Journal of Applied Mathematics & Bioinformatics, vol.3, no.3, 2013, 75-88
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2013
On strongly lacunary summability
of sequences of sets
Uğur Ulusu1 and Fatih Nuray2
Abstract
In this paper, we introduce the concept of Wijsman strongly lacunary summability for set sequences. Then, we discus its relation with
Wijsman strongly Cesàro summability. Furthermore, we also give its
relation with Wijsman almost convergence.
Mathematics Subject Classification: 40A05; 40A35
Keywords: Lacunary summability; Strongly lacunary summability; Strongly
Cesàro Summability; Sequence of sets; Wijsman convergence
1
Introduction
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
1
2
Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe
University, Afyonkarahisar, Turkey.
Department of Mathematics, Faculty of Science and Literature, Afyon Kocatepe
University, Afyonkarahisar, Turkey.
Article Info: Received : May 30, 2013. Revised : July 23, 2013
Published online : September 15, 2013
76
On strongly lacunary...
extensions considered in this paper is the concept of Wijsman convergence
(see, [2,3,4,11,14,15,16]). Baronti and Papini studied the relationship between
Wijsman convergence and other types of convergence for sequence of sets in [2].
In [11], Nuray and Rhoades extended the notion of convergence of set sequences
to statistical convergence, and gave some basic theorems. In [14], Ulusu and
Nuray defined the Wijsman lacunary statistical convergence of sequence of
sets, and considered its relation with Wiijsman statistical convergence, which
was defined by Nuray and Rhoades.
Freedman, et al. established the connection between the strongly Cesàro
summable sequences space |σ1 | and the strongly lacunary summable sequences
space Nθ in their work [7] published in 1978.
This paper is inspired by the paper [7] by Freedman, Sember and Raphael
entitled ”Some Cesàro type summability spaces”. We introduce the concept of
Wijsman strongly lacunary summability for sequences of sets. Then, we discus
its relation with Wijsman strongly Cesàro summability, which was defined by
Nuray and Rhoades. Furthermore, we also give its relation with Wijsman
almost convergence, which was again defined by Nuray and Rhoades.
Let us start with fundamental definitions from the literature;
By a lacunary sequence we mean an increasing integer sequence θ = {kr }
such that k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. Throughout this
paper the intervals determined by θ will be denoted by Ir = (kr−1 , kr ] and let
kr
qr = kr−1
.
Let (X, ρ) be a metric space. For any point x ∈ X and any non-empty
subset A of X, we define the distance from x to A by
d(x, A) = inf ρ(x, A).
a∈A
Let (X, ρ) be a metric space. For any non-empty closed subsets A, Ak ⊆ X,
we say that the sequence {Ak } is Wijsman convergent to A if
lim d(x, Ak ) = d(x, A)
k→∞
for each x ∈ X. In this case we write W − lim Ak = A.
The concepts of Wijsman strongly Cesàro summability was introduced by
Nuray and Rhoades as follows: Let (X, ρ) be a metric space. For any nonempty closed subsets A, Ak ⊆ X, we say that {Ak } is Wijsman strongly Cesàro
77
U. Ulusu and F. Nuray
summable to A if for each x ∈ X,
n
1X
lim
|d(x, Ak ) − d(x, A)| = 0.
n→∞ n
k=1
[W σ1 ]
In this case we write Ak → A[W σ1 ] or Ak → A.
Also the concept of Wijsman strongly almost convergence for sequences of
sets was given by Nuray and Rhoades as follows: Let (X, ρ) be a metric space.
For any non-empty closed subsets A, Ak ⊆ X, we say that {Ak } is Wijsman
strongly almost convergent to A if for each x ∈ X,
n
1X
|d(x, Ak+i ) − d(x, A)| = 0
lim
n→∞ n
k=1
uniformly in i.
2
Main results
In this section, we will define Wijsman strongly lacunary summability of
sequences of sets. Then, we will give its the relationship between the Wijsman
strongly Cesàro summability and the Wijsman almost convergence.
Definition 2.1. Let (X, ρ) be a metric space and θ = {kr } be a lacunary
sequence. For any non-empty closed subsets A, Ak ⊆ X, we say that the
sequence {Ak } is Wijsman lacunary summable to A if for each x ∈ X,
1 X
d(x, Ak ) = d(x, A).
r→∞ hr
k∈I
lim
r
In this case we write Ak → A(W Nθ ).
Definition 2.2. Let (X, ρ) be a metric space and θ = {kr } be a lacunary
sequence. For any non-empty closed subsets A, Ak ⊆ X, we say that the
sequence {Ak } is Wijsman strongly lacunary summable to A if for each x ∈ X,
1 X
|d(x, Ak ) − d(x, A)| = 0.
r→∞ hr
k∈I
lim
r
78
On strongly lacunary...
[W N ]
In this case we write Ak → A([W N ]θ ) or Ak → θ A.
The set of Wijsman strongly lacunary summable sequences will be denoted
)
(
1 X
|d(x, Ak ) − d(x, A)| = 0 .
[W N ]θ = {Ak } : lim
r hr
k∈I
r
Example 2.3. Let X = R2 and we define a sequence {Ak } as follows:
 ©
ª
£√ ¤
1
2
2
2

, if kr−1 < k < kr−1 + hr ,
 (x, y) ∈ R : x + (y − 1) = k
Ak :=


{(0, 0)}
, otherwise.
This sequence is Wijsman strongly lacunary summable to the set A =
{(0, 0)} since
p
1
1 X
|d(x, Ak ) − d(x, {(0, 0)})| ≤ lim
· 2 kr − kr−1 = 0.
r→∞ hr
r→∞ hr
k∈I
lim
r
i.e., {Ak } ∈ [W N ]θ .
Definition 2.4. Let (X, ρ) be a metric space and θ = {kr } be a lacunary
sequence. For any non-empty closed subsets A, Ak ⊆ X, we say that the
sequence {Ak } is Wijsman strongly p−lacunary summable to A if for each p
positive real number and for each x ∈ X,
1 X
|d(x, Ak ) − d(x, A)|p = 0.
r→∞ hr
k∈I
lim
r
In this case we write Ak → A([W N ]pθ ).
Lemma 2.5. For any lacunary sequence θ = {kr }, [W σ1 ] ⊆ [W N ]θ if and
only if lim inf r qr > 1.
Proof. Suppose first that lim inf r qr > 1, then there exists a λ > 0 such
that
qr ≥ 1 + λ for all r ≥ 1, which implies that
1+λ
kr
≤
hr
λ
and
kr−1
1
≤ .
hr
λ
79
U. Ulusu and F. Nuray
If {Ak } ∈ |W σ1 |, then we can write
1 X
|d(x, Ak ) − d(x, A)|
hr k∈I
r
kr−1
kr
1 X
1 X
=
|d(x, Ai ) − d(x, A)| −
|d(x, Ai ) − d(x, A)|
hr i=1
hr i=1
Ã
!
Ã
!
kr−1
kr
kr 1 X
kr−1
1 X
=
|d(x, Ai ) − d(x, A)| −
|d(x, Ai ) − d(x, A)| .
hr kr i=1
hr
kr−1 i=1
Since {Ak } ∈ |W σ1 |, the terms
kr
1 X
|d(x, Ai ) − d(x, A)|
kr i=1
and
kr−1
1 X
kr−1
|d(x, Ai ) − d(x, A)|
i=1
both convergent to 0, and it follows that
1 X
|d(x, Ak ) − d(x, A)| → 0,
hr k∈I
r
that is, {Ak } ∈ [W N ]θ .
Conversely, suppose that [W σ1 ] ⊆ [W N ]θ and lim inf r qr = 1. Since θ is
© ª
lacunary sequence, we can select a subsequence krj of θ satisfying
krj
1
<1+
krj −1
j
and
krj −1
> j,
krj−1
where rj ≥ rj−1 + 2.
Now we define a sequence {Ak } as follows:
 ©
ª
1
2
2
2

, if k ∈ Irj ,
 (x, y) ∈ R , x + (y − 1) = k4
Ak :=


{(0, 0)}
, otherwise.
j = 1, 2, · · · ,
Then, for at least one x ∈ X, we have
1 X
|d(x, Ak ) − d(x, {(0, 0)})| = T
hrj k∈I
for j = 1, 2, · · · ,
rj
and
1 X
|d(x, Ak ) − d(x, {(0, 0)})| = 0
hr k∈I
r
for r 6= rj .
(T ∈ R+ )
80
On strongly lacunary...
It follows that {Ak } ∈
/ [W Nθ ] since
1 X
|d(x, Ak ) − d(x, {(0, 0)})| 6= 0.
hr k∈I
r
However, {Ak } is Wijsman strongly Cesàro summable, since if n is any sufficiently large integer we can find the unique j for which krj −1 < n < krj+1 −1
and write
n
kr + hrj
1X
1 1
2
|d(x, Ak ) − d(x, {(0, 0)})| ≤ j−1
≤ + = .
n k=1
krj −1
j j
j
As n → ∞ it follows that also j → ∞. Hence {Ak } ∈ |W σ1 |. This contradicts
to our assumption. Therefore, lim inf r qr > 1.
Lemma 2.6. For any lacunary sequence θ = {kr }, [W N ]θ ⊆ [W σ1 ] if and
only if lim supr qr < ∞.
Proof. Suppose first that lim supr qr < ∞, then there exists M > 0 such that
qr < M for all r. Let {Ak } ∈ [W Nθ ] and ε > 0. Then we can find R > 0 and
K > 0 such that
sup τi < ε
and
τi < K
i≥R
where
τr =
for all i = 1, 2, · · · ,
1 X
|d(x, Ak ) − d(x, A)|.
hr I
r
If t is any integer with kr−1 < t ≤ kr , where r > R, we can write
t
1X
|d(x, Ai ) − d(x, A)|
t i=1
≤
=
kr
1 X
kr−1
|d(x, Ai ) − d(x, A)|
i=1
Ã
X
1
kr−1
+
|d(x, Ai ) − d(x, A)| +
X
I1
X
Ir−1
|d(x, Ai ) − d(x, A)| +
I2
X
Ir
|d(x, Ai ) − d(x, A)| + · · ·

|d(x, Ai ) − d(x, A)|
81
U. Ulusu and F. Nuray
k1
k2 − k1
kR − kR−1
.τ1 +
.τ2 + · · · +
τR
kr−1
kr−1
kr−1
kR+1 − kR
kr − kr−1
+
τR+1 + · · · +
τr
kr−1
kr−1
µ
¶
µ
¶
kR
kr − kR
≤ supτi
+ supτi
kr−1
kr−1
i≥1
i≥R
kR
+ ε · M.
≤K·
kr−1
≤
Since kr−1 → ∞ as t → ∞, it follows that
t
1X
|d(x, Ai ) − d(x, A)| → 0
t i=1
and, consequently, {Ak } ∈ |W σ1 |.
Conversely, we assume that [W Nθ ] ⊆ [W σ1 ] and lim supr qr = ∞. To
proceed we construct a sequence in [W Nθ ] that is not Wijsman strongly Cesàro
summable. First select a subsequence (krj ) of the lacunary sequence θ = {kr }
such that qrj > j, and then we define the sequence {Ak } as follows:


j = 1, 2, · · · ,
 {1} , if krj −1 < k ≤ 2krj −1 ,
Ak :=


{0} , otherwise.
Then, we have
τrj =
krj −1
1 X
1
|d(x, Ak ) − d(x, {0})| =
<
,
hrj I
krj −1 − krj−1
j−1
rj
and
τr =
1 X
|d(x, Ak ) − d(x, {0})| = 0,
hr I
for r 6= rj .
r
It follows that {Ak } ∈ [W Nθ ] since
1 X
lim
|d(x, Ak ) − d(x, {0})| = 0.
r→∞ hr
k∈I
r
Observe next that any sequence in |W σ1 | consisting of only {0}’s and {1}’s has
an associated Wijsman strongly limit A which is {0} or {1}. For the sequence
{Ak } above, and k = 1, 2, · · · , krj ,
(krj − 2krj −1 )
2krj −1
1 X
2
|d(x, Ak ) − d(x, {1})| ≥
=1−
>1−
krj k
krj
krj
j
82
On strongly lacunary...
which converges to {1}, and, for k = 1, 2, ...2krj −1 ,
1
X
2krj −1
k
|d(x, Ak ) − d(x, {0})| ≥
krj −1
1
= ,
2krj −1
2
and it follows that {Ak } ∈
/ |W σ1 |.
Combining Lemma 1 and Lemma 2 we have.
Theorem 2.7. Let θ = {kr } be a lacunary sequence, then [W N ]θ = [W σ1 ]
if and only if
1 < lim inf qr ≤ lim sup qr < ∞.
r
r
Proof. This follows from Lemma 1 and Lemma 2.
[W σ1 ]
[W N ]
Theorem 2.8. Let {Ak } ∈ [W σ1 ] ∩ [W N ]θ . If Ak → A and Ak → θ B,
then A = B.
[W N ]
[W σ1 ]
Proof. Let Ak → A, Ak → θ B, and suppose that A 6= B. We can write
υr + τr =
≥
1
hr
1
hr
P
|d(x, Ak ) − d(x, A)| +
k∈Ir
P
1
hr
P
|d(x, Ak ) − d(x, B)|
k∈Ir
|d(x, A) − d(x, B)|
k∈Ir
= |d(x, A) − d(x, B)|
where
υr =
1X
|d(x, Ak ) − d(x, A)|
hr k∈I
r
and
τr =
1X
|d(x, Ak ) − d(x, B)|.
hr k∈I
r
Since {Ak } ∈ [W N ]θ , τr → 0. Thus for sufficiently large r we must have
1
υr > |d(x, A) − d(x, B)|.
2
83
U. Ulusu and F. Nuray
Observe that
kr
1 X
1 X
|d(x, Ai ) − d(x, A)| ≥
|d(x, Ai ) − d(x, A)|
kr i=1
kr I
r
=
kr − kr−1
.υr
kr
µ
1
= 1−
qr
1
>
2
µ
¶
1
1−
qr
.υr
¶
. |d(x, A) − d(x, B)|
for sufficiently large r. Since {Ak } ∈ [W σ1 ], the left hand side of the inequality
above convergent to 0, so we must have qr → 1. But this implies, by proof of
Lemma 2, that
[W N ]θ ⊂ [W σ1 ] .
That is, we have
[W N ]
[W σ1 ]
Ak −→θ B ⇒ Ak −→ B,
and therefore
t
1X
|d(x, Ai ) − d(x, B)| → 0.
t i=1
Then we have
1
t
t
P
|d(x, Ai ) − d(x, B)| +
i=1
t
1P
|d(x, Ai ) − d(x, A)| ≥ |d(x, A) − d(x, B)| > 0,
t i=1
which yields a contradiction to our assumption, since both terms on the left
hand side convergent to 0. That is, for each x ∈ X,
|d(x, A) − d(x, B)| = 0,
and therefore A = B.
Nuray and Rhoades (Nuray and Rhoades, 2012) introduced the notion of
strongly almost convergence for sequence of sets as follows: For any non-empty
84
On strongly lacunary...
closed subsets A, Ak ⊆ X, we say that the sequence {Ak } is Wijsman strongly
almost convergent to A if for each x ∈ X,
n
1X
lim
|d(x, Ak+i ) − d(x, A)| = 0
n n
k=1
uniformly in i. In this case we write Ak → A([W AC]).
The set of Wijsman strongly almost convergent sequences will be denoted
(
)
n
1X
[W AC] := {Ak } : lim
|d(x, Ak+i ) − d(x, A)| = 0 .
n n
k=1
Lemma 2.9. For any lacunary sequence θ = {kr }, [W AC] ⊂ [W N ]θ .
Proof. If Ak ∈ [W AC], then there exist N > 0 and a non-empty closed subset
A ⊂ X such that for each ε > 0 and for each x ∈ X,
n
1X
|d(x, Ak+i ) − d(x, A)| < ε,
n k=1
for n > N ,
uniformly in i. We can choose R > 0 such that r ≥ R implies hr > N since θ
is lacunary sequence and, consequently,
1X
τr =
|d(x, Ak ) − d(x, A)| < ε.
hr I
r
Thus {Ak } ∈ [W N ]θ . To obtain a sequence in [W N ]θ and not in [W AC], we
define the sequence {Ak } as follows:

√

 {1} , if kr−1 < k ≤ kr−1 + hr
Ak =


{0} , otherwise.
Then there are arbitrarily long strings of consecutive {0}’s in the coordinates
of {Ak } , as well as arbitrarily long strings of consecutive {1}’s, from which
it follows that {Ak } is not Wijsman strongly almost convergent. However,
{Ak } ∈ [W N ]θ since
£¯√ ¯¤
¯ hr ¯
1X
|d(x, Ak ) − d(x, {0})| = lim
= 0.
lim τr = lim
r→∞
r→∞
r→∞ hr
hr
I
r
85
U. Ulusu and F. Nuray
Theorem 2.10. [W AC] =
T
[W N ]θ .
Proof. We need show only that if {Ak } ∈
/ [W AC] there exists a lacunary θ
such that {Ak } ∈
/ [W N ]θ . We may assume that {Ak } is Wijsman strongly
Cesàro Summable. (otherwise {Ak } ∈
/ [W N ](θ=2r ) ). Consequently, there exists
a unique non-empty closed subset A ⊆ X for which
t
1X
lim
|d(x, Ak ) − d(x, A)| = 0.
t→∞ t
k=1
Let {Ak } ∈
/ [W AC] . Then there exists ε > 0 for each x ∈ X such that for each
N there exist n > N and an integer m for which,
m+n
1 X
|d(x, Ak ) − d(x, A)| ≥ ε,
n k=m+1
m = 1, 2, · · · .
Then we can select sequences (mr ) and (nr ) such that nr → ∞ and
1
nr
mX
r +nr
|d(x, Ak ) − d(x, A)| ≥ ε.
k=mr +1
Since {Ak } ∈ [W σ1 ] it follows that mr → ∞ also. We construct a lacunary
sequence θ = {kr } as follows:
k2 = m1 + n1 ,
k4 = mr2 + nr2 ,
k6 = mr3 + nr3 ,
..
.
k1 = m1 ,
k3 = mr2 ,
k5 = mr3 ,
..
.
where mr2 > 2k2
where mr3 > 2k4
k2i−1 = mri ,
where mri > 2k2i−2
k2i = mri +nri ,
..
.
Then clearly θ = {kr } is lacunary and, for r = 2j,
m +n
rj
rj
1 X
|d(x, Ak ) − d(x, A)| ≥ ε.
τr =
nr k=m +1
rj
It follows from Theorem 2 that {Ak } ∈
/ [W N ]θ .
86
On strongly lacunary...
We now consider the inclusion of [W N ]θ0 , by [W N ]θ , where θ0 is lacunary
refinement of θ. Recall (Freedman, et al., 1978) that the lacunary sequence
θ0 = {kr0 } is called a lacunary refinement of the lacunary sequence θ = {kr } if
{kr } ⊆ {kr0 }.
Lemma 2.11. If θ0 , is a lacunary refinement of θ and if Ak ∈
/ [W N ]θ ,
then
Ak ∈
/ [W N ]θ0 .
Proof. Let Ak ∈
/ [W N ]θ . Then, for any non-empty closed subset A ⊆ X there
exists ε > 0 and a subsequence (krj ) of (kr ) such that
k
rj
1 X
|d(x, Ak ) − d(x, A)| ≥ ε.
τrj =
hrj k=1
Writing
0
0
0
Irj = Is+1
∪ Is+2
∪ ... ∪ Is+p
where
0
0
krj −1 = ks0 < ks+1
< ... < ks+p
= krj .
Then we have
P
P
P
|d(x, Ak ) − d(x, A)|
|d(x, Ak ) − d(x, A)| + ... +
|d(x, Ak ) − d(x, A)| +
τrj =
0
Is+2
0
Is+1
0
Is+p
h0s+1 + h0s+2 + ... + h0s+p
It follows from [Lemma 3.2, in (7)] that
1 X
|d(x, Ak ) − d(x, A)| ≥ ε
h0s+j 0
Is+j
for some j and, consequently, {Ak } ∈
/ [W N ]θ0 .
Theorem 2.12. [W AC] =
T
{[W N ]θ : limr qr = 1}
Proof. If {Ak } ∈
/ [W AC] , then there exists, by Theorem 3, a lacunary sequence
θ such that {Ak } ∈
/ [W N ]θ . If we define
©
ª
θ0 = (kr0 ) = (kr ) ∪ n2 : [(n − 1)2 , (n + 1)2 ] ∩ (kr ) = ∅ ,
/ [W N ]θ0 .
then θ0 is lacunary sequence, limr qr0 = 1, and, by Lemma 4, {Ak } ∈
.
U. Ulusu and F. Nuray
87
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