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Hindawi Publishing CorporationISRNMathematical AnalysisVolume 2013 Article ID 602963 6 pageshttpdxdoiorg1011552013602963
Research ArticleOn 119878119871
120582(119868)-Asymptotically Statistical Equivalence of
Sequences of Sets
Oumlmer KJGJ1 and FatJh Nuray2
1 Mathematics Education Department Faculty of Education Cumhuriyet University Sıvas Turkey2Department of Mathematics Faculty of Science and Literature Afyon Kocatepe University 03200 Afyonkarahısar Turkey
Correspondence should be addressed to Fatıh Nuray fnurayakuedutr
Received 10 June 2013 Accepted 13 August 2013
Academic Editors R Avery and G Schimperna
Copyright copy 2013 O Kısı and F NurayThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents the notion of 119878119871120582(119868)-asymptotically statistical equivalence which is a natural combination of asymptotic 119868-
equivalence and 120582-statistical equivalence for sequences of sets We find its relations to 119868-asymptotically statistical convergencestrong 120582
119868-asymptotically equivalence and strong Cesaro 119868-asymptotically equivalence for sequences of sets
1 Introduction and Background
Let 120582 = (120582119899) be a nondecreasing sequence of positive
numbers tending to infin such that 120582119899+1
le 120582119899+ 1 120582
1= 1
The generalized de la Vallee-Poussin mean is defined by
119905119899 (119909) =
1
120582119899
sum
119896isin119868119899
119909119896 (1)
where 119868119899= [119899 minus 120582
119899+ 1 119899]
A sequence 119909 = (119909119896) is said to be (119881 120582)-summable to a
number 119871 if
lim119899rarrinfin
119905119899 (119909) = 119871 (2)
If 120582119899
= 119899 then (119881 120582)-summability reduces to (119862 1)-summability
We write
[119862 1] = 119909 = (119909119899) exist119871 isin R lim119899rarrinfin
1
119899
119899
sum
119896=1
1003816100381610038161003816119909119896 minus 1198711003816100381610038161003816 = 0
[119881 120582] =
119909 = (119909119899) exist119871 isin R lim
119899rarrinfin
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119909119896 minus 1198711003816100381610038161003816 = 0
(3)
for the sets of sequences 119909 = (119909119896) which are strongly Cesaro
summable and strongly (119881 120582)-summable to 119871 that is 119909119896rarr
119871[119862 1] and 119909119896rarr 119871[119881 120582] respectively Let Λ denote the set
of all nondecreasing sequences 120582 = (120582119899) of positive numbers
tending toinfin such that 120582119899+1
le 120582119899and 120582
1= 1
Statistical convergence of sequences of points was intro-duced by Fast (see [1]) and under different names ithas been discussed in number theory trigonometric seriesand summability In 1993 Marouf presented definitions forasymptotically equivalent and asymptotic regular matricesIn 2003 Patterson extended these concepts by presentingan asymptotically statistical equivalent analog of these def-initions and natural regularity conditions for nonnegativesummability matrices Mursalen defined 120582-statistical conver-gence by using the 120582 sequence He denoted this new methodby 119878120582 and found its relation to statistical convergence [119862 1]-
summability and [119881 120582]-summability (see [2]) Savas intro-duced and studied the concepts of strongly 120582-summabilityand 120582-statistical convergence for fuzzy numbers (see [3])He also presented asymptotically 120582-statistical equivalentsequences of fuzzy numbers (see [4]) Kostyrko et al (see [56]) introduced the concept of 119868-convergence of sequences in ametric space and studied someproperties of this convergenceIn addition to these definitions natural inclusion theoremsare also presented The concept of convergence of sequencesof points has been extended by several authors to convergence
2 ISRNMathematical Analysis
of sequences of sets One of these extensions that we willconsider in this paper is Wijsman convergence The conceptof Wijsman statistical convergence is an implementation ofthe concept of statistical convergence presented byNuray andRhoades (see [7])
Definition 1 The sequence 119909 = (119909119896) is said to be statistically
convergent to the number 119871 if for every 120576 gt 0
lim119899rarrinfin
1
119899
1003816100381610038161003816119896 le 119899 1003816100381610038161003816119909119899 minus 119871
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0
(4)
In this case one writes 119904119905 minus lim119909119896= 119871 (see [8])
Definition 2 A family of sets 119868 sube 2N is called an ideal if and
only if(i) 0 isin 119868(ii) for each 119860 119861 isin 119868 one has 119860 cup 119861 isin 119868(iii) for each119860 isin 119868 and each 119861 sube 119860 one has 119861 isin 119868 (see [5])
An ideal is called nontrivial if N notin 119868 and nontrivial ideal iscalled admissible if 119899 isin 119868 for each 119899 isin N
Definition 3 A family of sets 119865 sube 2N is a filter inN if and only
if(i) 0 notin 119868(ii) for each 119860 119861 isin 119865 one has 119860 cap 119861 isin 119865(iii) for each 119860 isin 119865 and each 119861 supe 119860 one has 119861 isin 119865 (see
[5])
Proposition 4 119868 is a nontrivial ideal in N if and only if
119865 = 119865 (119868) = 119872 = N 119860 119860 isin 119868 (5)
is a filter in N (see [5])
Definition 5 Let 119868 be a nontrivial ideal of subsets ofN and let(119883 119889) be a metric space A sequence 119909
119899119899isinN of elements of119883
is said to be 119868-convergent to 119871 Therefore 119871 = 119868 minus lim119899rarrinfin
119909119899
if and only if for each 120576 gt 0 the set
119860 (120576) = 119899 isin N 1003816100381610038161003816119909119899 minus 119871
1003816100381610038161003816 ge 120576 (6)
belongs to 119868 The number 119871 is called the 119868 limit of thesequence 119909 = (119909
119899)119899isinN isin 119883 (see [5])
Definition 6 Let (119883 119889) be ametric space For any non-emptyclosed subsets 119860119860
119896sube 119883 one says that the sequence 119860
119896 is
Wijsman convergent to 119860
lim119896rarrinfin
119889 (119909 119860119896) = 119889 (119909 119860) (7)
for each 119909 isin 119883 In this case one writes119882 minus lim119896rarrinfin
119860119896= 119860
(see [9 10])
As an example consider the following sequence of circlesin the (119909 119910)-plane
119860119896= (119909 119910) 119909
2+ 1199102+ 2119896119909 = 0 (8)
As 119896 rarr infin the sequence isWijsman convergent to the119910-axis119860 = (119909 119910) 119909 = 0
Definition 7 Let (119883 119889) be a metric space For any non-emptyclosed subsets 119860119860
119896sube 119883 one says that the sequence 119860
119896 is
Wijsman statistically convergent to119860 if for 120576 gt 0 and for each119909 isin 119883
lim119899rarrinfin
1
119899
1003816100381610038161003816119896 le 119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0
(9)
In this case one writes 119904119905 minus lim119882119860119896= 119860 or 119860
119896rarr 119860(119882119878)
(see [7])
119882119878 = 119860119896 119904119905 minus lim
119882119860119896= 119860 (10)
where119882119878 denotes the set of Wijsman statistical convergencesequences
Also the concept of bounded sequence for sequences ofsets was given by Nuray and Rhoades (see [7]) Let (119883 120588) be ametric space For any non-empty closed subsets 119860
119896of 119883 we
say that the sequence 119860119896 is bounded if sup
119896119889(119909 119860
119896) lt infin
for each 119909 isin 119883
Definition 8 Let (119883 119889) be ametric space For any non-emptyclosed subsets 119860119860
119896sube 119883 we say that the sequence 119860
119896
is Wijsman Cesaro summable to 119860 if 119889(119909 119860119896) is Cesaro
summable to 119889(119909 119860) that is for each 119909 isin 119883
lim119899rarrinfin
1
119899
119899
sum
119896=1
119889 (119909 119860119896) = 119889 (119909 119860) (11)
and one says that the sequence 119860119896 is Wijsman strongly
Cesaro summable to 119860 if 119889(119909 119860119896) is strongly summable to
119889(119909 119860) that is for each 119909 isin 119883
lim119899rarrinfin
1
119899
119899
sum
119896=1
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (12)
(see [7])
Definition 9 Let (119883 119889) be ametric space For any non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are asymptotically equivalent (Wijsman sense)
if for each 119909 isin 119883
lim119896rarrinfin
119889 (119909 119860119896)
119889 (119909 119861119896)= 1 (13)
(denoted by 119860119896sim 119861119896) (see [11])
As an example consider the following sequences of circlesin the (119909 119910)-plane
119860119896= (119909 119910) isin R
2 1199092+ 1199102+ 2119896119910 = 0
119861119896= (119909 119910) isin R
2 1199092+ 1199102minus 2119896119910 = 0
(14)
Since
lim119896rarrinfin
119889 (119909 119860119896)
119889 (119909 119861119896)= 1 (15)
the sequences 119860119896 and 119861
119896 are asymptotically equivalent
(Wijsman sense) that is 119860119896sim 119861119896
ISRNMathematical Analysis 3
Definition 10 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences 119860
119896
and 119861119896 are asymptotically statistically equivalent (Wijsman
sense) ofmultiple 119871 provided that for every 120576 gt 0 and for each119909 isin 119883
lim119899rarrinfin
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
= 0 (16)
(denoted by 119860119896
119882119878119871
sim 119861119896) and simply asymptotically statisti-
cally equivalent (Wijsman sense) if 119871 = 1 (see [11])
2 Main Results
Definition 11 (see [12]) Let (119883 119889) be a metric space and let119868 sube 2
N be a proper ideal in N For any non-empty closedsubsets119860119860
119896sub 119883 we say that the sequence 119860
119896 is Wijsman
119868-convergent to 119860 if for each 120576 gt 0 and for each 119909 isin 119883 theset
119860 (119909 120576) = 119896 isin N 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 120576 (17)
belongs to 119868 In this case one writes 119868119882minus lim119860
119896= 119860 or119860
119896rarr
119860(119868119882) and the set of Wijsman 119868-convergent sequences of sets
will be denoted by
119868119882= 119860
119896 119896 isin N
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 ge 120576 isin 119868 (18)
As an example consider the following sequence Let119883 = R2and let 119860
119896 be the following sequence
119860119896=
(119909 119910) isin R2 1199092 + 1199102 minus 2119896119910 = 0 if 119896 = 1198992
(119909 119910) isin R2 119910 = minus1 if 119896 = 1198992(19)
and 119860 = (119909 119910) isin R2 119910 = 0 The sequence 119860119896 is not
Wijsman convergent to the set 119860 But if we take 119868 = 119868119889 then
119860119896 is Wijsman 119868-convergent to set 119860 where 119868
119889is the ideal
of sets that have zero density
Definition 12 Let (119883 119889) be a metric space For any non-empty closed subsets 119860119860
119896sube 119883 we say that the sequence
119860119896 is said to beWijsman 120582-statistically convergent or119882119878
120582-
convergent to 119860 if for every 120576 gt 0 and for each 119909 isin 119883
lim119899rarrinfin
1
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (20)
In this case one writes 119878120582minus lim119882119860119896= 119860 or 119860
119896 rarr 119860(119882119878
120582)
and
119882119878120582= 119860
119896 119860 sube 119883119882119878
120582minus lim119860
119896= 119860 (21)
If 120582119899= 119899 thenWijsman 120582-statistical convergence is the same
as Wijsman statistical convergence for the sequences of sets
Definition 13 Let (119883 119889)be ametric space For anynon-emptyclosed subsets 119860119860
119896sube 119883 we say that the sequence 119860
119896 is
said to beWijsman strongly (119881 120582) summable to119860 if for every120576 gt 0 and for each 119909 isin 119883
lim119899rarrinfin
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (22)
In this case one writes 119860119896 rarr 119860[119881 120582]
If 120582119899= 119899 then [119881 120582]-summability reduces to [119862 1]-
summability for sequences of sets
Theorem 14 Let 120582 isin Λ (119883 119889) be ametric space For any non-empty closed subsets 119860119860
119896sube 119883 then
(i) 119860119896 rarr 119860[119881 120582] rArr 119860
119896 rarr 119860(119882119878
120582) and the
inclusion [119881 120582] ⫅ (119882119878120582) is proper for sequences of sets
(ii) if 119860119896 is bounded (ie 119860
119896 isin 119871
infin) and 119860
119896 rarr
119860(119882119878120582) then 119860
119896 rarr 119860[119881 120582]
(iii) 119882119878120582cap 119871infin= [119881 120582] cap 119871
infin
where 119871infin
denotes the set of bounded sequences of sets
Proof (i) Let 120576 gt 0 and 119860119896 rarr 119860[119881 120582] One has
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
ge 120576 sdot1003816100381610038161003816119896 isin 119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 ge 120576
1003816100381610038161003816
(23)
Therefore 119860119896 rarr 119860[119881 120582] rArr 119860
119896 rarr 119860(119882119878
120582)
The following example shows that (119882119878120582) ⫋ [119881 120582] for
sequences of sets
119860119896=
119896 for 119899 minus [radic120582119899] + 1 le 119896 le 119899
0 otherwise(24)
Then 119860119896 notin 119871infin
and for every 120576 (0 lt 120576 le 1)
1
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 0)
1003816100381610038161003816 ge 1205761003816100381610038161003816
=
[radic120582119899]
120582119899
997888rarr 0 as 119899 997888rarr infin
(25)
that is 119860119896 rarr 0 (119882119878
120582) On the other hand
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 0)1003816100381610038161003816 999424999426999456 0 as 119899 997888rarr infin (26)
that is 119860119896 999424999426999456 0 [119881 120582]
4 ISRNMathematical Analysis
(ii) Suppose that 119860119896 is bounded and 119860
119896 rarr 119860(119882119878
120582)
Then there is a119872 such that |119889(119909 119860119896) minus 119889(119909 119860)| le 119872 for all
119896 Given 120576 gt 0 one has
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
le119872
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 + 120576
(27)
which implies that 119860119896 rarr 119860[119881 120582]
(iii) This immediately follows from (i) and (ii)
Definition 15 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin
119883 one says that the sequences 119860119896 and 119861
119896 are said to
be asymptotically Wijsman 119868-equivalent of multiple 119871 if forevery 120576 gt 0 and for each 119909 isin 119883
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (28)
This will be denoted by 119860119896
119868119882
sim 119861119896
Definition 16 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin 119883
one says that the sequences 119860119896 and 119861
119896 are said to be
strong Cesaro 119868-asymptotically equivalent (Wijsman sense)of multiple 119871 if every 120576 gt 0 and for each 119909 isin 119883
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (29)
This will be denoted by 119860119896
119862119871
1(119868119882)
sim 119861119896
Definition 17 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are Wijsman 119868-asymptotically statistically
equivalent of multiple 119871 if for every 120576 120575 gt 0 and for each119909 isin 119883
119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868 (30)
This will be denoted by 119860119896
119878119871
(119868119882)
sim 119861119896
Example 18 Let 119868 sube 2N be a proper ideal in N and let (119883 119889)be a metric space then 119860119860
119896sub 119883 are non-empty closed
subsets Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896
=
(119909 119910) isin R2 0le 119909le119899 0le119910le1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise
119861119896
= (119909 119910) isin R2 0 le 119909 le 119899 0 le 119910 le minus
1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise(31)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (32)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 1198601198961198681
119882
sim119861119896 where 119868
119889is the
ideal of sets that have zero density
Example 19 Let119868 sube 2N be a proper ideal inN and let (119883 119889) beametric space then119860119860
119896sub 119883 are non-empty closed subsets
Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896=
(119909 119910) isin R2 1199092 + (119910 minus 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise
119861119896=
(119909 119910) isin R2 1199092 + (119910 + 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise(33)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (34)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 119860119896
1198681
119882
sim 119861119896 where 119868
119889is
the ideal of sets which have zero density
Definition 20 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are strongly 120582
119868-asymptotically equivalent
(Wijsman sense) of multiple 119871 if for every 120576 gt 0 and for each119909 isin 119883
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
isin 119868 (35)
This will be denoted by 119860119896
119881119871
120582(119868119882)
sim 119861119896
ISRNMathematical Analysis 5
Definition 21 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are 119868-asymptotically 120582-statistically equivalent
(Wijsman sense) of multiple 119871 provided that for every 120576 120575 gt0 and for each 119909 isin 119883
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868
(36)
This will be denoted by 119860119896
119878119871
120582(119868119882)
sim 119861119896
Theorem 22 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that 119860119896
119881119871
120582(119868119882)
sim 119861119896and 120576 gt 0 Then
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(37)
and so
1
120576120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(38)
Then for any 120575 gt 0
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575
sube
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576120575
(39)
Since right hand belongs to 119868 then left hand also belongs to119868 and this completes the proof
Theorem 23 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896 and 119861
119896 are bounded and 119860
119896
119878119871
120582(119868119882)
sim 119861119896 then 119860
119896
119881119871
120582(119868119882)
sim
119861119896
Proof Let 119860119896 119861119896 be bounded sequences and let 119860
119896
119878119871
120582(119868119882)
sim
119861119896 Then there is an119872 such that
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 119872 (40)
for all 119896 For each 120576 gt 0
1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 1198721
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
+120576
2
(41)
Then
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
times
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
ge120576
2119872 isin 119868
(42)
Therefore 119860119896
119881119871
120582(119868119882)
sim 119861119896
The following example shows that if 119860119896 and 119861
119896 are not
bounded thenTheorem 23 cannot be true
Example 24 Take 119871 = 1 and define 119860119896 to be
119860119896=
119896 119896 = 119896119903minus1
+ 1 119896119903minus1
+ 2 119896119903minus1
+ [radic120582119899]
1 otherwise(43)
where lfloorsdotrfloor denotes the greatest integer function and 119861119896= 1
for all 119896 Note that 119860119896 is not boundedThen119860
119896
119878119871
120582(119868)
sim 119861119896 but
119860119896
119881119871
120582(119868)
sim 119861119896is not true
Theorem 25 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119862119871
1(119868119882)
sim 119861119896
6 ISRNMathematical Analysis
Proof Assume that 119860119896
119881119871
120582(119868)
sim 119861119896and 120576 gt 0 Then
1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
119899sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le1
120582119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le2
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
(44)
and so
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2 isin 119868
(45)
Hence 119860119896
119862119871
1(119868119882)
sim 119861119896
Theorem 26 If lim inf 120582119899119899 gt 0 then 119860
119896
119878119871
(119868119882)
sim 119861119896implies
119860119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that lim inf (120582119899119899) gt 0 and there exists a 120575 gt 0
such that 120582119899119899 ge 120575 for sufficiently large 119899 For given 120576 gt 0 one
has
1
119899119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
supe1
119899119896 isin 119868
119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
(46)
Therefore
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge120582119899
119899
1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 1205751
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(47)
then for any 120578 gt 0 we get
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578
sube 119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578120575 isin 119868
(48)
and this completes the proof
References
[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951
[2] Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slovacavol 50 no 1 pp 111ndash115 2000
[3] E Savas ldquoOn strongly 120582-summable sequences of fuzzy num-bersrdquo Information Sciences vol 125 no 1ndash4 pp 181ndash186 2000
[4] E Savas ldquoOn asymptotically 120582-statistical equivalent sequencesof fuzzy numbersrdquoNewMathematics and Natural Computationvol 3 no 3 pp 301ndash306 2007
[5] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[6] P Kostyrko M Macaj T Salat and M Sleziak ldquo119868-convergenceand extremal 119868-limit pointsrdquo Mathematica Slovaca vol 55 no4 pp 443ndash464 2005
[7] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012
[8] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985
[9] R AWijsman ldquoConvergence of sequences of convex sets conesand functionsrdquo Bulletin of the American Mathematical Societyvol 70 pp 186ndash188 1964
[10] R AWijsman ldquoConvergence of sequences of convex sets conesand functions IIrdquo Transactions of the American MathematicalSociety vol 123 pp 32ndash45 1966
[11] U Ulusu and F Nuray ldquoOn asymptotically Lacunary statisticalequivalent set sequencesrdquo Journal of Mathematics vol 2013Article ID 310438 5 pages 2013
[12] O Kısı and F Nuray ldquoA new convergence for sequences of setsrdquosubmitted for publication
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Stochastic AnalysisInternational Journal of
2 ISRNMathematical Analysis
of sequences of sets One of these extensions that we willconsider in this paper is Wijsman convergence The conceptof Wijsman statistical convergence is an implementation ofthe concept of statistical convergence presented byNuray andRhoades (see [7])
Definition 1 The sequence 119909 = (119909119896) is said to be statistically
convergent to the number 119871 if for every 120576 gt 0
lim119899rarrinfin
1
119899
1003816100381610038161003816119896 le 119899 1003816100381610038161003816119909119899 minus 119871
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0
(4)
In this case one writes 119904119905 minus lim119909119896= 119871 (see [8])
Definition 2 A family of sets 119868 sube 2N is called an ideal if and
only if(i) 0 isin 119868(ii) for each 119860 119861 isin 119868 one has 119860 cup 119861 isin 119868(iii) for each119860 isin 119868 and each 119861 sube 119860 one has 119861 isin 119868 (see [5])
An ideal is called nontrivial if N notin 119868 and nontrivial ideal iscalled admissible if 119899 isin 119868 for each 119899 isin N
Definition 3 A family of sets 119865 sube 2N is a filter inN if and only
if(i) 0 notin 119868(ii) for each 119860 119861 isin 119865 one has 119860 cap 119861 isin 119865(iii) for each 119860 isin 119865 and each 119861 supe 119860 one has 119861 isin 119865 (see
[5])
Proposition 4 119868 is a nontrivial ideal in N if and only if
119865 = 119865 (119868) = 119872 = N 119860 119860 isin 119868 (5)
is a filter in N (see [5])
Definition 5 Let 119868 be a nontrivial ideal of subsets ofN and let(119883 119889) be a metric space A sequence 119909
119899119899isinN of elements of119883
is said to be 119868-convergent to 119871 Therefore 119871 = 119868 minus lim119899rarrinfin
119909119899
if and only if for each 120576 gt 0 the set
119860 (120576) = 119899 isin N 1003816100381610038161003816119909119899 minus 119871
1003816100381610038161003816 ge 120576 (6)
belongs to 119868 The number 119871 is called the 119868 limit of thesequence 119909 = (119909
119899)119899isinN isin 119883 (see [5])
Definition 6 Let (119883 119889) be ametric space For any non-emptyclosed subsets 119860119860
119896sube 119883 one says that the sequence 119860
119896 is
Wijsman convergent to 119860
lim119896rarrinfin
119889 (119909 119860119896) = 119889 (119909 119860) (7)
for each 119909 isin 119883 In this case one writes119882 minus lim119896rarrinfin
119860119896= 119860
(see [9 10])
As an example consider the following sequence of circlesin the (119909 119910)-plane
119860119896= (119909 119910) 119909
2+ 1199102+ 2119896119909 = 0 (8)
As 119896 rarr infin the sequence isWijsman convergent to the119910-axis119860 = (119909 119910) 119909 = 0
Definition 7 Let (119883 119889) be a metric space For any non-emptyclosed subsets 119860119860
119896sube 119883 one says that the sequence 119860
119896 is
Wijsman statistically convergent to119860 if for 120576 gt 0 and for each119909 isin 119883
lim119899rarrinfin
1
119899
1003816100381610038161003816119896 le 119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0
(9)
In this case one writes 119904119905 minus lim119882119860119896= 119860 or 119860
119896rarr 119860(119882119878)
(see [7])
119882119878 = 119860119896 119904119905 minus lim
119882119860119896= 119860 (10)
where119882119878 denotes the set of Wijsman statistical convergencesequences
Also the concept of bounded sequence for sequences ofsets was given by Nuray and Rhoades (see [7]) Let (119883 120588) be ametric space For any non-empty closed subsets 119860
119896of 119883 we
say that the sequence 119860119896 is bounded if sup
119896119889(119909 119860
119896) lt infin
for each 119909 isin 119883
Definition 8 Let (119883 119889) be ametric space For any non-emptyclosed subsets 119860119860
119896sube 119883 we say that the sequence 119860
119896
is Wijsman Cesaro summable to 119860 if 119889(119909 119860119896) is Cesaro
summable to 119889(119909 119860) that is for each 119909 isin 119883
lim119899rarrinfin
1
119899
119899
sum
119896=1
119889 (119909 119860119896) = 119889 (119909 119860) (11)
and one says that the sequence 119860119896 is Wijsman strongly
Cesaro summable to 119860 if 119889(119909 119860119896) is strongly summable to
119889(119909 119860) that is for each 119909 isin 119883
lim119899rarrinfin
1
119899
119899
sum
119896=1
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (12)
(see [7])
Definition 9 Let (119883 119889) be ametric space For any non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are asymptotically equivalent (Wijsman sense)
if for each 119909 isin 119883
lim119896rarrinfin
119889 (119909 119860119896)
119889 (119909 119861119896)= 1 (13)
(denoted by 119860119896sim 119861119896) (see [11])
As an example consider the following sequences of circlesin the (119909 119910)-plane
119860119896= (119909 119910) isin R
2 1199092+ 1199102+ 2119896119910 = 0
119861119896= (119909 119910) isin R
2 1199092+ 1199102minus 2119896119910 = 0
(14)
Since
lim119896rarrinfin
119889 (119909 119860119896)
119889 (119909 119861119896)= 1 (15)
the sequences 119860119896 and 119861
119896 are asymptotically equivalent
(Wijsman sense) that is 119860119896sim 119861119896
ISRNMathematical Analysis 3
Definition 10 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences 119860
119896
and 119861119896 are asymptotically statistically equivalent (Wijsman
sense) ofmultiple 119871 provided that for every 120576 gt 0 and for each119909 isin 119883
lim119899rarrinfin
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
= 0 (16)
(denoted by 119860119896
119882119878119871
sim 119861119896) and simply asymptotically statisti-
cally equivalent (Wijsman sense) if 119871 = 1 (see [11])
2 Main Results
Definition 11 (see [12]) Let (119883 119889) be a metric space and let119868 sube 2
N be a proper ideal in N For any non-empty closedsubsets119860119860
119896sub 119883 we say that the sequence 119860
119896 is Wijsman
119868-convergent to 119860 if for each 120576 gt 0 and for each 119909 isin 119883 theset
119860 (119909 120576) = 119896 isin N 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 120576 (17)
belongs to 119868 In this case one writes 119868119882minus lim119860
119896= 119860 or119860
119896rarr
119860(119868119882) and the set of Wijsman 119868-convergent sequences of sets
will be denoted by
119868119882= 119860
119896 119896 isin N
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 ge 120576 isin 119868 (18)
As an example consider the following sequence Let119883 = R2and let 119860
119896 be the following sequence
119860119896=
(119909 119910) isin R2 1199092 + 1199102 minus 2119896119910 = 0 if 119896 = 1198992
(119909 119910) isin R2 119910 = minus1 if 119896 = 1198992(19)
and 119860 = (119909 119910) isin R2 119910 = 0 The sequence 119860119896 is not
Wijsman convergent to the set 119860 But if we take 119868 = 119868119889 then
119860119896 is Wijsman 119868-convergent to set 119860 where 119868
119889is the ideal
of sets that have zero density
Definition 12 Let (119883 119889) be a metric space For any non-empty closed subsets 119860119860
119896sube 119883 we say that the sequence
119860119896 is said to beWijsman 120582-statistically convergent or119882119878
120582-
convergent to 119860 if for every 120576 gt 0 and for each 119909 isin 119883
lim119899rarrinfin
1
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (20)
In this case one writes 119878120582minus lim119882119860119896= 119860 or 119860
119896 rarr 119860(119882119878
120582)
and
119882119878120582= 119860
119896 119860 sube 119883119882119878
120582minus lim119860
119896= 119860 (21)
If 120582119899= 119899 thenWijsman 120582-statistical convergence is the same
as Wijsman statistical convergence for the sequences of sets
Definition 13 Let (119883 119889)be ametric space For anynon-emptyclosed subsets 119860119860
119896sube 119883 we say that the sequence 119860
119896 is
said to beWijsman strongly (119881 120582) summable to119860 if for every120576 gt 0 and for each 119909 isin 119883
lim119899rarrinfin
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (22)
In this case one writes 119860119896 rarr 119860[119881 120582]
If 120582119899= 119899 then [119881 120582]-summability reduces to [119862 1]-
summability for sequences of sets
Theorem 14 Let 120582 isin Λ (119883 119889) be ametric space For any non-empty closed subsets 119860119860
119896sube 119883 then
(i) 119860119896 rarr 119860[119881 120582] rArr 119860
119896 rarr 119860(119882119878
120582) and the
inclusion [119881 120582] ⫅ (119882119878120582) is proper for sequences of sets
(ii) if 119860119896 is bounded (ie 119860
119896 isin 119871
infin) and 119860
119896 rarr
119860(119882119878120582) then 119860
119896 rarr 119860[119881 120582]
(iii) 119882119878120582cap 119871infin= [119881 120582] cap 119871
infin
where 119871infin
denotes the set of bounded sequences of sets
Proof (i) Let 120576 gt 0 and 119860119896 rarr 119860[119881 120582] One has
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
ge 120576 sdot1003816100381610038161003816119896 isin 119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 ge 120576
1003816100381610038161003816
(23)
Therefore 119860119896 rarr 119860[119881 120582] rArr 119860
119896 rarr 119860(119882119878
120582)
The following example shows that (119882119878120582) ⫋ [119881 120582] for
sequences of sets
119860119896=
119896 for 119899 minus [radic120582119899] + 1 le 119896 le 119899
0 otherwise(24)
Then 119860119896 notin 119871infin
and for every 120576 (0 lt 120576 le 1)
1
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 0)
1003816100381610038161003816 ge 1205761003816100381610038161003816
=
[radic120582119899]
120582119899
997888rarr 0 as 119899 997888rarr infin
(25)
that is 119860119896 rarr 0 (119882119878
120582) On the other hand
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 0)1003816100381610038161003816 999424999426999456 0 as 119899 997888rarr infin (26)
that is 119860119896 999424999426999456 0 [119881 120582]
4 ISRNMathematical Analysis
(ii) Suppose that 119860119896 is bounded and 119860
119896 rarr 119860(119882119878
120582)
Then there is a119872 such that |119889(119909 119860119896) minus 119889(119909 119860)| le 119872 for all
119896 Given 120576 gt 0 one has
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
le119872
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 + 120576
(27)
which implies that 119860119896 rarr 119860[119881 120582]
(iii) This immediately follows from (i) and (ii)
Definition 15 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin
119883 one says that the sequences 119860119896 and 119861
119896 are said to
be asymptotically Wijsman 119868-equivalent of multiple 119871 if forevery 120576 gt 0 and for each 119909 isin 119883
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (28)
This will be denoted by 119860119896
119868119882
sim 119861119896
Definition 16 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin 119883
one says that the sequences 119860119896 and 119861
119896 are said to be
strong Cesaro 119868-asymptotically equivalent (Wijsman sense)of multiple 119871 if every 120576 gt 0 and for each 119909 isin 119883
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (29)
This will be denoted by 119860119896
119862119871
1(119868119882)
sim 119861119896
Definition 17 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are Wijsman 119868-asymptotically statistically
equivalent of multiple 119871 if for every 120576 120575 gt 0 and for each119909 isin 119883
119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868 (30)
This will be denoted by 119860119896
119878119871
(119868119882)
sim 119861119896
Example 18 Let 119868 sube 2N be a proper ideal in N and let (119883 119889)be a metric space then 119860119860
119896sub 119883 are non-empty closed
subsets Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896
=
(119909 119910) isin R2 0le 119909le119899 0le119910le1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise
119861119896
= (119909 119910) isin R2 0 le 119909 le 119899 0 le 119910 le minus
1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise(31)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (32)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 1198601198961198681
119882
sim119861119896 where 119868
119889is the
ideal of sets that have zero density
Example 19 Let119868 sube 2N be a proper ideal inN and let (119883 119889) beametric space then119860119860
119896sub 119883 are non-empty closed subsets
Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896=
(119909 119910) isin R2 1199092 + (119910 minus 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise
119861119896=
(119909 119910) isin R2 1199092 + (119910 + 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise(33)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (34)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 119860119896
1198681
119882
sim 119861119896 where 119868
119889is
the ideal of sets which have zero density
Definition 20 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are strongly 120582
119868-asymptotically equivalent
(Wijsman sense) of multiple 119871 if for every 120576 gt 0 and for each119909 isin 119883
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
isin 119868 (35)
This will be denoted by 119860119896
119881119871
120582(119868119882)
sim 119861119896
ISRNMathematical Analysis 5
Definition 21 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are 119868-asymptotically 120582-statistically equivalent
(Wijsman sense) of multiple 119871 provided that for every 120576 120575 gt0 and for each 119909 isin 119883
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868
(36)
This will be denoted by 119860119896
119878119871
120582(119868119882)
sim 119861119896
Theorem 22 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that 119860119896
119881119871
120582(119868119882)
sim 119861119896and 120576 gt 0 Then
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(37)
and so
1
120576120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(38)
Then for any 120575 gt 0
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575
sube
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576120575
(39)
Since right hand belongs to 119868 then left hand also belongs to119868 and this completes the proof
Theorem 23 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896 and 119861
119896 are bounded and 119860
119896
119878119871
120582(119868119882)
sim 119861119896 then 119860
119896
119881119871
120582(119868119882)
sim
119861119896
Proof Let 119860119896 119861119896 be bounded sequences and let 119860
119896
119878119871
120582(119868119882)
sim
119861119896 Then there is an119872 such that
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 119872 (40)
for all 119896 For each 120576 gt 0
1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 1198721
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
+120576
2
(41)
Then
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
times
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
ge120576
2119872 isin 119868
(42)
Therefore 119860119896
119881119871
120582(119868119882)
sim 119861119896
The following example shows that if 119860119896 and 119861
119896 are not
bounded thenTheorem 23 cannot be true
Example 24 Take 119871 = 1 and define 119860119896 to be
119860119896=
119896 119896 = 119896119903minus1
+ 1 119896119903minus1
+ 2 119896119903minus1
+ [radic120582119899]
1 otherwise(43)
where lfloorsdotrfloor denotes the greatest integer function and 119861119896= 1
for all 119896 Note that 119860119896 is not boundedThen119860
119896
119878119871
120582(119868)
sim 119861119896 but
119860119896
119881119871
120582(119868)
sim 119861119896is not true
Theorem 25 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119862119871
1(119868119882)
sim 119861119896
6 ISRNMathematical Analysis
Proof Assume that 119860119896
119881119871
120582(119868)
sim 119861119896and 120576 gt 0 Then
1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
119899sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le1
120582119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le2
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
(44)
and so
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2 isin 119868
(45)
Hence 119860119896
119862119871
1(119868119882)
sim 119861119896
Theorem 26 If lim inf 120582119899119899 gt 0 then 119860
119896
119878119871
(119868119882)
sim 119861119896implies
119860119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that lim inf (120582119899119899) gt 0 and there exists a 120575 gt 0
such that 120582119899119899 ge 120575 for sufficiently large 119899 For given 120576 gt 0 one
has
1
119899119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
supe1
119899119896 isin 119868
119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
(46)
Therefore
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge120582119899
119899
1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 1205751
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(47)
then for any 120578 gt 0 we get
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578
sube 119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578120575 isin 119868
(48)
and this completes the proof
References
[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951
[2] Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slovacavol 50 no 1 pp 111ndash115 2000
[3] E Savas ldquoOn strongly 120582-summable sequences of fuzzy num-bersrdquo Information Sciences vol 125 no 1ndash4 pp 181ndash186 2000
[4] E Savas ldquoOn asymptotically 120582-statistical equivalent sequencesof fuzzy numbersrdquoNewMathematics and Natural Computationvol 3 no 3 pp 301ndash306 2007
[5] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[6] P Kostyrko M Macaj T Salat and M Sleziak ldquo119868-convergenceand extremal 119868-limit pointsrdquo Mathematica Slovaca vol 55 no4 pp 443ndash464 2005
[7] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012
[8] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985
[9] R AWijsman ldquoConvergence of sequences of convex sets conesand functionsrdquo Bulletin of the American Mathematical Societyvol 70 pp 186ndash188 1964
[10] R AWijsman ldquoConvergence of sequences of convex sets conesand functions IIrdquo Transactions of the American MathematicalSociety vol 123 pp 32ndash45 1966
[11] U Ulusu and F Nuray ldquoOn asymptotically Lacunary statisticalequivalent set sequencesrdquo Journal of Mathematics vol 2013Article ID 310438 5 pages 2013
[12] O Kısı and F Nuray ldquoA new convergence for sequences of setsrdquosubmitted for publication
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 3
Definition 10 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences 119860
119896
and 119861119896 are asymptotically statistically equivalent (Wijsman
sense) ofmultiple 119871 provided that for every 120576 gt 0 and for each119909 isin 119883
lim119899rarrinfin
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
= 0 (16)
(denoted by 119860119896
119882119878119871
sim 119861119896) and simply asymptotically statisti-
cally equivalent (Wijsman sense) if 119871 = 1 (see [11])
2 Main Results
Definition 11 (see [12]) Let (119883 119889) be a metric space and let119868 sube 2
N be a proper ideal in N For any non-empty closedsubsets119860119860
119896sub 119883 we say that the sequence 119860
119896 is Wijsman
119868-convergent to 119860 if for each 120576 gt 0 and for each 119909 isin 119883 theset
119860 (119909 120576) = 119896 isin N 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 120576 (17)
belongs to 119868 In this case one writes 119868119882minus lim119860
119896= 119860 or119860
119896rarr
119860(119868119882) and the set of Wijsman 119868-convergent sequences of sets
will be denoted by
119868119882= 119860
119896 119896 isin N
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 ge 120576 isin 119868 (18)
As an example consider the following sequence Let119883 = R2and let 119860
119896 be the following sequence
119860119896=
(119909 119910) isin R2 1199092 + 1199102 minus 2119896119910 = 0 if 119896 = 1198992
(119909 119910) isin R2 119910 = minus1 if 119896 = 1198992(19)
and 119860 = (119909 119910) isin R2 119910 = 0 The sequence 119860119896 is not
Wijsman convergent to the set 119860 But if we take 119868 = 119868119889 then
119860119896 is Wijsman 119868-convergent to set 119860 where 119868
119889is the ideal
of sets that have zero density
Definition 12 Let (119883 119889) be a metric space For any non-empty closed subsets 119860119860
119896sube 119883 we say that the sequence
119860119896 is said to beWijsman 120582-statistically convergent or119882119878
120582-
convergent to 119860 if for every 120576 gt 0 and for each 119909 isin 119883
lim119899rarrinfin
1
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (20)
In this case one writes 119878120582minus lim119882119860119896= 119860 or 119860
119896 rarr 119860(119882119878
120582)
and
119882119878120582= 119860
119896 119860 sube 119883119882119878
120582minus lim119860
119896= 119860 (21)
If 120582119899= 119899 thenWijsman 120582-statistical convergence is the same
as Wijsman statistical convergence for the sequences of sets
Definition 13 Let (119883 119889)be ametric space For anynon-emptyclosed subsets 119860119860
119896sube 119883 we say that the sequence 119860
119896 is
said to beWijsman strongly (119881 120582) summable to119860 if for every120576 gt 0 and for each 119909 isin 119883
lim119899rarrinfin
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (22)
In this case one writes 119860119896 rarr 119860[119881 120582]
If 120582119899= 119899 then [119881 120582]-summability reduces to [119862 1]-
summability for sequences of sets
Theorem 14 Let 120582 isin Λ (119883 119889) be ametric space For any non-empty closed subsets 119860119860
119896sube 119883 then
(i) 119860119896 rarr 119860[119881 120582] rArr 119860
119896 rarr 119860(119882119878
120582) and the
inclusion [119881 120582] ⫅ (119882119878120582) is proper for sequences of sets
(ii) if 119860119896 is bounded (ie 119860
119896 isin 119871
infin) and 119860
119896 rarr
119860(119882119878120582) then 119860
119896 rarr 119860[119881 120582]
(iii) 119882119878120582cap 119871infin= [119881 120582] cap 119871
infin
where 119871infin
denotes the set of bounded sequences of sets
Proof (i) Let 120576 gt 0 and 119860119896 rarr 119860[119881 120582] One has
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
ge 120576 sdot1003816100381610038161003816119896 isin 119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 ge 120576
1003816100381610038161003816
(23)
Therefore 119860119896 rarr 119860[119881 120582] rArr 119860
119896 rarr 119860(119882119878
120582)
The following example shows that (119882119878120582) ⫋ [119881 120582] for
sequences of sets
119860119896=
119896 for 119899 minus [radic120582119899] + 1 le 119896 le 119899
0 otherwise(24)
Then 119860119896 notin 119871infin
and for every 120576 (0 lt 120576 le 1)
1
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 0)
1003816100381610038161003816 ge 1205761003816100381610038161003816
=
[radic120582119899]
120582119899
997888rarr 0 as 119899 997888rarr infin
(25)
that is 119860119896 rarr 0 (119882119878
120582) On the other hand
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 0)1003816100381610038161003816 999424999426999456 0 as 119899 997888rarr infin (26)
that is 119860119896 999424999426999456 0 [119881 120582]
4 ISRNMathematical Analysis
(ii) Suppose that 119860119896 is bounded and 119860
119896 rarr 119860(119882119878
120582)
Then there is a119872 such that |119889(119909 119860119896) minus 119889(119909 119860)| le 119872 for all
119896 Given 120576 gt 0 one has
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
le119872
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 + 120576
(27)
which implies that 119860119896 rarr 119860[119881 120582]
(iii) This immediately follows from (i) and (ii)
Definition 15 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin
119883 one says that the sequences 119860119896 and 119861
119896 are said to
be asymptotically Wijsman 119868-equivalent of multiple 119871 if forevery 120576 gt 0 and for each 119909 isin 119883
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (28)
This will be denoted by 119860119896
119868119882
sim 119861119896
Definition 16 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin 119883
one says that the sequences 119860119896 and 119861
119896 are said to be
strong Cesaro 119868-asymptotically equivalent (Wijsman sense)of multiple 119871 if every 120576 gt 0 and for each 119909 isin 119883
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (29)
This will be denoted by 119860119896
119862119871
1(119868119882)
sim 119861119896
Definition 17 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are Wijsman 119868-asymptotically statistically
equivalent of multiple 119871 if for every 120576 120575 gt 0 and for each119909 isin 119883
119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868 (30)
This will be denoted by 119860119896
119878119871
(119868119882)
sim 119861119896
Example 18 Let 119868 sube 2N be a proper ideal in N and let (119883 119889)be a metric space then 119860119860
119896sub 119883 are non-empty closed
subsets Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896
=
(119909 119910) isin R2 0le 119909le119899 0le119910le1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise
119861119896
= (119909 119910) isin R2 0 le 119909 le 119899 0 le 119910 le minus
1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise(31)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (32)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 1198601198961198681
119882
sim119861119896 where 119868
119889is the
ideal of sets that have zero density
Example 19 Let119868 sube 2N be a proper ideal inN and let (119883 119889) beametric space then119860119860
119896sub 119883 are non-empty closed subsets
Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896=
(119909 119910) isin R2 1199092 + (119910 minus 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise
119861119896=
(119909 119910) isin R2 1199092 + (119910 + 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise(33)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (34)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 119860119896
1198681
119882
sim 119861119896 where 119868
119889is
the ideal of sets which have zero density
Definition 20 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are strongly 120582
119868-asymptotically equivalent
(Wijsman sense) of multiple 119871 if for every 120576 gt 0 and for each119909 isin 119883
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
isin 119868 (35)
This will be denoted by 119860119896
119881119871
120582(119868119882)
sim 119861119896
ISRNMathematical Analysis 5
Definition 21 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are 119868-asymptotically 120582-statistically equivalent
(Wijsman sense) of multiple 119871 provided that for every 120576 120575 gt0 and for each 119909 isin 119883
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868
(36)
This will be denoted by 119860119896
119878119871
120582(119868119882)
sim 119861119896
Theorem 22 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that 119860119896
119881119871
120582(119868119882)
sim 119861119896and 120576 gt 0 Then
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(37)
and so
1
120576120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(38)
Then for any 120575 gt 0
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575
sube
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576120575
(39)
Since right hand belongs to 119868 then left hand also belongs to119868 and this completes the proof
Theorem 23 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896 and 119861
119896 are bounded and 119860
119896
119878119871
120582(119868119882)
sim 119861119896 then 119860
119896
119881119871
120582(119868119882)
sim
119861119896
Proof Let 119860119896 119861119896 be bounded sequences and let 119860
119896
119878119871
120582(119868119882)
sim
119861119896 Then there is an119872 such that
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 119872 (40)
for all 119896 For each 120576 gt 0
1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 1198721
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
+120576
2
(41)
Then
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
times
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
ge120576
2119872 isin 119868
(42)
Therefore 119860119896
119881119871
120582(119868119882)
sim 119861119896
The following example shows that if 119860119896 and 119861
119896 are not
bounded thenTheorem 23 cannot be true
Example 24 Take 119871 = 1 and define 119860119896 to be
119860119896=
119896 119896 = 119896119903minus1
+ 1 119896119903minus1
+ 2 119896119903minus1
+ [radic120582119899]
1 otherwise(43)
where lfloorsdotrfloor denotes the greatest integer function and 119861119896= 1
for all 119896 Note that 119860119896 is not boundedThen119860
119896
119878119871
120582(119868)
sim 119861119896 but
119860119896
119881119871
120582(119868)
sim 119861119896is not true
Theorem 25 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119862119871
1(119868119882)
sim 119861119896
6 ISRNMathematical Analysis
Proof Assume that 119860119896
119881119871
120582(119868)
sim 119861119896and 120576 gt 0 Then
1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
119899sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le1
120582119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le2
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
(44)
and so
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2 isin 119868
(45)
Hence 119860119896
119862119871
1(119868119882)
sim 119861119896
Theorem 26 If lim inf 120582119899119899 gt 0 then 119860
119896
119878119871
(119868119882)
sim 119861119896implies
119860119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that lim inf (120582119899119899) gt 0 and there exists a 120575 gt 0
such that 120582119899119899 ge 120575 for sufficiently large 119899 For given 120576 gt 0 one
has
1
119899119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
supe1
119899119896 isin 119868
119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
(46)
Therefore
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge120582119899
119899
1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 1205751
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(47)
then for any 120578 gt 0 we get
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578
sube 119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578120575 isin 119868
(48)
and this completes the proof
References
[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951
[2] Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slovacavol 50 no 1 pp 111ndash115 2000
[3] E Savas ldquoOn strongly 120582-summable sequences of fuzzy num-bersrdquo Information Sciences vol 125 no 1ndash4 pp 181ndash186 2000
[4] E Savas ldquoOn asymptotically 120582-statistical equivalent sequencesof fuzzy numbersrdquoNewMathematics and Natural Computationvol 3 no 3 pp 301ndash306 2007
[5] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[6] P Kostyrko M Macaj T Salat and M Sleziak ldquo119868-convergenceand extremal 119868-limit pointsrdquo Mathematica Slovaca vol 55 no4 pp 443ndash464 2005
[7] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012
[8] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985
[9] R AWijsman ldquoConvergence of sequences of convex sets conesand functionsrdquo Bulletin of the American Mathematical Societyvol 70 pp 186ndash188 1964
[10] R AWijsman ldquoConvergence of sequences of convex sets conesand functions IIrdquo Transactions of the American MathematicalSociety vol 123 pp 32ndash45 1966
[11] U Ulusu and F Nuray ldquoOn asymptotically Lacunary statisticalequivalent set sequencesrdquo Journal of Mathematics vol 2013Article ID 310438 5 pages 2013
[12] O Kısı and F Nuray ldquoA new convergence for sequences of setsrdquosubmitted for publication
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Stochastic AnalysisInternational Journal of
4 ISRNMathematical Analysis
(ii) Suppose that 119860119896 is bounded and 119860
119896 rarr 119860(119882119878
120582)
Then there is a119872 such that |119889(119909 119860119896) minus 119889(119909 119860)| le 119872 for all
119896 Given 120576 gt 0 one has
1
120582119899
sum
119896isin119868119899
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816
le119872
120582119899
1003816100381610038161003816119896 isin 119868119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)
1003816100381610038161003816 ge 1205761003816100381610038161003816 + 120576
(27)
which implies that 119860119896 rarr 119860[119881 120582]
(iii) This immediately follows from (i) and (ii)
Definition 15 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin
119883 one says that the sequences 119860119896 and 119861
119896 are said to
be asymptotically Wijsman 119868-equivalent of multiple 119871 if forevery 120576 gt 0 and for each 119909 isin 119883
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (28)
This will be denoted by 119860119896
119868119882
sim 119861119896
Definition 16 Let (119883 119889) be a metric space and let 119868 be anadmissible ideal For non-empty closed subsets 119860
119896 119861119896sub 119883
such that 119889(119909 119860119896) gt 0 and 119889(119909 119861
119896) gt 0 for each 119909 isin 119883
one says that the sequences 119860119896 and 119861
119896 are said to be
strong Cesaro 119868-asymptotically equivalent (Wijsman sense)of multiple 119871 if every 120576 gt 0 and for each 119909 isin 119883
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (29)
This will be denoted by 119860119896
119862119871
1(119868119882)
sim 119861119896
Definition 17 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are Wijsman 119868-asymptotically statistically
equivalent of multiple 119871 if for every 120576 120575 gt 0 and for each119909 isin 119883
119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868 (30)
This will be denoted by 119860119896
119878119871
(119868119882)
sim 119861119896
Example 18 Let 119868 sube 2N be a proper ideal in N and let (119883 119889)be a metric space then 119860119860
119896sub 119883 are non-empty closed
subsets Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896
=
(119909 119910) isin R2 0le 119909le119899 0le119910le1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise
119861119896
= (119909 119910) isin R2 0 le 119909 le 119899 0 le 119910 le minus
1
119899sdot 119909 if 119896 = 119899
2
0 0 otherwise(31)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (32)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 1198601198961198681
119882
sim119861119896 where 119868
119889is the
ideal of sets that have zero density
Example 19 Let119868 sube 2N be a proper ideal inN and let (119883 119889) beametric space then119860119860
119896sub 119883 are non-empty closed subsets
Let119883 = R2 119860119896 119861119896 be the following sequences
119860119896=
(119909 119910) isin R2 1199092 + (119910 minus 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise
119861119896=
(119909 119910) isin R2 1199092 + (119910 + 1)2=1
119896 if 119896 = 119899
2
0 0 otherwise(33)
If we take 119868 = 119868119889we have
119896 isin N
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 1
100381610038161003816100381610038161003816100381610038161003816
ge 120576 isin 119868 (34)
Thus the sequences 119860119896 and 119861
119896 are asymptotically 119868-
equivalent (Wijsman sense) that is 119860119896
1198681
119882
sim 119861119896 where 119868
119889is
the ideal of sets which have zero density
Definition 20 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are strongly 120582
119868-asymptotically equivalent
(Wijsman sense) of multiple 119871 if for every 120576 gt 0 and for each119909 isin 119883
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
isin 119868 (35)
This will be denoted by 119860119896
119881119871
120582(119868119882)
sim 119861119896
ISRNMathematical Analysis 5
Definition 21 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are 119868-asymptotically 120582-statistically equivalent
(Wijsman sense) of multiple 119871 provided that for every 120576 120575 gt0 and for each 119909 isin 119883
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868
(36)
This will be denoted by 119860119896
119878119871
120582(119868119882)
sim 119861119896
Theorem 22 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that 119860119896
119881119871
120582(119868119882)
sim 119861119896and 120576 gt 0 Then
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(37)
and so
1
120576120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(38)
Then for any 120575 gt 0
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575
sube
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576120575
(39)
Since right hand belongs to 119868 then left hand also belongs to119868 and this completes the proof
Theorem 23 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896 and 119861
119896 are bounded and 119860
119896
119878119871
120582(119868119882)
sim 119861119896 then 119860
119896
119881119871
120582(119868119882)
sim
119861119896
Proof Let 119860119896 119861119896 be bounded sequences and let 119860
119896
119878119871
120582(119868119882)
sim
119861119896 Then there is an119872 such that
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 119872 (40)
for all 119896 For each 120576 gt 0
1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 1198721
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
+120576
2
(41)
Then
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
times
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
ge120576
2119872 isin 119868
(42)
Therefore 119860119896
119881119871
120582(119868119882)
sim 119861119896
The following example shows that if 119860119896 and 119861
119896 are not
bounded thenTheorem 23 cannot be true
Example 24 Take 119871 = 1 and define 119860119896 to be
119860119896=
119896 119896 = 119896119903minus1
+ 1 119896119903minus1
+ 2 119896119903minus1
+ [radic120582119899]
1 otherwise(43)
where lfloorsdotrfloor denotes the greatest integer function and 119861119896= 1
for all 119896 Note that 119860119896 is not boundedThen119860
119896
119878119871
120582(119868)
sim 119861119896 but
119860119896
119881119871
120582(119868)
sim 119861119896is not true
Theorem 25 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119862119871
1(119868119882)
sim 119861119896
6 ISRNMathematical Analysis
Proof Assume that 119860119896
119881119871
120582(119868)
sim 119861119896and 120576 gt 0 Then
1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
119899sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le1
120582119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le2
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
(44)
and so
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2 isin 119868
(45)
Hence 119860119896
119862119871
1(119868119882)
sim 119861119896
Theorem 26 If lim inf 120582119899119899 gt 0 then 119860
119896
119878119871
(119868119882)
sim 119861119896implies
119860119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that lim inf (120582119899119899) gt 0 and there exists a 120575 gt 0
such that 120582119899119899 ge 120575 for sufficiently large 119899 For given 120576 gt 0 one
has
1
119899119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
supe1
119899119896 isin 119868
119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
(46)
Therefore
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge120582119899
119899
1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 1205751
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(47)
then for any 120578 gt 0 we get
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578
sube 119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578120575 isin 119868
(48)
and this completes the proof
References
[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951
[2] Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slovacavol 50 no 1 pp 111ndash115 2000
[3] E Savas ldquoOn strongly 120582-summable sequences of fuzzy num-bersrdquo Information Sciences vol 125 no 1ndash4 pp 181ndash186 2000
[4] E Savas ldquoOn asymptotically 120582-statistical equivalent sequencesof fuzzy numbersrdquoNewMathematics and Natural Computationvol 3 no 3 pp 301ndash306 2007
[5] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[6] P Kostyrko M Macaj T Salat and M Sleziak ldquo119868-convergenceand extremal 119868-limit pointsrdquo Mathematica Slovaca vol 55 no4 pp 443ndash464 2005
[7] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012
[8] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985
[9] R AWijsman ldquoConvergence of sequences of convex sets conesand functionsrdquo Bulletin of the American Mathematical Societyvol 70 pp 186ndash188 1964
[10] R AWijsman ldquoConvergence of sequences of convex sets conesand functions IIrdquo Transactions of the American MathematicalSociety vol 123 pp 32ndash45 1966
[11] U Ulusu and F Nuray ldquoOn asymptotically Lacunary statisticalequivalent set sequencesrdquo Journal of Mathematics vol 2013Article ID 310438 5 pages 2013
[12] O Kısı and F Nuray ldquoA new convergence for sequences of setsrdquosubmitted for publication
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Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 5
Definition 21 Let (119883 119889) be a metric space For non-emptyclosed subsets 119860
119896 119861119896sub 119883 such that 119889(119909 119860
119896) gt 0 and
119889(119909 119861119896) gt 0 for each 119909 isin 119883 one says that the sequences
119860119896 and 119861
119896 are 119868-asymptotically 120582-statistically equivalent
(Wijsman sense) of multiple 119871 provided that for every 120576 120575 gt0 and for each 119909 isin 119883
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575 isin 119868
(36)
This will be denoted by 119860119896
119878119871
120582(119868119882)
sim 119861119896
Theorem 22 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that 119860119896
119881119871
120582(119868119882)
sim 119861119896and 120576 gt 0 Then
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(37)
and so
1
120576120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(38)
Then for any 120575 gt 0
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120575
sube
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576120575
(39)
Since right hand belongs to 119868 then left hand also belongs to119868 and this completes the proof
Theorem 23 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896 and 119861
119896 are bounded and 119860
119896
119878119871
120582(119868119882)
sim 119861119896 then 119860
119896
119881119871
120582(119868119882)
sim
119861119896
Proof Let 119860119896 119861119896 be bounded sequences and let 119860
119896
119878119871
120582(119868119882)
sim
119861119896 Then there is an119872 such that
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 119872 (40)
for all 119896 For each 120576 gt 0
1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|ge120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
|119889(119909119860119896)minus119889(119909119860)|lt120576
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le 1198721
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
+120576
2
(41)
Then
119899 isin N 1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
times
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2
100381610038161003816100381610038161003816100381610038161003816
ge120576
2119872 isin 119868
(42)
Therefore 119860119896
119881119871
120582(119868119882)
sim 119861119896
The following example shows that if 119860119896 and 119861
119896 are not
bounded thenTheorem 23 cannot be true
Example 24 Take 119871 = 1 and define 119860119896 to be
119860119896=
119896 119896 = 119896119903minus1
+ 1 119896119903minus1
+ 2 119896119903minus1
+ [radic120582119899]
1 otherwise(43)
where lfloorsdotrfloor denotes the greatest integer function and 119861119896= 1
for all 119896 Note that 119860119896 is not boundedThen119860
119896
119878119871
120582(119868)
sim 119861119896 but
119860119896
119881119871
120582(119868)
sim 119861119896is not true
Theorem 25 Let 120582 isin Λ and let 119868 be an admissible ideal in NIf 119860119896
119881119871
120582(119868119882)
sim 119861119896 then 119860
119896
119862119871
1(119868119882)
sim 119861119896
6 ISRNMathematical Analysis
Proof Assume that 119860119896
119881119871
120582(119868)
sim 119861119896and 120576 gt 0 Then
1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
119899sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le1
120582119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le2
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
(44)
and so
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2 isin 119868
(45)
Hence 119860119896
119862119871
1(119868119882)
sim 119861119896
Theorem 26 If lim inf 120582119899119899 gt 0 then 119860
119896
119878119871
(119868119882)
sim 119861119896implies
119860119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that lim inf (120582119899119899) gt 0 and there exists a 120575 gt 0
such that 120582119899119899 ge 120575 for sufficiently large 119899 For given 120576 gt 0 one
has
1
119899119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
supe1
119899119896 isin 119868
119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
(46)
Therefore
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge120582119899
119899
1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 1205751
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(47)
then for any 120578 gt 0 we get
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578
sube 119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578120575 isin 119868
(48)
and this completes the proof
References
[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951
[2] Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slovacavol 50 no 1 pp 111ndash115 2000
[3] E Savas ldquoOn strongly 120582-summable sequences of fuzzy num-bersrdquo Information Sciences vol 125 no 1ndash4 pp 181ndash186 2000
[4] E Savas ldquoOn asymptotically 120582-statistical equivalent sequencesof fuzzy numbersrdquoNewMathematics and Natural Computationvol 3 no 3 pp 301ndash306 2007
[5] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[6] P Kostyrko M Macaj T Salat and M Sleziak ldquo119868-convergenceand extremal 119868-limit pointsrdquo Mathematica Slovaca vol 55 no4 pp 443ndash464 2005
[7] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012
[8] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985
[9] R AWijsman ldquoConvergence of sequences of convex sets conesand functionsrdquo Bulletin of the American Mathematical Societyvol 70 pp 186ndash188 1964
[10] R AWijsman ldquoConvergence of sequences of convex sets conesand functions IIrdquo Transactions of the American MathematicalSociety vol 123 pp 32ndash45 1966
[11] U Ulusu and F Nuray ldquoOn asymptotically Lacunary statisticalequivalent set sequencesrdquo Journal of Mathematics vol 2013Article ID 310438 5 pages 2013
[12] O Kısı and F Nuray ldquoA new convergence for sequences of setsrdquosubmitted for publication
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Stochastic AnalysisInternational Journal of
6 ISRNMathematical Analysis
Proof Assume that 119860119896
119881119871
120582(119868)
sim 119861119896and 120576 gt 0 Then
1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
=1
119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
119899sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le1
120582119899
119899minus120582119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
+1
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
le2
120582119899
sum
119896isin119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
(44)
and so
119899 isin N 1
119899
119899
sum
119896=1
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
sube 119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge120576
2 isin 119868
(45)
Hence 119860119896
119862119871
1(119868119882)
sim 119861119896
Theorem 26 If lim inf 120582119899119899 gt 0 then 119860
119896
119878119871
(119868119882)
sim 119861119896implies
119860119896
119878119871
120582(119868119882)
sim 119861119896
Proof Assume that lim inf (120582119899119899) gt 0 and there exists a 120575 gt 0
such that 120582119899119899 ge 120575 for sufficiently large 119899 For given 120576 gt 0 one
has
1
119899119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
supe1
119899119896 isin 119868
119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
(46)
Therefore
1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge120582119899
119899
1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 1205751
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
(47)
then for any 120578 gt 0 we get
119899 isin N 1
120582119899
100381610038161003816100381610038161003816100381610038161003816
119896 isin 119868119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578
sube 119899 isin N 1
119899
100381610038161003816100381610038161003816100381610038161003816
119896 le 119899
100381610038161003816100381610038161003816100381610038161003816
119889 (119909 119860119896)
119889 (119909 119861119896)minus 119871
100381610038161003816100381610038161003816100381610038161003816
ge 120576
100381610038161003816100381610038161003816100381610038161003816
ge 120578120575 isin 119868
(48)
and this completes the proof
References
[1] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951
[2] Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slovacavol 50 no 1 pp 111ndash115 2000
[3] E Savas ldquoOn strongly 120582-summable sequences of fuzzy num-bersrdquo Information Sciences vol 125 no 1ndash4 pp 181ndash186 2000
[4] E Savas ldquoOn asymptotically 120582-statistical equivalent sequencesof fuzzy numbersrdquoNewMathematics and Natural Computationvol 3 no 3 pp 301ndash306 2007
[5] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000
[6] P Kostyrko M Macaj T Salat and M Sleziak ldquo119868-convergenceand extremal 119868-limit pointsrdquo Mathematica Slovaca vol 55 no4 pp 443ndash464 2005
[7] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012
[8] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985
[9] R AWijsman ldquoConvergence of sequences of convex sets conesand functionsrdquo Bulletin of the American Mathematical Societyvol 70 pp 186ndash188 1964
[10] R AWijsman ldquoConvergence of sequences of convex sets conesand functions IIrdquo Transactions of the American MathematicalSociety vol 123 pp 32ndash45 1966
[11] U Ulusu and F Nuray ldquoOn asymptotically Lacunary statisticalequivalent set sequencesrdquo Journal of Mathematics vol 2013Article ID 310438 5 pages 2013
[12] O Kısı and F Nuray ldquoA new convergence for sequences of setsrdquosubmitted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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