Research ArticleInvariant Statistical Convergence of Sequences of Sets withrespect to a Modulus Function

Nimet Pancaroglu and Fatih Nuray

Department of Mathematics Afyon Kocatepe University Afyonkarahisar Turkey

Correspondence should be addressed to Fatih Nuray fnurayakuedutr

Received 15 December 2013 Accepted 21 February 2014 Published 23 March 2014

Academic Editor Irena Rachunkova

Copyright copy 2014 N Pancaroglu and F Nuray This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We introduce and study the concept of invariant convergence for sequences of sets with respect to modulus function 119891 and givesome inclusion relations

1 Introduction

The concept of statistical convergence for sequences of realnumbers was introduced by Fast [1] and studied by Salat [2]and others Let 119870 sube N and 119870

119899= 119896 le 119899 119896 isin 119870 Then the

natural density of 119870 is defined by 120575(119870) = lim119899119899minus1|119870

119899| if the

limit exists where |119870119899| denotes the cardinality of119870

119899

A sequence 119909 = (119909119896) complex numbers is said to be

statistically convergent to 119871 if for each 120576 gt 0

lim119899

1

119899

1003816100381610038161003816119896 le 119899 1003816100381610038161003816119909119896 minus 119871

1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (1)

Convergence concept for sequences of set had beenstudied by Beer [3] Aubin and Frankowska [4] and Barontiand Papini [5] The concept of statistical convergence ofsequences of set was introduced by Nuray and Rhoades [6] in2012Ulusu andNuray [7] introduced the concept ofWijsmanlacunary statistical convergence of sequences of set Similarlythe concepts of Wijsman invariant statistical and Wijsmanlacunary invariant statistical convergence were introduced byPancaroglu and Nuray [8] in 2013

A function 119891 [0infin) rarr [0infin) is called a modulus if

(1) 119891(119909) = 0 if and if only if 119909 = 0

(2) 119891(119909 + 119910) le 119891(119909) + 119891(119910)

(3) 119891 is increasing

(4) 119891 is continuous from the right at 0

A modulus may be unbounded (eg 119891(119909) = 119909119901 0 lt 119901 lt1) or bounded (eg 119891(119909) = 119909(119909 + 1))

Modulus function was introduced by Nakano [9] in 1953Ruckle [10] used in idea of modulus function 119891 to constructa class of FK spaces Consider

119871 (119891) = 119909 = (119909119896) infin

sum119896=1

119891 (10038161003816100381610038161199091198961003816100381610038161003816) lt infin (2)

The space 119871(119891) is closely related to the space ℓ1which is a

119871(119891) space with 119891(119909) = 119909 for all real 119909 ge 0Maddox [11] defined the following spaces by using a

modulus function 119891

1199080(119891) = 119909 isin 119904 lim

119899rarrinfin

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161199091198961003816100381610038161003816) = 0

119908 (119891) = 119909 isin 119904 lim119899rarrinfin

1

119899

119899

sum119896=1

119891 (1003816100381610038161003816119909119896 minus 119871

1003816100381610038161003816) = 0 for some 119871

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 818020 5 pageshttpdxdoiorg1011552014818020

2 Abstract and Applied Analysis

119908infin(119891) = 119909 isin 119904 sup

119899

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161199091198961003816100381610038161003816) lt infin

(3)

where 119904 is space of all complex sequencesLater Connor [12] extended his definition by replacing

the Cesaro matrix with an arbitrary nonnegative matrixsummability method 119861 = (119887

119899119896) as follow

119908 (119861 119891)

= 119909 isin 119904 lim119899rarrinfin

infin

sum119896=1

119887119899119896119891 (1003816100381610038161003816119909119896 minus 119871

1003816100381610038161003816) = 0 for some 119871

(4)

2 Definitions and Notations

Let 120590 be a mapping of the positive integers into itself Acontinuous linear functional 120593 on ℓ

infin the space of real

bounded sequences is said to be an invariant mean or a 120590mean if and only if

(1) 120601(119909) ge 0 for all sequences 119909 = (119909119899) with 119909

119899ge 0 for

all 119899

(2) 120601(119890) = 1 where 119890 = (1 1 1 )

(3) 120601(119909120590(119899)) = 120601(119909) for all 119909 isin ℓ

infin

The mappings 120601 are assumed to be one-to-one such that120590119898(119899) = 119899 for all positive integers 119899 and 119898 where 120590119898(119899)denotes the119898th iterate of themapping 120590 at 119899Thus 120601 extendsthe limit functional on 119888 the space of convergent sequencesin the sense that 120601(119909) = lim119909 for all 119909 isin 119888 In case 120590is translation mapping 120590(119899) = 119899 + 1 the 120590 mean is oftencalled a Banach limit and 119881

120590 the set of bounded sequences

all of whose invariant means are equal is the set of almostconvergent sequences

It can be shown that

119881120590= 119909 = (119909

119899) lim119899

119905119898119899(119909) = 119871

uniformly in 119898 119871 = 120590 minus lim119909 (5)

where

119905119898119899(119909) =

119909119898+ 119909120590(119898)

+ 1199091205902(119898)+ sdot sdot sdot + 119909

120590119899(119898)

119899 + 1 (6)

Nuray and Savas [13] defined the following sequencespaces by using a modulus function 119891 and a nonnegativeregular matrix 119861 = (119887

119899119896)

1199080(119861120590 119891) = 119909 isin 119904 lim

119899rarrinfin

infin

sum119896=1

119887119899119896119891 (10038161003816100381610038161003816119909120590119896(119898)

10038161003816100381610038161003816) = 0

uniformly in 119898

119908 (119861120590 119891)

= 119909 isin 119904 lim119899rarrinfin

infin

sum119896=1

119887119899119896119891 (10038161003816100381610038161003816119909120590119896(119898) minus 119871

10038161003816100381610038161003816) = 0

for some 119871 uniformly in 119898

119908infin(119861120590 119891) = 119909 isin 119904 sup

119899119898

infin

sum119896=1

119887119899119896119891 (10038161003816100381610038161003816119909120590119896(119898)

10038161003816100381610038161003816) lt infin

(7)

Definition 1 (see [14]) A set 119864 of positive integers is said tohave uniform invariant density of zero if

lim119899rarrinfin

1

119899

10038161003816100381610038161003816119896 le 119899 119864 cap 120590 (119898) 1205902

(119898) 120590119896

(119898)10038161003816100381610038161003816 = 0 (8)

uniformly in119898

By using uniform invariant density the following defini-tion was given

Definition 2 (see [10]) A complex number sequence 119909 = (119909119896)

is said to be 120590-statistically convergent to 119871 if for every 120576 gt 0

lim119899

1

119899

100381610038161003816100381610038160 le 119896 le 119899 10038161003816100381610038161003816119909120590119896(119898) minus 119871

10038161003816100381610038161003816 ge 12057610038161003816100381610038161003816 = 0

uniformly in 119898 = 1 2 (9)

This will be denoted by 119878120590minus lim119909 = 119871 or 119909

119896rarr 119871(119878

120590)

Let (119883 120588) be a metric space For any point 119909 isin 119883 andnonempty subset 119860 of 119883 we define the distance from 119909 to 119860by

119889 (119909 119860) = inf119886isin119860

120588 (119909 119860) (10)

The concept of Wijsman convergence was introduced byWijsman [7] as follows

Let (119883 120588) be a metric space For any nonempty closedsubsets119860119860

119896sube 119883 we say that the sequence 119860

119896 is Wijsman

convergent to 119860 if

lim119896rarrinfin

119889 (119909 119860119896) = 119889 (119909 119860) (11)

for each 119909 isin 119883 This will be denoted by119882minus lim119860119896= 119860

Convergence concept for sequences of set had been stud-ied by Beer [3] Aubin and Frankowska [4] and Baronti and

Abstract and Applied Analysis 3

Papini [5] The concepts of Wijsman statistical convergenceandWijsman strong Cesaro summability were introduced byNuray and Rhoades [6] as follows

Let (119883 120588) be a metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 the sequence 119860

119896 is said to be Wijsman

strongly Cesaro summable to 119860 if for each 119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (12)

Let (119883 120588) be a metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 the sequence 119860

119896 is said to be Wijsman

statistically convergent to 119860 if for 120576 gt 0 and each 119909 isin 119883

lim119899rarrinfin

1

119899

1003816100381610038161003816119896 le 119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)

1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (13)

In this case we write 119904119905 minus lim119882119860119896= 119860 or 119860

119896rarr 119860(119882119878)

3 Main Result

The purpose of this paper is by using a modulus functionto introduce and study new sequence spaces of sequences ofsets The following three definitions were given in [8]

Definition 3 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman invariant convergent to 119860 if for each 119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

119889 (119909 119860120590119896(119898)) = 119889 (119909 119860) (14)

uniformly in119898

In this case we write 119860119896rarr 119860(119882119881

120590) and the set of

all Wijsman invariant convergent sequences of sets will bedenoted119882119881

120590

Definition 4 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman strongly invariant convergent to 119860 if for each 119909 isin119883

lim119899rarrinfin

1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 = 0 (15)

uniformly in119898

In this case we write 119860119896rarr 119860([119882119881

120590]) and the set of all

Wijsman strongly invariant convergent sequences of sets willbe denoted [119882119881

120590]

Definition 5 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman invariant statistically convergent to 119860 if for each120576 gt 0 and for each 119909 isin 119883

lim119899rarrinfin

1

119899

100381610038161003816100381610038160 le 119896 le 119899 10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816 ge 12057610038161003816100381610038161003816 = 0

(16)

uniformly in119898

In this case we write 119860119896rarr 119860(119882119878

120590) and the set of all

Wijsman invariant statistically convergent sequences of setswill be denoted119882119878

120590

Let (119883 120588) be metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 and 119909 isin 119883 we define the sequences of

sets space [119882119881120590]infin

as follows

[119882119881120590]infin= 119860

119896 sup119898119899

1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898))10038161003816100381610038161003816 lt infin (17)

Now by using a modulus function we introduce thefollowing new sequence spaces of sequences of sets

Definition 6 Let (119883 120588) be a metric space and119891 be a modulusfunction For any nonempty closed subsets 119860 119860

119896sube 119883 we

say that the sequence 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to the modulus 119891 if for each119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) = 0 (18)

uniformly in119898

In this case we write 119860119896rarr 119860([119882119881

120590(119891)]) and the set of

all Wijsman strongly invariant convergent sequences of setswith respect to the modulus 119891 will be denoted [119882119881

120590(119891)]

Let (119883 120588) be metric space and 119891 be a modulus functionFor any nonempty closed subsets 119860 119860

119896sube 119883 and 119909 isin 119883 we

define the sequences of sets space [119882119881120590(119891)]infin

as follows

[119882119881120590(119891)]infin= 119860

119896 sup119898119899

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816) lt infin

(19)

If 119891(119909) = 119909 then the spaces [119882119881120590(119891)] and [119882119881

120590(119891)]infin

reduce to [119882119881120590] and [119882119881

120590]infin respectively

Now we study the relation between 119882119878120590and [119882119881

120590(119891)]

convergence

Theorem 7 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 Then

(i) 119860119896rarr 119860([119882119881

120590(119891)]) implies 119860

119896rarr 119860(119882119878

120590)

(ii) 119891 is bounded and 119860119896rarr 119860(119882119878

120590) implies 119860

119896rarr

119860([119882119881120590(119891)])

(iii) 119882119878120590= [119882119881

120590(119891)] if 119891 is bounded

Proof (i) Let 120576 gt 0 and 119860119896rarr 119860([119882119881

120590(119891)]) Then we can

write119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

ge119899

sum119896=1

|119889(119909119860120590119896(119898))minus119889(119909119860)|ge120576

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

ge 120576 sdot10038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge 120576

10038161003816100381610038161003816

(20)

which yields the result

4 Abstract and Applied Analysis

(ii) Suppose that 119860119896rarr 119860(119882119878

120590) and 119891 is bounded for

each119898 ge 1 set

119866 = sup119896119898

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816 + |119889 (119909 119860)|) (21)

Let 120576 gt 0 and select119873120576such that

1

119899

1003816100381610038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge

120576

21003816100381610038161003816100381610038161003816lt120576

2119866 (22)

for all 119898 and 119899 gt 119873120576 and set 119871(119899119898 119909) = 119896 le 119899

|119889(119909 119860120590119896(119898)) minus119889(119909 119860)| ge 1205762 Now for all119898 and 119899 gt 119873

120576 we

have that

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

= sum119896isin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

+ sum119896notin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

lt1

119899(119899

120576

2119866)119866 +

1

119899(120576

2) 119899 =

120576

2+120576

2= 120576

(23)

Hence 119860119896 is strongly invariant convergent to 119860 with

respect to the modulus function 119891(iii)This is an immediate consequence of (i) and (ii)This completes the proof of the theorem

Theorem 8 Let 119891 be a modulus function Then [119882119881120590(119891)] sub

[119882119881120590(119891)]infin

Proof Suppose that 119860119896 isin [119882119881

120590(119891)] Then we can write

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816)

=1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860) + 119889 (119909 119860)

10038161003816100381610038161003816)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) +1

119899

119899

sum119896=1

119891 (|119889 (119909 119860)|)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) + 1198721

119899

119899

sum119896=1

119891 (1)

(24)

where 119872 is an integer such that 119889(119909 119860) lt 119872 Therefore119860119896 isin [119882119881

120590(119891)]infin

Theorem 9 If 119891 is a modulus function and 119860119896 is strongly

invariant convergent to 119860 then 119860119896 is strongly invariant

convergent to119860with respect to themodulus119891 that is [119882119881120590] sub

[119882119881120590(119891)]

Proof Let 119860119896 isin [119882119881

120590] Then we can write

119878119899119898=1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(25)

uniformly in119898Let 120576 gt 0 and choose 120575 with 0 lt 120575 lt 1 such that 119891(119905) lt 120576

for 0 le 119905 le 120575 Write 119886119896119898= |119889(119909 119860

120590119896(119898)) minus 119889(119909 119860)|

119899

sum119896=1

119891 (119886119896119898) = Σ1+ Σ2 (26)

where the first summation is over 119886119896119898le 120575 and second over

119886119896119898gt 120575 Then Σ

1le 120576119899 and for 119886

119896119898gt 120575

119891 (119886119896119898) lt

119886119896119898

120575lt 1 + [

119886119896119898

120575] (27)

where [119911] denotes the integer part of 119911 By definitionmodulusfunction we have for 119886

119896119898gt 120575

119891 (119886119896119898) le (1 + [

119886119896119898

120575])119891 (1) le 2119891 (1)

119886119896119898

120575 (28)

Hence Σ2le 2119891(1)(119886

119896119898120575) which together with Σ

1le 120576119899

yields [119882119881120590] sub [119882119881

120590(119891)]

Lemma 10 (see [15]) Let 119891 be a modulus function Let 120572 gt 0be given constant Then there is a constant 119888 gt 0 such that119891(119909) gt 119888119909 (0 lt 119909 lt 120572)

Theorem 11 Let 119860119896 be a bounded sequence and 119891 be a

modulus function Then 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to modulus 119891 if and only if 119860119896

is Wijsman strongly invariant convergent to 119860 that is

[119882119881120590(119891)] cap 119871

infin= [119882119881

120590] (29)

where 119871infin

denotes the set of bounded sequences of sets

Proof The proof of the theorem follows fromTheorem 9 andLemma 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Fast ldquoSur la convergence statistiquerdquoColloquiumMathemat-icae vol 2 no 3-4 pp 241ndash244 1951

[2] T Salat ldquoOn statistically convergent sequences of real numbersrdquoMathematica Slovaca vol 30 no 2 pp 139ndash150 1980

[3] G Beer ldquoConvergence of continuous linear functionals andtheir level setsrdquo Archiv der Mathematik vol 52 no 5 pp 482ndash491 1989

[4] J P Aubin and H Frankowska Set Valued Analysis BirkhauserBoston Mass USA 1990

Abstract and Applied Analysis 5

[5] M Baronti and P L Papini ldquoConvergence of sequences of setsrdquoinMethods of Functional Analysis in ApproximationTheory vol76 pp 135ndash155 Birkhauser Basel Switzerland 1986

[6] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012

[7] U Ulusu and F Nuray ldquoLacunary statistical convergencesequences of setsrdquo Progress in Applied Mathematics vol 4 no2 pp 99ndash109 2012

[8] N Pancaroglu and F Nuray ldquoOn invariant statistically con-vergence and lacu-nary invariant statistically convergence ofsequences of setsrdquo Progress in Applied Mathematics vol 5 no2 pp 23ndash29 2013

[9] H Nakano ldquoConcave modularsrdquo Journal of the MathematicalSociety of Japan vol 5 no 1 pp 29ndash49 1953

[10] W H Ruckle ldquo119865119870 spaces in which the sequence of coordinatevectors is boundedrdquo Canadian Journal of Mathematics vol 25pp 973ndash978 1973

[11] I J Maddox ldquoSequence spaces defined by a modulusrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol100 no 1 pp 161ndash166 1986

[12] J Connor ldquoOn strong matrix summability with respect to amodulus and statistical convergencerdquo Canadian MathematicalBulletin vol 32 no 2 pp 194ndash198 1989

[13] F Nuray and E Savas ldquoSome new sequence spaces definedby a modulus functionrdquo Indian Journal of Pure and AppliedMathematics vol 24 no 11 pp 657ndash663 1993

[14] E Savas ldquoStrongly 120590-convergent sequencesrdquo Bulletin of theCalcutta Mathematical Society vol 81 no 4 pp 295ndash300 1989

[15] E Savas ldquoOn strong almost 119860-summability with respect to amodulus and statistical convergencerdquo Indian Journal of Pure andApplied Mathematics vol 23 no 3 pp 217ndash222 1992

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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

2 Abstract and Applied Analysis

119908infin(119891) = 119909 isin 119904 sup

119899

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161199091198961003816100381610038161003816) lt infin

(3)

where 119904 is space of all complex sequencesLater Connor [12] extended his definition by replacing

the Cesaro matrix with an arbitrary nonnegative matrixsummability method 119861 = (119887

119899119896) as follow

119908 (119861 119891)

= 119909 isin 119904 lim119899rarrinfin

infin

sum119896=1

119887119899119896119891 (1003816100381610038161003816119909119896 minus 119871

1003816100381610038161003816) = 0 for some 119871

(4)

2 Definitions and Notations

Let 120590 be a mapping of the positive integers into itself Acontinuous linear functional 120593 on ℓ

infin the space of real

bounded sequences is said to be an invariant mean or a 120590mean if and only if

(1) 120601(119909) ge 0 for all sequences 119909 = (119909119899) with 119909

119899ge 0 for

all 119899

(2) 120601(119890) = 1 where 119890 = (1 1 1 )

(3) 120601(119909120590(119899)) = 120601(119909) for all 119909 isin ℓ

infin

The mappings 120601 are assumed to be one-to-one such that120590119898(119899) = 119899 for all positive integers 119899 and 119898 where 120590119898(119899)denotes the119898th iterate of themapping 120590 at 119899Thus 120601 extendsthe limit functional on 119888 the space of convergent sequencesin the sense that 120601(119909) = lim119909 for all 119909 isin 119888 In case 120590is translation mapping 120590(119899) = 119899 + 1 the 120590 mean is oftencalled a Banach limit and 119881

120590 the set of bounded sequences

all of whose invariant means are equal is the set of almostconvergent sequences

It can be shown that

119881120590= 119909 = (119909

119899) lim119899

119905119898119899(119909) = 119871

uniformly in 119898 119871 = 120590 minus lim119909 (5)

where

119905119898119899(119909) =

119909119898+ 119909120590(119898)

+ 1199091205902(119898)+ sdot sdot sdot + 119909

120590119899(119898)

119899 + 1 (6)

Nuray and Savas [13] defined the following sequencespaces by using a modulus function 119891 and a nonnegativeregular matrix 119861 = (119887

119899119896)

1199080(119861120590 119891) = 119909 isin 119904 lim

119899rarrinfin

infin

sum119896=1

119887119899119896119891 (10038161003816100381610038161003816119909120590119896(119898)

10038161003816100381610038161003816) = 0

uniformly in 119898

119908 (119861120590 119891)

= 119909 isin 119904 lim119899rarrinfin

infin

sum119896=1

119887119899119896119891 (10038161003816100381610038161003816119909120590119896(119898) minus 119871

10038161003816100381610038161003816) = 0

for some 119871 uniformly in 119898

119908infin(119861120590 119891) = 119909 isin 119904 sup

119899119898

infin

sum119896=1

119887119899119896119891 (10038161003816100381610038161003816119909120590119896(119898)

10038161003816100381610038161003816) lt infin

(7)

Definition 1 (see [14]) A set 119864 of positive integers is said tohave uniform invariant density of zero if

lim119899rarrinfin

1

119899

10038161003816100381610038161003816119896 le 119899 119864 cap 120590 (119898) 1205902

(119898) 120590119896

(119898)10038161003816100381610038161003816 = 0 (8)

uniformly in119898

By using uniform invariant density the following defini-tion was given

Definition 2 (see [10]) A complex number sequence 119909 = (119909119896)

is said to be 120590-statistically convergent to 119871 if for every 120576 gt 0

lim119899

1

119899

100381610038161003816100381610038160 le 119896 le 119899 10038161003816100381610038161003816119909120590119896(119898) minus 119871

10038161003816100381610038161003816 ge 12057610038161003816100381610038161003816 = 0

uniformly in 119898 = 1 2 (9)

This will be denoted by 119878120590minus lim119909 = 119871 or 119909

119896rarr 119871(119878

120590)

Let (119883 120588) be a metric space For any point 119909 isin 119883 andnonempty subset 119860 of 119883 we define the distance from 119909 to 119860by

119889 (119909 119860) = inf119886isin119860

120588 (119909 119860) (10)

The concept of Wijsman convergence was introduced byWijsman [7] as follows

Let (119883 120588) be a metric space For any nonempty closedsubsets119860119860

119896sube 119883 we say that the sequence 119860

119896 is Wijsman

convergent to 119860 if

lim119896rarrinfin

119889 (119909 119860119896) = 119889 (119909 119860) (11)

for each 119909 isin 119883 This will be denoted by119882minus lim119860119896= 119860

Convergence concept for sequences of set had been stud-ied by Beer [3] Aubin and Frankowska [4] and Baronti and

Abstract and Applied Analysis 3

Papini [5] The concepts of Wijsman statistical convergenceandWijsman strong Cesaro summability were introduced byNuray and Rhoades [6] as follows

Let (119883 120588) be a metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 the sequence 119860

119896 is said to be Wijsman

strongly Cesaro summable to 119860 if for each 119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (12)

Let (119883 120588) be a metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 the sequence 119860

119896 is said to be Wijsman

statistically convergent to 119860 if for 120576 gt 0 and each 119909 isin 119883

lim119899rarrinfin

1

119899

1003816100381610038161003816119896 le 119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)

1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (13)

In this case we write 119904119905 minus lim119882119860119896= 119860 or 119860

119896rarr 119860(119882119878)

3 Main Result

The purpose of this paper is by using a modulus functionto introduce and study new sequence spaces of sequences ofsets The following three definitions were given in [8]

Definition 3 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman invariant convergent to 119860 if for each 119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

119889 (119909 119860120590119896(119898)) = 119889 (119909 119860) (14)

uniformly in119898

In this case we write 119860119896rarr 119860(119882119881

120590) and the set of

all Wijsman invariant convergent sequences of sets will bedenoted119882119881

120590

Definition 4 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman strongly invariant convergent to 119860 if for each 119909 isin119883

lim119899rarrinfin

1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 = 0 (15)

uniformly in119898

In this case we write 119860119896rarr 119860([119882119881

120590]) and the set of all

Wijsman strongly invariant convergent sequences of sets willbe denoted [119882119881

120590]

Definition 5 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman invariant statistically convergent to 119860 if for each120576 gt 0 and for each 119909 isin 119883

lim119899rarrinfin

1

119899

100381610038161003816100381610038160 le 119896 le 119899 10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816 ge 12057610038161003816100381610038161003816 = 0

(16)

uniformly in119898

In this case we write 119860119896rarr 119860(119882119878

120590) and the set of all

Wijsman invariant statistically convergent sequences of setswill be denoted119882119878

120590

Let (119883 120588) be metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 and 119909 isin 119883 we define the sequences of

sets space [119882119881120590]infin

as follows

[119882119881120590]infin= 119860

119896 sup119898119899

1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898))10038161003816100381610038161003816 lt infin (17)

Now by using a modulus function we introduce thefollowing new sequence spaces of sequences of sets

Definition 6 Let (119883 120588) be a metric space and119891 be a modulusfunction For any nonempty closed subsets 119860 119860

119896sube 119883 we

say that the sequence 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to the modulus 119891 if for each119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) = 0 (18)

uniformly in119898

In this case we write 119860119896rarr 119860([119882119881

120590(119891)]) and the set of

all Wijsman strongly invariant convergent sequences of setswith respect to the modulus 119891 will be denoted [119882119881

120590(119891)]

Let (119883 120588) be metric space and 119891 be a modulus functionFor any nonempty closed subsets 119860 119860

119896sube 119883 and 119909 isin 119883 we

define the sequences of sets space [119882119881120590(119891)]infin

as follows

[119882119881120590(119891)]infin= 119860

119896 sup119898119899

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816) lt infin

(19)

If 119891(119909) = 119909 then the spaces [119882119881120590(119891)] and [119882119881

120590(119891)]infin

reduce to [119882119881120590] and [119882119881

120590]infin respectively

Now we study the relation between 119882119878120590and [119882119881

120590(119891)]

convergence

Theorem 7 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 Then

(i) 119860119896rarr 119860([119882119881

120590(119891)]) implies 119860

119896rarr 119860(119882119878

120590)

(ii) 119891 is bounded and 119860119896rarr 119860(119882119878

120590) implies 119860

119896rarr

119860([119882119881120590(119891)])

(iii) 119882119878120590= [119882119881

120590(119891)] if 119891 is bounded

Proof (i) Let 120576 gt 0 and 119860119896rarr 119860([119882119881

120590(119891)]) Then we can

write119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

ge119899

sum119896=1

|119889(119909119860120590119896(119898))minus119889(119909119860)|ge120576

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

ge 120576 sdot10038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge 120576

10038161003816100381610038161003816

(20)

which yields the result

4 Abstract and Applied Analysis

(ii) Suppose that 119860119896rarr 119860(119882119878

120590) and 119891 is bounded for

each119898 ge 1 set

119866 = sup119896119898

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816 + |119889 (119909 119860)|) (21)

Let 120576 gt 0 and select119873120576such that

1

119899

1003816100381610038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge

120576

21003816100381610038161003816100381610038161003816lt120576

2119866 (22)

for all 119898 and 119899 gt 119873120576 and set 119871(119899119898 119909) = 119896 le 119899

|119889(119909 119860120590119896(119898)) minus119889(119909 119860)| ge 1205762 Now for all119898 and 119899 gt 119873

120576 we

have that

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

= sum119896isin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

+ sum119896notin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

lt1

119899(119899

120576

2119866)119866 +

1

119899(120576

2) 119899 =

120576

2+120576

2= 120576

(23)

Hence 119860119896 is strongly invariant convergent to 119860 with

respect to the modulus function 119891(iii)This is an immediate consequence of (i) and (ii)This completes the proof of the theorem

Theorem 8 Let 119891 be a modulus function Then [119882119881120590(119891)] sub

[119882119881120590(119891)]infin

Proof Suppose that 119860119896 isin [119882119881

120590(119891)] Then we can write

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816)

=1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860) + 119889 (119909 119860)

10038161003816100381610038161003816)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) +1

119899

119899

sum119896=1

119891 (|119889 (119909 119860)|)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) + 1198721

119899

119899

sum119896=1

119891 (1)

(24)

where 119872 is an integer such that 119889(119909 119860) lt 119872 Therefore119860119896 isin [119882119881

120590(119891)]infin

Theorem 9 If 119891 is a modulus function and 119860119896 is strongly

invariant convergent to 119860 then 119860119896 is strongly invariant

convergent to119860with respect to themodulus119891 that is [119882119881120590] sub

[119882119881120590(119891)]

Proof Let 119860119896 isin [119882119881

120590] Then we can write

119878119899119898=1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(25)

uniformly in119898Let 120576 gt 0 and choose 120575 with 0 lt 120575 lt 1 such that 119891(119905) lt 120576

for 0 le 119905 le 120575 Write 119886119896119898= |119889(119909 119860

120590119896(119898)) minus 119889(119909 119860)|

119899

sum119896=1

119891 (119886119896119898) = Σ1+ Σ2 (26)

where the first summation is over 119886119896119898le 120575 and second over

119886119896119898gt 120575 Then Σ

1le 120576119899 and for 119886

119896119898gt 120575

119891 (119886119896119898) lt

119886119896119898

120575lt 1 + [

119886119896119898

120575] (27)

where [119911] denotes the integer part of 119911 By definitionmodulusfunction we have for 119886

119896119898gt 120575

119891 (119886119896119898) le (1 + [

119886119896119898

120575])119891 (1) le 2119891 (1)

119886119896119898

120575 (28)

Hence Σ2le 2119891(1)(119886

119896119898120575) which together with Σ

1le 120576119899

yields [119882119881120590] sub [119882119881

120590(119891)]

Lemma 10 (see [15]) Let 119891 be a modulus function Let 120572 gt 0be given constant Then there is a constant 119888 gt 0 such that119891(119909) gt 119888119909 (0 lt 119909 lt 120572)

Theorem 11 Let 119860119896 be a bounded sequence and 119891 be a

modulus function Then 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to modulus 119891 if and only if 119860119896

is Wijsman strongly invariant convergent to 119860 that is

[119882119881120590(119891)] cap 119871

infin= [119882119881

120590] (29)

where 119871infin

denotes the set of bounded sequences of sets

Proof The proof of the theorem follows fromTheorem 9 andLemma 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Fast ldquoSur la convergence statistiquerdquoColloquiumMathemat-icae vol 2 no 3-4 pp 241ndash244 1951

[2] T Salat ldquoOn statistically convergent sequences of real numbersrdquoMathematica Slovaca vol 30 no 2 pp 139ndash150 1980

[3] G Beer ldquoConvergence of continuous linear functionals andtheir level setsrdquo Archiv der Mathematik vol 52 no 5 pp 482ndash491 1989

[4] J P Aubin and H Frankowska Set Valued Analysis BirkhauserBoston Mass USA 1990

Abstract and Applied Analysis 5

[5] M Baronti and P L Papini ldquoConvergence of sequences of setsrdquoinMethods of Functional Analysis in ApproximationTheory vol76 pp 135ndash155 Birkhauser Basel Switzerland 1986

[6] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012

[7] U Ulusu and F Nuray ldquoLacunary statistical convergencesequences of setsrdquo Progress in Applied Mathematics vol 4 no2 pp 99ndash109 2012

[8] N Pancaroglu and F Nuray ldquoOn invariant statistically con-vergence and lacu-nary invariant statistically convergence ofsequences of setsrdquo Progress in Applied Mathematics vol 5 no2 pp 23ndash29 2013

[9] H Nakano ldquoConcave modularsrdquo Journal of the MathematicalSociety of Japan vol 5 no 1 pp 29ndash49 1953

[10] W H Ruckle ldquo119865119870 spaces in which the sequence of coordinatevectors is boundedrdquo Canadian Journal of Mathematics vol 25pp 973ndash978 1973

[11] I J Maddox ldquoSequence spaces defined by a modulusrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol100 no 1 pp 161ndash166 1986

[12] J Connor ldquoOn strong matrix summability with respect to amodulus and statistical convergencerdquo Canadian MathematicalBulletin vol 32 no 2 pp 194ndash198 1989

[13] F Nuray and E Savas ldquoSome new sequence spaces definedby a modulus functionrdquo Indian Journal of Pure and AppliedMathematics vol 24 no 11 pp 657ndash663 1993

[14] E Savas ldquoStrongly 120590-convergent sequencesrdquo Bulletin of theCalcutta Mathematical Society vol 81 no 4 pp 295ndash300 1989

[15] E Savas ldquoOn strong almost 119860-summability with respect to amodulus and statistical convergencerdquo Indian Journal of Pure andApplied Mathematics vol 23 no 3 pp 217ndash222 1992

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Papini [5] The concepts of Wijsman statistical convergenceandWijsman strong Cesaro summability were introduced byNuray and Rhoades [6] as follows

Let (119883 120588) be a metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 the sequence 119860

119896 is said to be Wijsman

strongly Cesaro summable to 119860 if for each 119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)1003816100381610038161003816 = 0 (12)

Let (119883 120588) be a metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 the sequence 119860

119896 is said to be Wijsman

statistically convergent to 119860 if for 120576 gt 0 and each 119909 isin 119883

lim119899rarrinfin

1

119899

1003816100381610038161003816119896 le 119899 1003816100381610038161003816119889 (119909 119860119896) minus 119889 (119909 119860)

1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (13)

In this case we write 119904119905 minus lim119882119860119896= 119860 or 119860

119896rarr 119860(119882119878)

3 Main Result

The purpose of this paper is by using a modulus functionto introduce and study new sequence spaces of sequences ofsets The following three definitions were given in [8]

Definition 3 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman invariant convergent to 119860 if for each 119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

119889 (119909 119860120590119896(119898)) = 119889 (119909 119860) (14)

uniformly in119898

In this case we write 119860119896rarr 119860(119882119881

120590) and the set of

all Wijsman invariant convergent sequences of sets will bedenoted119882119881

120590

Definition 4 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman strongly invariant convergent to 119860 if for each 119909 isin119883

lim119899rarrinfin

1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 = 0 (15)

uniformly in119898

In this case we write 119860119896rarr 119860([119882119881

120590]) and the set of all

Wijsman strongly invariant convergent sequences of sets willbe denoted [119882119881

120590]

Definition 5 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 we say that the sequence 119860

119896 is

Wijsman invariant statistically convergent to 119860 if for each120576 gt 0 and for each 119909 isin 119883

lim119899rarrinfin

1

119899

100381610038161003816100381610038160 le 119896 le 119899 10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816 ge 12057610038161003816100381610038161003816 = 0

(16)

uniformly in119898

In this case we write 119860119896rarr 119860(119882119878

120590) and the set of all

Wijsman invariant statistically convergent sequences of setswill be denoted119882119878

120590

Let (119883 120588) be metric space For any nonempty closedsubsets 119860 119860

119896sube 119883 and 119909 isin 119883 we define the sequences of

sets space [119882119881120590]infin

as follows

[119882119881120590]infin= 119860

119896 sup119898119899

1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898))10038161003816100381610038161003816 lt infin (17)

Now by using a modulus function we introduce thefollowing new sequence spaces of sequences of sets

Definition 6 Let (119883 120588) be a metric space and119891 be a modulusfunction For any nonempty closed subsets 119860 119860

119896sube 119883 we

say that the sequence 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to the modulus 119891 if for each119909 isin 119883

lim119899rarrinfin

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) = 0 (18)

uniformly in119898

In this case we write 119860119896rarr 119860([119882119881

120590(119891)]) and the set of

all Wijsman strongly invariant convergent sequences of setswith respect to the modulus 119891 will be denoted [119882119881

120590(119891)]

Let (119883 120588) be metric space and 119891 be a modulus functionFor any nonempty closed subsets 119860 119860

119896sube 119883 and 119909 isin 119883 we

define the sequences of sets space [119882119881120590(119891)]infin

as follows

[119882119881120590(119891)]infin= 119860

119896 sup119898119899

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816) lt infin

(19)

If 119891(119909) = 119909 then the spaces [119882119881120590(119891)] and [119882119881

120590(119891)]infin

reduce to [119882119881120590] and [119882119881

120590]infin respectively

Now we study the relation between 119882119878120590and [119882119881

120590(119891)]

convergence

Theorem 7 Let (119883 120588) be a metric space For any nonemptyclosed subsets 119860 119860

119896sube 119883 Then

(i) 119860119896rarr 119860([119882119881

120590(119891)]) implies 119860

119896rarr 119860(119882119878

120590)

(ii) 119891 is bounded and 119860119896rarr 119860(119882119878

120590) implies 119860

119896rarr

119860([119882119881120590(119891)])

(iii) 119882119878120590= [119882119881

120590(119891)] if 119891 is bounded

Proof (i) Let 120576 gt 0 and 119860119896rarr 119860([119882119881

120590(119891)]) Then we can

write119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

ge119899

sum119896=1

|119889(119909119860120590119896(119898))minus119889(119909119860)|ge120576

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

ge 120576 sdot10038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge 120576

10038161003816100381610038161003816

(20)

which yields the result

4 Abstract and Applied Analysis

(ii) Suppose that 119860119896rarr 119860(119882119878

120590) and 119891 is bounded for

each119898 ge 1 set

119866 = sup119896119898

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816 + |119889 (119909 119860)|) (21)

Let 120576 gt 0 and select119873120576such that

1

119899

1003816100381610038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge

120576

21003816100381610038161003816100381610038161003816lt120576

2119866 (22)

for all 119898 and 119899 gt 119873120576 and set 119871(119899119898 119909) = 119896 le 119899

|119889(119909 119860120590119896(119898)) minus119889(119909 119860)| ge 1205762 Now for all119898 and 119899 gt 119873

120576 we

have that

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

= sum119896isin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

+ sum119896notin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

lt1

119899(119899

120576

2119866)119866 +

1

119899(120576

2) 119899 =

120576

2+120576

2= 120576

(23)

Hence 119860119896 is strongly invariant convergent to 119860 with

respect to the modulus function 119891(iii)This is an immediate consequence of (i) and (ii)This completes the proof of the theorem

Theorem 8 Let 119891 be a modulus function Then [119882119881120590(119891)] sub

[119882119881120590(119891)]infin

Proof Suppose that 119860119896 isin [119882119881

120590(119891)] Then we can write

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816)

=1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860) + 119889 (119909 119860)

10038161003816100381610038161003816)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) +1

119899

119899

sum119896=1

119891 (|119889 (119909 119860)|)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) + 1198721

119899

119899

sum119896=1

119891 (1)

(24)

where 119872 is an integer such that 119889(119909 119860) lt 119872 Therefore119860119896 isin [119882119881

120590(119891)]infin

Theorem 9 If 119891 is a modulus function and 119860119896 is strongly

invariant convergent to 119860 then 119860119896 is strongly invariant

convergent to119860with respect to themodulus119891 that is [119882119881120590] sub

[119882119881120590(119891)]

Proof Let 119860119896 isin [119882119881

120590] Then we can write

119878119899119898=1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(25)

uniformly in119898Let 120576 gt 0 and choose 120575 with 0 lt 120575 lt 1 such that 119891(119905) lt 120576

for 0 le 119905 le 120575 Write 119886119896119898= |119889(119909 119860

120590119896(119898)) minus 119889(119909 119860)|

119899

sum119896=1

119891 (119886119896119898) = Σ1+ Σ2 (26)

where the first summation is over 119886119896119898le 120575 and second over

119886119896119898gt 120575 Then Σ

1le 120576119899 and for 119886

119896119898gt 120575

119891 (119886119896119898) lt

119886119896119898

120575lt 1 + [

119886119896119898

120575] (27)

where [119911] denotes the integer part of 119911 By definitionmodulusfunction we have for 119886

119896119898gt 120575

119891 (119886119896119898) le (1 + [

119886119896119898

120575])119891 (1) le 2119891 (1)

119886119896119898

120575 (28)

Hence Σ2le 2119891(1)(119886

119896119898120575) which together with Σ

1le 120576119899

yields [119882119881120590] sub [119882119881

120590(119891)]

Lemma 10 (see [15]) Let 119891 be a modulus function Let 120572 gt 0be given constant Then there is a constant 119888 gt 0 such that119891(119909) gt 119888119909 (0 lt 119909 lt 120572)

Theorem 11 Let 119860119896 be a bounded sequence and 119891 be a

modulus function Then 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to modulus 119891 if and only if 119860119896

is Wijsman strongly invariant convergent to 119860 that is

[119882119881120590(119891)] cap 119871

infin= [119882119881

120590] (29)

where 119871infin

denotes the set of bounded sequences of sets

Proof The proof of the theorem follows fromTheorem 9 andLemma 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Fast ldquoSur la convergence statistiquerdquoColloquiumMathemat-icae vol 2 no 3-4 pp 241ndash244 1951

[2] T Salat ldquoOn statistically convergent sequences of real numbersrdquoMathematica Slovaca vol 30 no 2 pp 139ndash150 1980

[3] G Beer ldquoConvergence of continuous linear functionals andtheir level setsrdquo Archiv der Mathematik vol 52 no 5 pp 482ndash491 1989

[4] J P Aubin and H Frankowska Set Valued Analysis BirkhauserBoston Mass USA 1990

Abstract and Applied Analysis 5

[5] M Baronti and P L Papini ldquoConvergence of sequences of setsrdquoinMethods of Functional Analysis in ApproximationTheory vol76 pp 135ndash155 Birkhauser Basel Switzerland 1986

[6] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012

[7] U Ulusu and F Nuray ldquoLacunary statistical convergencesequences of setsrdquo Progress in Applied Mathematics vol 4 no2 pp 99ndash109 2012

[8] N Pancaroglu and F Nuray ldquoOn invariant statistically con-vergence and lacu-nary invariant statistically convergence ofsequences of setsrdquo Progress in Applied Mathematics vol 5 no2 pp 23ndash29 2013

[9] H Nakano ldquoConcave modularsrdquo Journal of the MathematicalSociety of Japan vol 5 no 1 pp 29ndash49 1953

[10] W H Ruckle ldquo119865119870 spaces in which the sequence of coordinatevectors is boundedrdquo Canadian Journal of Mathematics vol 25pp 973ndash978 1973

[11] I J Maddox ldquoSequence spaces defined by a modulusrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol100 no 1 pp 161ndash166 1986

[12] J Connor ldquoOn strong matrix summability with respect to amodulus and statistical convergencerdquo Canadian MathematicalBulletin vol 32 no 2 pp 194ndash198 1989

[13] F Nuray and E Savas ldquoSome new sequence spaces definedby a modulus functionrdquo Indian Journal of Pure and AppliedMathematics vol 24 no 11 pp 657ndash663 1993

[14] E Savas ldquoStrongly 120590-convergent sequencesrdquo Bulletin of theCalcutta Mathematical Society vol 81 no 4 pp 295ndash300 1989

[15] E Savas ldquoOn strong almost 119860-summability with respect to amodulus and statistical convergencerdquo Indian Journal of Pure andApplied Mathematics vol 23 no 3 pp 217ndash222 1992

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

4 Abstract and Applied Analysis

(ii) Suppose that 119860119896rarr 119860(119882119878

120590) and 119891 is bounded for

each119898 ge 1 set

119866 = sup119896119898

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816 + |119889 (119909 119860)|) (21)

Let 120576 gt 0 and select119873120576such that

1

119899

1003816100381610038161003816100381610038161003816119896 le 119899

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 ge

120576

21003816100381610038161003816100381610038161003816lt120576

2119866 (22)

for all 119898 and 119899 gt 119873120576 and set 119871(119899119898 119909) = 119896 le 119899

|119889(119909 119860120590119896(119898)) minus119889(119909 119860)| ge 1205762 Now for all119898 and 119899 gt 119873

120576 we

have that

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

= sum119896isin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

+ sum119896notin119871(119899119898119909)

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816)

lt1

119899(119899

120576

2119866)119866 +

1

119899(120576

2) 119899 =

120576

2+120576

2= 120576

(23)

Hence 119860119896 is strongly invariant convergent to 119860 with

respect to the modulus function 119891(iii)This is an immediate consequence of (i) and (ii)This completes the proof of the theorem

Theorem 8 Let 119891 be a modulus function Then [119882119881120590(119891)] sub

[119882119881120590(119891)]infin

Proof Suppose that 119860119896 isin [119882119881

120590(119891)] Then we can write

1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898))

10038161003816100381610038161003816)

=1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860) + 119889 (119909 119860)

10038161003816100381610038161003816)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) +1

119899

119899

sum119896=1

119891 (|119889 (119909 119860)|)

le1

119899

119899

sum119896=1

119891 (10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)

10038161003816100381610038161003816) + 1198721

119899

119899

sum119896=1

119891 (1)

(24)

where 119872 is an integer such that 119889(119909 119860) lt 119872 Therefore119860119896 isin [119882119881

120590(119891)]infin

Theorem 9 If 119891 is a modulus function and 119860119896 is strongly

invariant convergent to 119860 then 119860119896 is strongly invariant

convergent to119860with respect to themodulus119891 that is [119882119881120590] sub

[119882119881120590(119891)]

Proof Let 119860119896 isin [119882119881

120590] Then we can write

119878119899119898=1

119899

119899

sum119896=1

10038161003816100381610038161003816119889 (119909 119860120590119896(119898)) minus 119889 (119909 119860)10038161003816100381610038161003816 997888rarr 0 (119899 997888rarr infin)

(25)

uniformly in119898Let 120576 gt 0 and choose 120575 with 0 lt 120575 lt 1 such that 119891(119905) lt 120576

for 0 le 119905 le 120575 Write 119886119896119898= |119889(119909 119860

120590119896(119898)) minus 119889(119909 119860)|

119899

sum119896=1

119891 (119886119896119898) = Σ1+ Σ2 (26)

where the first summation is over 119886119896119898le 120575 and second over

119886119896119898gt 120575 Then Σ

1le 120576119899 and for 119886

119896119898gt 120575

119891 (119886119896119898) lt

119886119896119898

120575lt 1 + [

119886119896119898

120575] (27)

where [119911] denotes the integer part of 119911 By definitionmodulusfunction we have for 119886

119896119898gt 120575

119891 (119886119896119898) le (1 + [

119886119896119898

120575])119891 (1) le 2119891 (1)

119886119896119898

120575 (28)

Hence Σ2le 2119891(1)(119886

119896119898120575) which together with Σ

1le 120576119899

yields [119882119881120590] sub [119882119881

120590(119891)]

Lemma 10 (see [15]) Let 119891 be a modulus function Let 120572 gt 0be given constant Then there is a constant 119888 gt 0 such that119891(119909) gt 119888119909 (0 lt 119909 lt 120572)

Theorem 11 Let 119860119896 be a bounded sequence and 119891 be a

modulus function Then 119860119896 is Wijsman strongly invariant

convergent to 119860 with respect to modulus 119891 if and only if 119860119896

is Wijsman strongly invariant convergent to 119860 that is

[119882119881120590(119891)] cap 119871

infin= [119882119881

120590] (29)

where 119871infin

denotes the set of bounded sequences of sets

Proof The proof of the theorem follows fromTheorem 9 andLemma 10

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H Fast ldquoSur la convergence statistiquerdquoColloquiumMathemat-icae vol 2 no 3-4 pp 241ndash244 1951

[2] T Salat ldquoOn statistically convergent sequences of real numbersrdquoMathematica Slovaca vol 30 no 2 pp 139ndash150 1980

[3] G Beer ldquoConvergence of continuous linear functionals andtheir level setsrdquo Archiv der Mathematik vol 52 no 5 pp 482ndash491 1989

[4] J P Aubin and H Frankowska Set Valued Analysis BirkhauserBoston Mass USA 1990

Abstract and Applied Analysis 5

[5] M Baronti and P L Papini ldquoConvergence of sequences of setsrdquoinMethods of Functional Analysis in ApproximationTheory vol76 pp 135ndash155 Birkhauser Basel Switzerland 1986

[6] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012

[7] U Ulusu and F Nuray ldquoLacunary statistical convergencesequences of setsrdquo Progress in Applied Mathematics vol 4 no2 pp 99ndash109 2012

[8] N Pancaroglu and F Nuray ldquoOn invariant statistically con-vergence and lacu-nary invariant statistically convergence ofsequences of setsrdquo Progress in Applied Mathematics vol 5 no2 pp 23ndash29 2013

[9] H Nakano ldquoConcave modularsrdquo Journal of the MathematicalSociety of Japan vol 5 no 1 pp 29ndash49 1953

[10] W H Ruckle ldquo119865119870 spaces in which the sequence of coordinatevectors is boundedrdquo Canadian Journal of Mathematics vol 25pp 973ndash978 1973

[11] I J Maddox ldquoSequence spaces defined by a modulusrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol100 no 1 pp 161ndash166 1986

[12] J Connor ldquoOn strong matrix summability with respect to amodulus and statistical convergencerdquo Canadian MathematicalBulletin vol 32 no 2 pp 194ndash198 1989

[13] F Nuray and E Savas ldquoSome new sequence spaces definedby a modulus functionrdquo Indian Journal of Pure and AppliedMathematics vol 24 no 11 pp 657ndash663 1993

[14] E Savas ldquoStrongly 120590-convergent sequencesrdquo Bulletin of theCalcutta Mathematical Society vol 81 no 4 pp 295ndash300 1989

[15] E Savas ldquoOn strong almost 119860-summability with respect to amodulus and statistical convergencerdquo Indian Journal of Pure andApplied Mathematics vol 23 no 3 pp 217ndash222 1992

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

Abstract and Applied Analysis 5

[5] M Baronti and P L Papini ldquoConvergence of sequences of setsrdquoinMethods of Functional Analysis in ApproximationTheory vol76 pp 135ndash155 Birkhauser Basel Switzerland 1986

[6] F Nuray and B E Rhoades ldquoStatistical convergence ofsequences of setsrdquoFasciculiMathematici no 49 pp 87ndash99 2012

[7] U Ulusu and F Nuray ldquoLacunary statistical convergencesequences of setsrdquo Progress in Applied Mathematics vol 4 no2 pp 99ndash109 2012

[8] N Pancaroglu and F Nuray ldquoOn invariant statistically con-vergence and lacu-nary invariant statistically convergence ofsequences of setsrdquo Progress in Applied Mathematics vol 5 no2 pp 23ndash29 2013

[9] H Nakano ldquoConcave modularsrdquo Journal of the MathematicalSociety of Japan vol 5 no 1 pp 29ndash49 1953

[10] W H Ruckle ldquo119865119870 spaces in which the sequence of coordinatevectors is boundedrdquo Canadian Journal of Mathematics vol 25pp 973ndash978 1973

[11] I J Maddox ldquoSequence spaces defined by a modulusrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol100 no 1 pp 161ndash166 1986

[12] J Connor ldquoOn strong matrix summability with respect to amodulus and statistical convergencerdquo Canadian MathematicalBulletin vol 32 no 2 pp 194ndash198 1989

[13] F Nuray and E Savas ldquoSome new sequence spaces definedby a modulus functionrdquo Indian Journal of Pure and AppliedMathematics vol 24 no 11 pp 657ndash663 1993

[14] E Savas ldquoStrongly 120590-convergent sequencesrdquo Bulletin of theCalcutta Mathematical Society vol 81 no 4 pp 295ndash300 1989

[15] E Savas ldquoOn strong almost 119860-summability with respect to amodulus and statistical convergencerdquo Indian Journal of Pure andApplied Mathematics vol 23 no 3 pp 217ndash222 1992

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