Journal of Mathematical Analysis
ISSN: 2217-3412, URL: http://www.ilirias.com
Volume 4 Issue 3(2013), Pages 33-46.
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL
CONVERGENCE OF SET SEQUENCES
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
Abstract. In this paper we presents three definitions which is a natural combination of the definition of asymptotic equivalence, statistical convergence,
lacunary statistical convergence, σ-statistical convergence and Wijsman convergence. In addition, we also present asymptotically equivalent sequences of
sets in sense of Wijsman and study some properties of this concept.
1. Introduction
Marouf in [20], presented definitions for asymptotically equivalent and asymptotic regular matrices. Pobyvancts in [28], introduced the concepts of asymptotically regular matrices, which preserve the asymptotic equivalence of two nonnegative numbers sequences. Patterson in [26], extend these concepts by presenting
an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Patterson and Savaş in
[27], introduced the concepts of an asymptotically lacunary statistical equivalent
sequences of real numbers. The concepts of σ-asymptotically lacunary statistical
equivalent sequences introduced by Patterson and Savas in [32]. Braha, in [7] introduced the concepts of asymptotically ∆m -lacunary statistical equivalent sequences
of real numbers.
The concept of convergence of sequences of points has been extended by several authors to convergence of sequences of sets. The one of these such extensions
considered in this paper is the concept of Wijsman convergence. The concept of Wijsman statistical convergence which is implementation of the concept of statistical
convergence to sequences of sets presented by Nuray and Rhoades in [25]. Similar
to the concept, the concept of Wijsman lacunary statistical convergence presented
Ulusu and Nuray in [35]. Hazarika and Esi in [14], introduced the notion asymptotically Wijsman generalized statistical convergence of sequences of sets. For more
works on convergence of sequences of sets, we refer to ([1], [2], [3], [4], [5], [36], [37],
[38]).
The idea of statistical convergence was formerly given under the name “almost
convergence” by Zygmund in the first edition of his celebrated monograph published
in Warsaw in 1935 [39]. The concept was formally introduced by Steinhaus [34] and
2010 Mathematics Subject Classification. 40A35, 40G15.
Key words and phrases. Asymptotic equivalence; statistical convergence; lacunary statistical
convergence; strongly σ-convergence; Wijsman convergence.
c
2013
Ilirias Publications, Prishtinë, Kosovë.
Submitted Jun 22, 2013. Published November 8, 2013.
33
34
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
Fast [9] and later was introduced by Schoenberg [33], and also independently by
Buck [6]. A lot of developments have been made in this areas after the works
of S̆alát [29] and Fridy [11]. Over the years and under different names statistical
convergence has been discussed in the theory of Fourier analysis, ergodic theory and
number theory. Fridy and Orhan [12] introduced the concept of lacunary statistical
convergence. Mursaleen and Mohiuddine in [24], introduced the concept of lacunary
statistical convergence with respect to the intuitionistic fuzzy normed space. For
details related to lacunary statistical convergence, we refer to ([8], [12], [13], [15],
[17], [21]).
By a lacunary sequence θ = (kr ), where k0 = 0 , we shall mean an increasing
sequence of non-negative integers with hr : kr − kr−1 → ∞ as r → ∞. The intervals
kr
will be
determined by θ will be denoted by Jr = (kr−1 , kr ] and the ratio kr−1
defined by qr (see [10]).
In this paper we define asymptotically lacunary σ-statistical equivalent sequences
of sets in sense of Wijsman and establish some basic results regarding the notions
asymptotically lacunary σ-statistical equivalent sequences of sets in sense of Wijsman and asymptotically Wijsman lacunary statistical equivalent sequences of sets.
Now we recall the definitions of statistical convergence, lacunary statistical convergence, σ-statistical convergence and Wijsman convergence.
Definition 1.1. A real or complex number sequence x = (xk ) is said to be statistically convergent to L if for every ε > 0
1
lim |{k ≤ n : |xk − L| ≥ ε}| = 0.
n n
In this case, we write S − lim x = L or xk → L(S) and S denotes the set of all
statistically convergent sequences.
Definition 1.2. ([12]) A sequence x = (xk ) is said to be lacunary statistically
convergent to the number L if for every ε > 0
1
lim
|{k ∈ Jr : |xk − L| ≥ ε}| = 0.
r→∞ hr
Let Sθ denotes the set of all lacunary statistically convergent sequences. If θ = (2r ),
then Sθ is the same as S.
Let σ be a one-to-one mapping from the set of natural numbers into itself. A
continuous linear functional φ on `∞ is said to be an invariant mean or a σ-mean
if and only if
(1) φ(x) ≥ 0 when the sequence x = (xk ) is such that xk ≥ 0 for all k,
(2) φ(e) = 1 where e = (1, 1, 1, ...), and
(3) φ(x) = φ(xσ(k) ) for all x ∈ `∞ .
Throughout this paper we shall consider the mapping σ has having on finite
orbits, that is, σ m (k) 6= k for all nonnegative integers with m ≥ 1, where σ m (k)
is the m-th iterate of σ at k. We denote Vσ is the set of bounded sequences all of
whose σ-mean are equal. If σ(k) = k + 1, then it is the set of almost convergent
sequences in [18].
If x = (xk ), the set T x = (T xk ) = (xσ(k) ). Savas [30], introduced the following
notion.
n
o
Vσ = x = (xk ) : lim tnk (x) = Le uniformly in k, L = σ − lim x ,
n
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL ...
where tnk (x) =
35
(xk +T xk +...+T n xk )
.
n+1
Definition 1.3. ([31]) A sequence x = (xk ) is said to be σ-statistically convergent
to the number L if for every ε > 0
1 k ≤ n : xσk (m) − L ≥ ε = 0, uniformly on m.
lim
n→∞ n
Let Sσ denotes the set of all σ-statistically convergent sequences.
Definition 1.4. ([30], [32]) A sequence x = (xk ) is said to be strongly lacunary
σ-convergent to the number L if
(
)
1 X Lθ = x = (xk ) : lim
xσk (m) − L = 0, uniformly on m .
r→∞ hr
k∈Jr
Definition 1.5. ([31]) A sequence x = (xk ) is said to be lacunary σ-statistically
convergent to the number L if for every ε > 0
1 lim
k ∈ Jr : xσk (m) − L ≥ ε = 0, uniformly on m.
r→∞ hr
Let Sθ,σ denotes the set of all lacunary σ-statistically convergent sequences. If
θ = (2r ), then Sθ,σ is the same as Sσ .
Let (X, ρ) be a metric space. For any point x ∈ X and any non-empty subset
A ⊂ X, the distance from x to A is defined by
d(x, A) = inf ρ (x, y) .
y∈A
Definition 1.6. ([2]) Let (X,ρ) be a metric space. For any non-empty closed subsets
A, Ak ⊂ X (k ∈ N) , we say that the sequence (Ak ) is Wijsman convergent to A if
limk d(x, Ak ) = d(x, A) for each x ∈ X. In this case we write W − lim Ak = A.
2. Definitions and Notations
Definition 2.1. ([20]) Two nonnegative sequences x = (xk ) and y = (yk ) are said
to be asymptotically equivalent if
xk
lim
= 1,
k yk
denoted by x ∼ y.
Definition 2.2. ([26]) Two nonnegative sequences x = (xk ) and y = (yk ) are said
to be asymptotically statistical equivalent of multiple L provided that for every ε > 0
xk
1 lim k ≤ n : − L ≥ ε = 0,
n n
yk
L
denoted by x ∼S y and simply asymptotically statistical equivalent if L = 1.
Savas and Nuray in [31], defined the asymptotically σ-statistical equivalent and
strongly asymptotically σ-statistical equivalent sequences as follows:
Definition 2.3. Two nonnegative sequences x = (xk ) and y = (yk ) are said to be
asymptotically σ-statistical equivalent of multiple L provided that for every ε > 0
xσk (m)
1 − L ≥ ε = 0, uniformly on m,
lim k ≤ n : n n
yσk (m)
L
denoted by x ∼Sσ y and simply asymptotically σ-statistical equivalent if L = 1.
36
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
Definition 2.4. Two nonnegative sequences x = (xk ) and y = (yk ) are said to be
strongly asymptotically σ-statistical equivalent of multiple L provided that
n 1 X xσk (m)
lim
− L = 0, uniformly on m,
y k
n n
σ (m)
k=1
L
denoted by x ∼[Vσ ] y and simply strongly asymptotically σ-statistical equivalent if
L = 1.
The concepts of Wijsman statistical convergence and boundedless for the sequence (Ak ) were given by Nuray and Rhoades [25] as follows:
Definition 2.5. ([25]) Let (X, ρ) be a metric space. For any non-empty closed
subsets A, Ak ⊂ X (k ∈ N) , we say that the sequence (Ak ) is Wijsman statistical
convergent to A if the sequence (d(x, Ak )) is statistically convergent to d(x, A), i.e.,
for ε > 0 and for each x ∈ X
lim
n
1
|{k ≤ n : |d(x, Ak ) − d(x, A)| ≥ ε}| = 0.
n
In this case, we write st − limk Ak = A or Ak → A (W S). The sequence (Ak ) is
bounded if supk d(x, Ak ) < ∞ for each x ∈ X. The set of all bounded sequences of
sets denoted by L∞ .
Ulusu and Nuray in [36], defined asymptotically equivalent and asymptotically
statistical equivalent sequences of sets as follows:
Definition 2.6. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. We say that
the sequences (Ak ) and (Bk ) are asymptotically equivalent (Wijsman sense) if for
each x ∈ X,
d(x, Ak )
lim
= 1,
k d(x, Bk )
denoted by (Ak ) ∼ (Bk ).
Definition 2.7. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. We say
that the sequences (Ak ) and (Bk ) are asymptotically statistical equivalent (Wijsman
sense) if for every ε > 0 and for each x ∈ X,
d(x, Ak )
1
lim k ≤ n : − L ≥ ε = 0
n n
d(x, Bk )
L
denoted by (Ak ) ∼W S (Bk ) and simply asymptotically statistical equivalent (Wijsman sense) if L = 1.
3. Asymptotically Wijsman lacunary σ-statistical equivalent
sequences
In this section, we define asymptotically Wijsman σ-statistical and asymptotically Wijsman lacunary σ-statistical equivalent sequences of sets and proved some
interesting results.
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL ...
37
Definition 3.1. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. Two sequences
(Ak ) and (Bk ) are said to be asymptotically Wijsman σ-statistical equivalent of
multiple L provided that for every ε > 0
d(x, Aσk (m) )
1 lim k ≤ n : − L ≥ ε = 0, uniformly on m,
n n
d(x, B k )
σ (m)
L
W Sσ
denoted by Ak ∼
if L = 1.
Bk and simply asymptotically Wijsman σ-statistical equivalent
Definition 3.2. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. Two sequences (Ak ) and (Bk ) are said to be strongly asymptotically Wijsman σ-statistical
equivalent of multiple L provided that
n 1 X d(x, Aσk (m) )
lim
d(x, B k ) − L = 0, uniformly on m,
n n
σ (m)
k=1
L
denoted by Ak ∼W [Vσ ] Bk and simply strongly asymptotically Wijsman σ-statistical
equivalent if L = 1.
Definition 3.3. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. We say that
the sequences (Ak ) and (Bk ) are asymptotically Wijsman lacunary σ-equivalent of
multiple L if for each x ∈ X,
1 X d(x, Aσk (m) )
lim
= L, uniformly in m
r hr
d(x, Bσk (m) )
k∈Jr
L
denoted by (Ak ) ∼W (Lθ )
equivalent if L = 1.
(Bk ) and simply asymptotically Wijsman lacunary σ-
Definition 3.4. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. We say
that the sequences (Ak ) and (Bk ) are strongly asymptotically Wijsman lacunary
σ-statistical equivalent if for every ε > 0 and for each x ∈ X,
1 X d(x, Aσk (m) )
lim
− L = 0 uniformly in m
r hr
d(x, B k )
k∈Jr
σ (m)
L
denoted by (Ak ) ∼W [Lθ ] (Bk ) and simply strongly asymptotically Wijsman lacunary σ-equivalent if L = 1.
Definition 3.5. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. We say
that the sequences (Ak ) and (Bk ) are asymptotically Wijsman lacunary σ-statistical
equivalent if for every ε > 0 and for each x ∈ X,
d(x, Aσk (m) )
1 − L ≥ ε = 0 uniformly in m
lim
k ∈ Jr : r hr d(x, B k )
σ (m)
L
W Sθ,σ
denoted by (Ak ) ∼
(Bk ) and simply asymptotically Wijsman lacunary σstatistical equivalent if L = 1.
38
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
Theorem 3.6. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed subsets
X (k ∈ N) . Then
L
L
(a) (Ak ) ∼W [Lθ ] (Bk ) ⇒ (Ak ) ∼W Sθ,σ (Bk ).
L
(b) W [Lθ ]L is a proper subset of W Sθ,σ
.
L
L
(c) Let (Ak ) ∈ L∞ and (Ak ) ∼W Sθ,σ (Bk ), then (Ak ) ∼W [Lθ ] (Bk ).
L
(d) W Sθ,σ
∩ L∞ = W [Lθ ]L ∩ L∞ .
L
Proof. (a) Let ε > 0 and (Ak ) ∼W [Lθ ] (Bk ). Then we can write
X
X d(x, Ak )
d(x, Aσk (m) )
≥
−
L
−
L
d(x, B k )
d(x, Bk )
σ (m)
k∈Jr
d(x,A
)
σ k (m)
−L≥ε
d(x,B
)
σ k (m)
k∈Jr
d(x, Aσk (m) )
≥ ε k ∈ Jr : − L ≥ ε d(x, Bσk (m) )
which gives the result.
L
(b) Suppose that W [Lθ ]L ⊂ W Sθ,σ
. Let (Ak ) and (Bk ) be two sequences defined
as follows:
√
{k} , if kr−1 < k ≤ kr−1 + [ hr ], r = 1, 2, 3, ...;
Ak =
{0} ,
otherwise
and
Bk = {0} for all k ∈ N.
It is clear that (Ak ) ∈
/ L∞ and for ε > 0 and for each x ∈ X,
√
d(x, Aσk (m) )
[ hr ]
1 lim
− 1 ≥ ε = lim
k ∈ Jr : = 0.
r
r hr d(x, Bσk (m) )
hr
1
So (Ak ) ∼W Sθ,σ (Bk ), but
1 X d(x, Aσk (m) )
lim
d(x, B k ) − L 6= 0.
r hr
σ (m)
k∈Jr
W [Lθ ]L
Therefore (Ak ) 6∼
(Bk ).
L
(c) Suppose that (Ak ) ∼W Sλ,σ (Bk ) and (Ak ) ∈ L∞ . We assume that
d(x,A
)
k
d(x,Bσk (m) ) − L ≤ M for each x ∈ X and for all k ∈ N. Given ε > 0, we get
σ (m)
1 X d(x, Aσk (m) )
d(x, B k ) − L
hr
σ
(m)
k∈Jr
X
X
1
d(x, Aσk (m) )
d(x, Aσk (m) )
1
− L +
− L
=
hr
d(x, Bσk (m) )
hr
d(x, Bσk (m) )
k∈Jr
d(x,A
)
σ k (m)
−L≥ε
d(x,B
)
σ k (m)
M ≤
k ∈ Jr :
hr from which the result follows.
(d) It follows from (a), (b) and
k∈Jr
d(x,A
)
σ k (m)
−L<ε
d(x,B
)
σ k (m)
d(x, Aσk (m) )
≥ ε + ε,
−
L
d(x, B k )
σ (m)
(c).
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL ...
39
Theorem 3.7. Suppose for given η1 > 0 and every ε > 0 there exist n0 and m0
such that
d(x, Aσk (m) )
1 ≥ ε < η1
0
≤
k
≤
n
−
1
:
−
L
n
d(x, Bσk (m) )
L
for all n ≥ n0 and m ≥ m0 , then (Ak ) ∼W Sσ (Bk ).
Proof. Let η1 be given. For every ε > 0, choose n1 and m0 such that
d(x, Aσk (m) )
1 ≥ ε < η1 , for all n ≥ n1 ; m ≥ m0 .
0
≤
k
≤
n
−
1
:
−
L
n
d(x, B k )
2
σ (m)
(3.1)
It is sufficient to show that there exists n2 such that for n ≥ n1 and 0 ≤ m ≤ m0
d(x, Aσk (m) )
1 ≥ ε < η1 .
(3.2)
0
≤
k
≤
n
−
1
:
−
L
n
d(x, B k )
σ (m)
Let n0 = max{n1 , n2 }. The the relation (3.2) will be true for n > n0 and for all m.
If m0 chosen fixed, then we get
d(x, Aσk (m) )
0 ≤ k ≤ m0 − 1 : d(x, B k ) − L ≥ ε = M.
σ (m)
Now for 0 ≤ m ≤ m0 and n > m0 we have
d(x, Aσk (m) )
1 ≥ε 0
≤
k
≤
n
−
1
:
−
L
n
d(x, Bσk (m) )
d(x, Aσk (m) )
1 − L ≥ ε ≤ 0 ≤ k ≤ m0 − 1 : n
d(x, Bσk (m) )
d(x, Aσk (m) )
1 + m0 ≤ k ≤ n − 1 : − L ≥ ε n
d(x, Bσk (m) )
M
d(x, Aσk (m) )
M
1
η1
≤
+ m0 ≤ k ≤ n − 1 : − L ≥ ε ≤
+ .
n
n
d(x, Bσk (m) )
n
2
Thus for sufficiently large n
M
d(x, Aσk (m) )
1 η1
m0 ≤ k ≤ n − 1 : − L ≥ ε ≤
+
< η1 .
n
d(x, Bσk (m) )
n
2
This established the result.
Theorem 3.8. For every lacunary sequence θ, W Sθ,σ = W Sσ .
Proof. Let (Ak ) ∈ W Sθ,σ , then for given η1 > 0 there exist ε > 0 and L such that
d(x, Aσk (m) )
1
k ∈ Jr : − L ≥ ε < η1 ,
hr
d(x, Bσk (m) )
for r ≥ r0 and m = kr−1 + 1 + v where v ≥ 0.
Let n ≥ hr and write n = shr + u with 0 ≤ u ≤ hr and s an integer. Since
n ≥ hr and s ≥ 0. We have
d(x, Aσk (m) )
1 ≥ε −
L
0
≤
k
≤
n
−
1
:
n
d(x, Bσk (m) )
d(x, Aσk (m) )
1
− L ≥ ε ≤ 0 ≤ k ≤ (s + 1)hr − 1 : n
d(x, Bσk (m) )
40
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
s d(x, Aσk (m) )
1 X =
ihr ≤ k ≤ (i + 1)hr − 1 : − L ≥ ε n i=1 d(x, Bσk (m) )
(s + 1)hr
2shr η1
η1 ≤
for s ≥ 1.
n
n
≤ 1, since shnr ≤ 1. Therefore
d(x, Aσk (m) )
1 0≤k ≤n−1:
− L ≥ ε ≤ 2η1 .
n
d(x, Bσk (m) )
≤
For
hr
n
Hence by Theorem 3.7, we have W Sθ,σ ⊆ W Sσ . Also W Sσ ⊆ W Sθ,σ for every
lacunary sequence θ. This completes the proof of the theorem.
Theorem 3.9. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed subsets X (k ∈ N) . Let θ = (kr ) be a lacunary sequence with lim inf r qr > 1. Then
L
L
(Ak ) ∼W Sσ (Bk ) ⇒ (Ak ) ∼W Sθ,σ (Bk ).
Proof. Suppose that lim inf r qr > 1, then there exists a α > 0 such that qr ≥ 1 + α
for sufficiently large r. Then we have
α
hr
.
≥
kr
1+α
L
If (Ak ) ∼W Sσ (Bk ), then for every ε > 0 and for sufficiently large r we have
d(x, Aσk (m) )
1 ≥ε k
≤
k
:
−
L
r
hr
d(x, Bσk (m) )
d(x, Aσk (m) )
1 ≥
k ∈ Jr : − L ≥ ε kr
d(x, Bσk (m) )
d(x, Aσk (m) )
α
1 ≥
. k ∈ Jr : − L ≥ ε .
1 + α hr
d(x, Bσk (m) )
This completes the proof.
Theorem 3.10. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed
subsets X (k ∈ N) . Let θ = (kr ) be a lacunary sequence with lim supr qr < ∞. Then
L
L
(Ak ) ∼W Sθ,σ (Bk ) ⇒ (Ak ) ∼W Sσ (Bk ).
Proof. If lim supr qr < ∞. Then there exists an K > 0 such that qr < K for all
L
r ≥ 1. Let (Ak ) ∼W Sθ,σ (Bk ) and η1 > 0. Then there exists B > 0 and ε > 0 such
that for every j ≥ B
d(x, Aσk (m) )
1 Mj =
k ∈ Jj : − L ≥ ε < η1 for all m.
hj
d(x, B k )
σ (m)
Also we can find A > 0 such that Mj < A for all j = 1, 2, 3, ... Now let i be an
integer with satisfying kr−1 < i ≤ kr , where r > B. Then we can write
d(x, Aσk (m) )
1 ≥ε k
≤
i
:
−
L
n
d(x, Bσk (m) )
d(x, Aσk (m) )
1 ≥ε ≤
k
≤
k
:
−
L
r
kr−1
d(x, Bσk (m) )
d(x, Aσk (m) )
1 k ∈ J1 : − L ≥ ε =
kr−1
d(x, Bσk (m) )
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL ...
d(x, Aσk (m) )
k ∈ J2 : ≥ε −
L
kr−1
d(x, Bσk (m) )
d(x, Aσk (m) )
1 +... +
k ∈ Jr : − L ≥ ε kr−1 d(x, Bσk (m) )
d(x, Aσk (m) )
k1 =
k
∈
J
:
≥
ε
−
L
1 kr−1 k1
d(x, Bσk (m) )
d(x, Aσk (m) )
k2 − k1
+
k ∈ J2 : − L ≥ ε kr−1 (k2 − k1 ) d(x, Bσk (m) )
d(x, Aσk (m) )
kB − kB−1
≥ε +... +
k
∈
J
:
−
L
B
kr−1 (kB − kB−1 )
d(x, Bσk (m) )
d(x, Aσk (m) )
kr − kr−1
k ∈ Jr : +... +
− L ≥ ε kr−1 (kr − kr−1 )
d(x, Bσk (m) )
k1
k2 − k1
kB − kB−1
=
+
M2 + ... +
MB
kr−1 M1
kr−1
kr−1
kB+1 − kB
kr − kr−1
+
MB+1 + ... +
Mr
kr−1
kr−1
kB
kB
kr − kB
≤ sup Mi
+ sup Mi
≤A
+ η1 K.
kr−1
kr−1
kr−1
i≥1
i≥B
This completes the proof of the theorem.
+
41
1
Theorem 3.11. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed
subsets X (k ∈ N) . Let θ = (kr ) be a lacunary sequence with 1 < lim inf r qr ≤
L
L
lim supr qr < ∞. Then (Ak ) ∼W Sθ,σ (Bk ) ⇔ (Ak ) ∼W Sσ (Bk ).
Proof. The result is an immediate consequence of Theorem 3.9 and Theorem 3.10.
Definition 3.12. Let p ∈ (0, ∞). Let (X, ρ) be a metric space. For any nonempty closed subsets Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for
each x ∈ X. We say that the sequences (Ak ) and (Bk ) are strongly asymptotically
Wijsman lacunary σp -statistical equivalent if for every ε > 0 and for each x ∈ X,
p
1 X d(x, Aσk (m) )
lim
d(x, B k ) − L = 0 uniformly in m
r hr
k∈Jr
σ (m)
L
denoted by (Ak ) ∼W [Lθ ]p (Bk ) and simply strongly asymptotically Wijsman lacunary σ-equivalent if L = 1.
Definition 3.13. Let p ∈ (0, ∞). Let (X, ρ) be a metric space. For any non-empty
closed subsets Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X.
We say that the sequences (Ak ) and (Bk ) are asymptotically Wijsman lacunary
σp -statistical equivalent if for every ε > 0 and for each x ∈ X,
p
d(x, Aσk (m) )
1 ≥ ε = 0 uniformly in m
lim
−
L
k
∈
J
:
r
r hr
d(x, Bσk (m) )
W SL
denoted by (Ak ) ∼ θ,σp (Bk ) and simply asymptotically Wijsman lacunary σstatistical equivalent if L = 1.
42
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
The proof of the following theorem follows from Theorem 3.6.
Theorem 3.14. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed
subsets X (k ∈ N) . Then
W SL
L
(a) (Ak ) ∼W [Lθ ]p (Bk ) ⇒ (Ak ) ∼ θ,σp (Bk ).
L
(b) W [Lθ ]L
p is a proper subset of W Sθ,σp .
W SL
L
(c) Let (Ak ) ∈ L∞ and (Ak ) ∼ θ,σp (Bk ), then (Ak ) ∼W [Lθ ]p (Bk ).
L
(d) W Sθ,σ
∩ L∞ = W [Lθ ]L
p ∩ L∞ .
p
4. Orlicz asymptotically Wijsman lacunary σ-statistical equivalent
sequences
In this section we define the notion of Orlicz asymptotically Wijsman lacunary
σ-statistical equivalence sequences of sets. An Orlicz function is a function M :
[0, ∞) → [0, ∞) which is continuous, nondecreasing and convex with M (0) = 0,
M (x) > 0 for x > 0 and M (x) → ∞ as x → ∞. An Orlicz function M is said to
satisfy the ∆2 − condition for all values of u, if there exists a constant K > 0 such
that M (2u) ≤ KM (u), u ≥ 0. Note that, if 0 < λ < 1,then M (λx) ≤ λM (x) , for
all x ≥ 0 (see [16]).
Also we will introduce Cesaro Orlicz asymptotically Wijsman σ-statistical equivalence between two sequences.
Definition 4.1. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. Two sequences
(Ak ) and (Bk ) are said to be M -strongly Cesaro Orlicz asymptotically Wijsman σstatistical equivalent of multiple L provided that for every
n
d(x, Aσk (m) )
1X
M − L = 0,
lim
n n
d(x, B k )
k=1
σ (m)
|σ1 |W M
(denoted by Ak ∼ Bk ), and simply W M -strongly Cesaro Orlicz asymptotically
Wijsman σ- equivalent if L = 1.
Definition 4.2. Let (X, ρ) be a metric space. For any non-empty closed subsets
Ak , Bk ⊆ X such that d(x, Ak ) > 0 and d(x, Bk ) > 0 for each x ∈ X. Two sequences
(Ak ) and (Bk ) are said to be M -lacunary strongly Cesaro Orlicz asymptotically
Wijsman σ-statistical equivalent of multiple L provided that for every
d(x, Aσk (m) )
1 X
lim
M − L = 0,
r hr
d(x, B k )
k∈Jr
σ (m)
|σ1 |LW M
(denoted by Ak
∼
Bk ), and simply LW M -strongly Cesaro Orlicz asymptotically Wijsman σ- equivalent if L = 1.
Theorem 4.3. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed subsets of X, (k ∈ N) . M be an Orlicz function which satisfies the ∆2 conditions and
L
W SMθ,σ
M (1) = 1. Then for two sequences (An ) and (Bn ) are said to be Ak
∼
Bk of
multiple L provided that for every > 0,
d(x, Aσk (m) )
1
the number of k ∈ Jr , : M − L ≥ = 0.
lim
r hr
d(x, Bσk (m) )
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL ...
If limr inf qr > 1, then Ak
L
W Sθ,σ
∼
Bk implies x
L
W SMθ,σ
∼
43
Bk .
Proof. Suppose that limr inf qr > 1, then there exists a δ > 0, such that qr ≥ 1 + α
for sufficiently large. Then we have
α
hr
≥
.
kr
1+α
If Ak
L
W Sθ,σ
∼
Bk , then for every > 0 and for sufficiently large r, we have
d(x, Aσk (m) )
1
the number of k ∈ Jr : M − L ≥ =
hr
d(x, Bσk (m) )
d(x, Aσk (m) )
kr 1
·
the number of k ∈ Jr : M − L ≥ ≤
hr kr
d(x, Bσk (m) )
d(x, Aσk (m) )
1+α 1
·
− L ≥ .
the number of k ∈ Jr : M α
kr
d(x, Bσk (m) )
(4.1)
In other side from fact that M satisfies the ∆2 conditions it follows that
d(x, Aσk (m) )
d(x, Aσk (m) )
M − L ≤ K · − L ,
d(x, Bσk (m) )
d(x, Bσk (m) )
d(x,A
)
k
for some constant K > 0, in both cases where d(x,Bσk (m) ) − L ≤ 1 and
σ (m)
d(x,A
)
k
d(x,Bσk (m) ) − L ≥ 1. Really in first case it follows directly from definition of the
σ (m)
Orlicz function. In second case we have
d(x, Aσk (m) )
(1)
2
(2)
s
(s)
d(x, B k ) − L = 2 · L = 2 · L = · · · = 2 · L ,
σ (m)
such that L(s) ≤ 1. Now taking into consideration ∆2 conditions of Orlicz functions,
we get the following estimation:
d(x, Aσk (m) )
d(x, Aσk (m) )
M (4.2)
− L ≤ T · L(s) · M (1) = K · − L ,
d(x, Bσk (m) )
d(x, Bσk (m) )
where T and K are constants. Now proof of Theorem follows from relations (4.1)
and (4.2).
Theorem 4.4. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed subsets
of X, for any k ∈ N and M be an Orlicz function. If lim supr qr < ∞, then from
Ak
L
W SMθ,σ
∼
Bk implies x
L
W Sθ,σ
∼
Bk .
Proof. Proof of the theorem is similar to Theorem 3.10, so we omit them.
The proof of the following results we omit, because they are similar to the fact
which we have prove above.
Theorem 4.5. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed subsets
of X, for any k ∈ N and M be an Orlicz function. The following relations are valid:
(i): If lim inf r qr > 1, then Ak
|σ1 |W M
(ii): If lim supr qr < ∞, then Ak
∼
B k ⇒ Ak
|σ1 |LW M
∼
|σ1 |LW M
B k ⇒ Ak
∼
Bk .
|σ1 |W M
∼
Bk .
44
BIPAN HAZARIKA, AYHAN ESI, N. L. BRAHA
(iii): If 1 < lim inf r qr ≤ lim supr qr < ∞, then Ak
Bk .
|σ1 |LW M
∼
Bk ⇐⇒ Ak
|σ1 |W M
∼
Theorem 4.6. Let (X, ρ) be a metric space and Ak , Bk be non-empty closed subsets
of X, for any k ∈ N and M be an Orlicz function. Then
W SL
|σ1 |W M
1: (a)If Ak ∼ Bk then Ak ∼σ Bk .
: (b) W SσL is a proper subset of |σ1 |W M .
W SL
2: If M satisfies the ∆2 condition and Ak ∈ l∞ (|σ1 |W M ) such that Ak ∼σ
|σ1 |W M
Bk , then Ak ∼ Bk .
3: If M satisfies the ∆2 condition, then SWσL ∩ l∞ (|σ1 |W M ) = (|σ1 |W M ) ∩
l∞ (|σ1 |W M ), where l∞ (|σ1 |W M ) = {(Ak ) : M (d(x, Aσk (m) )) ∈ l∞ , x ∈ X}
|σ1 |W M
Proof. 1: (a) If > 0 and Ak ∼ Bk , then
d(x, Aσk (m) )
1
− L ≥ ≤
the number of k ≤ n : n
d(x, Bσk (m) )
d(x, Aσk (m) )
1
the number of k ≤ n : M − L ≥ M ()
n
d(x, Bσk (m) )
n
n
X
d(x, Aσk (m) )
d(x, Aσk (m) )
1
1X
≤
− L ≤
M − L
M n
d(x, B k )
n
d(x, B k )
σ (m)
d(x,A k=1)
σ k (m)
M d(x,B
−L ≥M ()
)
σ k (m)
k=1
σ (m)
(b) To prove this fact it is enough to find a sequence of sets (Ak ) such that Ak ∈
/ |σ1 |W M . Let L = 1 and M = x. We define d(x, Aσk (m) ) to be
W SθL and Ak ∈
√
1, 2, · · · , [ n]
√
for k = i, for i ∈ {1, 2, · · · , [ n]}, and d(x, Aσk (m) ) = 1 otherwise. d(x, Bσk (m) ) = 1
for all k. It seems that (Ak ) is not d−metric bounded. In what follows we will prove
/ |σ1 |W M .
that Ak ∈ W SθL and Ak ∈
Let > 0 be given, then we have:
√
d(x, Aσk (m) )
1
[ n]
the number of k ≤ n : − L ≥ =
→ 0,
n
d(x, B k )
n
σ (m)
W SθL .
when n → ∞, hence Ak ∈
On the other hand
√
√
n
X
d(x, Aσk (m) )
[ n] ([ n] − 1)
√
1
M − L = 0 + 1 + 2 + · · · + [ n] − 1 =
,
n
d(x, Bσk (m) )
2n
k=1
from which follows that Ak ∈
/ |σ1 |W N.
2: Proof of this part is similar to proof of the above Theorems.
3: Proof follows from part 1 and 2.
Theorem 4.7. Let M be an Orlicz function and sup qr < ∞. Then
L
W Sθ,σ
|σ1 |LW M
1: (a)If Ak
∼
Bk then Ak ∼ Bk .
L
: (b) Sθ,σ
is a proper subset of |σ1 |LW M .
L
Sθ,σ
2: If M satisfies the ∆2 condition and Ak ∈ l∞ (|σ1 |LW M ), such that Ak ∼
Bk , then Ak
|σ1 |LW M
∼
Bk .
ON ASYMPTOTICALLY WIJSMAN LACUNARY σ-STATISTICAL ...
45
L
3: If M satisfies the ∆2 condition, then Sθ,σ
∩ l∞ (|σ1 |LW M ) = |σ1 |LW M ∩
l∞ (|σ1 |LW M ).
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Bipan Hazarika
Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791112,
Arunachal Pradesh, India.
E-mail address: bh [email protected]
Ayhan Esi
Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040,
Adiyaman, Turkey.
E-mail address: [email protected]
N. L. Braha
Department of Mathematics and Computer Sciences, Avenue ”Mother Theresa”, 5, Prishtinë, 10000, Kosova.
E-mail address: [email protected]