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

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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Mathematical Problems in Engineering

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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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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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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