1
,
r k ∈ Ir : d (tkn (X) , Yo ) ≥
2 which yields that that Sbθσ − lim (Xk + Yk ) = Xo + Yo .
The following result is a consequence of Theorem 2.1.
Corollary 2.2 Let X = (Xk ) and Y = (Yk ) be sequences of fuzzy numbers.
(iii) If Sbσ − lim Xk = Xo and a ∈ R, then Sbσ − lim aXk = aXo .
(iv) Sbσ − lim Xk = Xo and Sbσ − lim Yk = Yo , then Sbσ − lim (Xk + Yk ) =
Xo + Yo .
Theorem 2.3 Let θ = (kr ) be lacunary sequence and let X = (Xk ) be a
sequence of fuzzy numbers. Then, cσ , p .
(i) For lim inf qr > 1, we have Cb σ , p ⊂ M
θ
cσ , p .
(ii) For lim sup qr < ∞, we have Cb σ , p ⊃ M
θ
c σ , p if 1 < lim inf q ≤ lim sup q < ∞.
(iii) Cb σ , p = M
r
r
θ
54
Mehmet Açıkgöz et al.
Proof. (i) Let lim inf qr > 1, then
there
exists δ > 0 such that qr ≥ 1 + δ
σ
b
for all r ≥ 1. Then for X = (Xk ) ∈ C , p , we write
h−1
r
X
kr
X
[d (tkn (X) , Xo )]pk = h−1
r
[d (tkn (X) , Xo )]pk
k=1
k∈Ir
kr−1
−h−1
r
X
[d (tkn (X) , Xo )]pk
k=1

kr
X
k −1
= kr h−1
r
r

[d (tkn (X) , Xo )]pk 
k=1


kr−1
k −1
−kr−1 h−1
r
r−1
X
[d (tkn (X) , Xo )]pk  .
k=1
−1
≤ (1 + δ)δ−1 and kr−1 h−1
Since hr = kr −kr−1 , we have kr h−1
r ≤ δ . The terms
r
pk
pk
−1 Pkr
−1 Pkr
and kr−1 k=1 [d (tkn (X) , Xo )] both converge to
kr
k=1 [d (tkn (X) , Xo )]
cσ , p .
0 as r → ∞ uniformly in n. Therefore X = (Xk ) ∈ M
θ
(ii) Now suppose that lim sup qr < ∞, then
there exists A > 0 such that
σ
c
qr < A for all r ≥ 1. Let X = (Xk ) ∈ Mθ , p and ε > 0 be given. Then there
exists a number R > 0 such that
Ai = h−1
i
X
[d (tkn (X) , Xo )]pk < ε
k=Ir
for every i ≥ R and all n. We can also find K > 0 such that Ai < K for all
i = 1, 2, 3, · · ·. Now let m be any integer with kr−1 < m ≤ kr , where r > R.
Then
m−1
m
X
−1
[d (tkn (X) , Xo )]pk ≤ kr−1
k=1
=
−1
kr−1
kr
X
[d (tkn (X) , Xo )]pk
k=1
" P
pk
k=I1
[d (tkn (X) , Xo )] + k=I2 [d (tkn (X) , Xo )]pk + · · ·
P
+ k=Ir [d (tkn (X) , Xo )]pk
−1 −1
= (k1 − ko ) kr−1
k1
X
P
[d (tkn (X) , Xo )]pk
k=I1
+ (k2 −
−1 −1
k1 ) kr−1
k2
+ (kR −
−1
kR−1 ) kr−1
+ (kr −
−1
kr−1 ) kr−1
(k2 − k1 )−1
X
[d (tkn (X) , Xo )]pk + · · ·
k=I2
X
−1
(kR − kR−1 )
[d (tkn (X) , Xo )]pk + · · ·
k=IR
−1
(kr − kr−1 )
X
[d (tkn (X) , Xo )]pk
k=Ir
=
−1
k1 kr−1
A1
−1
k1 )kr−1
A2
+ (k2 −
−1
+ (kr − kr−1 ) kr−1
Ar
−1
AR + · · ·
+ · · · + (kR − kR−1 ) kr−1
#
55
Lacunary Invariant Statistical...
!
≤
!
−1
kR kr−1
sup Ai
i≥1
−1
−1
+ sup Ai (kr − kR ) kr−1
< KkR kr−1
+ εA.
i≥R
−1
Since kr−1 → ∞ as m → ∞, it follows
that
m
uniformly in n. Hence X = (Xk ) ∈ Cb σ , p .
(iii) Follows from (i) and (ii).
Pm
k=1
[d (tkn (X) , Xo )]pk → 0
Theorem 2.4 Let θ = (kr ) be lacunary sequence and let X = (Xk ) be a
sequence of fuzzynumbers.
Then,
σ
c
(i) Xk → Xo Mθ , p implies Xk → Xo Sbθσ .
F
cσ , p .
(ii) X = (Xk ) ∈ l∞
and Xk → Xo Sbθσ imply Xk → Xo M
θ
c σ , p if X = (X ) ∈ lF ,
(iii) Sbθσ = M
k
∞
θ
where 0 < h = inf k pk ≤ pk ≤ supk pk = H < ∞.
c σ , p . Then we can write
Proof. (i) ε > 0 and Xk → X0 M
θ
h−1
r
X
[d (tkn (X) , Xo )]pk = h−1
r
k∈Ir
X
[d (tkn (X) , Xo )]pk
k∈Ir , d(tkn (X),Xo )≥ε
+h−1
r
[d (tkn (X) , Xo )]pk
X
k∈Ir , d(tkn (X),Xo )<ε
≥
h−1
r
≥
h−1
r
X
[d (tkn (X) , Xo )]pk
k∈Ir , d(tkn (X),Xo )≥ε
X
[ε]pk
k∈Ir , d(tkn (X),Xo )≥ε
≥ h−1
r
X
h
min εh , εH
ipk
k∈Ir , d(tkn (X),Xo )≥ε
h
i
h H
≥ h−1
.
r |{k ∈ Ir : d (tkn (X), Xo ) ≥ ε}| min ε , ε
Hence Xk → Xo Sbθσ .
F
(ii) Suppose that X = (Xk ) ∈ l∞
and Xk → Xo Sbθσ . Since X = (Xk ) ∈
F
l∞
, there is a constant B > 0 such that d (tkn (X) , Xo ) ≤ B. Given ε > 0, we
have
h−1
r
X
k∈Ir
[d (tkn (X) , Xo )]pk = h−1
r
X
[d (tkn (X) , Xo )]pk
k∈Ir , d(tkn (X),Xo )≥ε
+h−1
r
X
[d (tkn (X) , Xo )]pk
k∈Ir , d(tkn (X),Xo )<ε
≤ h−1
r
h
max B h , B H
X
k∈Ir , d(tkn (X),Xo )≥ε
+h−1
r
X
k∈Ir , d(tkn (X),Xo )<ε
[ε]pk
i
56
Mehmet Açıkgöz et al.
h
i
≤ max B h , B H h−1
r |{k ∈ Ir : d (tkn (X) , Xo ) ≥ ε}|
h
i
+ max εh , εH .
cσ , p .
Hence Xk → Xo M
θ
(iii) Follows from (i) and (ii).
Theorem 2.5 Let θ = (kr ) be lacunary sequence and let X = (Xk ) be a
sequence of fuzzy numbers. Then,
(i) For lim sup qr < ∞, we have Sbθσ ⊂ Sbσ .
(ii) For lim inf qr > 1, we have Sbσ ⊂ Sbθσ .
(iii) if 1 < lim inf qr ≤ lim sup qr < ∞, then Sbθσ = Sbσ .
Proof. (i) If lim sup qr < ∞, then there
is a B > 0 such that qr < B
σ
b
for all r ≥ 1. Suppose that Xk → Xo Sθ and for each n ≥ 1 set Nrn =
|{k ∈ Ir : d (tkn (X) , Xo ) ≥ ε}|. Then there exists an ro ∈ N such that
Nrn h−1
r < ε f or all r > ro and n ≥ 1.
(5)
Now let M = max {Nrn : 1 ≤ r ≤ ro } and choose m such that kr−1 < m ≤ kr ,
then for each n ≥ 1 we have
m−1 |{k ≤ m : d (tkn (X), Xo ) ≥ ε}|
−1
≤ kr−1
|{k ≤ kr : d (tkn (X), Xo ) ≥ ε}|
n
−1
= kr−1
N1n + N2n + · · · + Nro n + N(ro +1)n + · · · + Nrn
n
o
−1
−1
≤ kr−1
M ro + kr−1
hro +1 (hro +1 )−1 N(ro +1)n + · · · + hr h−1
r Nrn
o
!
≤
−1
kr−1
M ro
+
−1
kr−1
sup
r>ro
h−1
r Nrn
{hro +1 + · · · + hr }
−1
−1
≤ kr−1
M ro + kr−1
ε (kr − kro ) by(5)
−1
≤ kr−1 M ro + εqr
−1
≤ kr−1
M ro + Bε.
This completes the proof.
(ii) Suppose that lim inf qr > 1. Then there exists a δ > 0 such that
−1
qr ≥ 1 + δ for sufficiently large
r ≥ 1. Since hr = kr − kr−1 , we have hr kr ≤
−1
(1 + δ) δ. Let Xk → Xo Sbσ . Then for every ε > 0 and for all n, we have
kr−1 |{k ≤ kr : d (tkn (X) , Xo ) ≥ ε}| ≥ kr−1 |{k ∈ Ir : d (tkn (X) , Xo ) ≥ ε}|
≥ (1 + δ)−1 δh−1
r |{k ∈ Ir : d (tkn (X), Xo ) ≥ ε}|
Hence Sbσ ⊂ Sbθσ .
(iii) Follows from (i) and (iii).
57
Lacunary Invariant Statistical...
cσ , q ⊂
Theorem 2.6 Let 0 < pk ≤ qk and pk qk−1 be bounded. Then M
θ
cσ , p .
M
θ
qk
c σ , q . Let w
and
Proof. Let X = (xk ) ∈ M
k,n = (d (tkn (X) , Xo ))
θ
−1
λk = pk qk for all k ∈ N . We define the sequences (uk,n ) as follows: For
wk,n ≥ 1, let uk,n = wk,n and vk,n = 0 and for wk,n < 1, let uk,n = 0 and
vk,n = wk,n . Then it is clear that for all k ∈ N , we have wk,n = uk,n + vk,n
λk
λk
λk
λ
k
k
and wk,n
= uλk,n
+ vk,n
. Now it follows that uλk,n
≤ uk,n ≤ wk,n and vk,n
≤ vk,n
.
Therefore
h−1
r
X
λn
wn,m
= h−1
r
k∈Ir
X
(un,m + vn,m )λn ≤ h−1
r
k∈Ir
λ
wn,m + h−1
r vn,m .
X
k∈Ir
Now for each r,
h−1
r
X
X
λ
vn,m
=
k∈Ir
λ
hr−1 vn,m
λ h−1
r
1−λ
k∈Ir

≤ 
X λ
h−1
r vn,m
λ λ1
λ 
 
k∈Ir
λ
h−1
r vn,m
λ λ1
λ

k∈Ir
λ

= h−1
r
X X
vn,m 
k∈Ir
and so
λ

h−1
r
X
λn
wn,m
≤ h−1
r
k∈Ir
X
wn,m + h−1
r
k∈Ir
X
vn,m  .
k∈Ir
c σ , p , i.e.
cσ , q
Hence X = (Xk ) ∈ M
M
θ
θ
following result is easy and thus is omitted.
⊂
c σ , p . The proof of the
M
θ
cσ , p ⊂ M
cσ .
Theorem 2.7 (i) Let 0 < inf k pk ≤ 1, then M
θ
θ
cσ ⊂ M
cσ , p .
(ii) Let 0 < pk ≤ supk pk ≤ ∞, then M
θ
θ
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