Research ArticleSome Generalized Difference Sequence Spaces Defined bya Sequence of Moduli in 119899-Normed Spaces

Abdullah Alotaibi1 Kuldip Raj2 and S A Mohiuddine1

1Operator Theory and Applications Research Group Department of Mathematics Faculty of Science King Abdulaziz UniversityPO Box 80203 Jeddah 21589 Saudi Arabia2School of Mathematics Shri Mata Vaishno Devi University Katra Jammu and Kashmir 182320 India

Correspondence should be addressed to S A Mohiuddine mohiuddinegmailcom

Received 12 December 2014 Revised 17 February 2015 Accepted 25 February 2015

Academic Editor Mohamed-Aziz Taoudi

Copyright copy 2015 Abdullah Alotaibi et al 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 some new generalized difference sequence spaces by means of ideal convergence infinite matrix and a sequence ofmodulus functions over 119899-normed spaces We also make an effort to study several properties relevant to topological algebraic andinclusion relations between these spaces

1 Introduction and Preliminaries

The concept of 2-normed spaces was initially developed byGahler [1] in the middle of 1960s while that of 119899-normedspaces one can see in Misiak [2] Since then many othershave studied this concept and obtained various results seeGunawan [3 4] and Gunawan and Mashadi [5] Let 119899 isin N

and119883 be a linear space over the fieldR of reals of dimension119889 where 119889 ge 119899 ge 2 A real valued function sdot sdot on 119883119899satisfying the following four conditions

(1) 1199091 1199092 119909

119899 = 0 if and only if 119909

1 1199092 119909

119899are

linearly dependent in119883(2) 119909

1 1199092 119909

119899 is invariant under permutation

(3) 1205721199091 1199092 119909

119899 = |120572|119909

1 1199092 119909

119899 for any 120572 isin R

(4) 119909+1199091015840 1199092 119909

119899 le 119909 119909

2 119909

119899+1199091015840

1199092 119909

119899

is called a 119899-norm on 119883 and the pair (119883 sdot sdot) is called a119899-normed space over the field R For more details about 119899-normed spaces see [6 7] and references therein

The notion of difference sequence spaces was introducedby Kızmaz [8] who studied the difference sequence spaces119897infin(Δ) 119888(Δ) and 119888

0(Δ) The notion was further generalized

by Et and Colak [9] by introducing the spaces 119897infin(Δ119898

) 119888(Δ119898)and 1198880(Δ119898

) Later the concept has been studied by Bektas etal [10] and Et and Esi [11] Another type of generalization of

the difference sequence spaces is due to Tripathy and Esi [12]who studied the spaces 119897

infin(Δ V) 119888(Δ V) and 1198880(Δ V) Recently

Esi et al [13] and Tripathy et al [14] have introduced a newtype of generalized difference operators and unified those asfollows

Let 119898 and V be nonnegative integers then for 119885 a givensequence space we have

119885 (Δ119898

V ) = 119909 = (119909119896) isin 119908 (Δ119898

V 119909119896) isin 119885 (1)

for 119885 = 119888 1198880 and 119897

infinwhere Δ119898V 119909 = (Δ

119898

V 119909119896) = (Δ119898minus1

V 119909119896minus

Δ119898minus1

V 119909119896+V) and Δ

0

V119909119896 = 119909119896 for all 119896 isin N which is equivalentto the following binomial representation

Δ119898

V 119909119896 =

119898

sum

]=0(minus1)

](

119898

])119909119896+V] (2)

Taking V = 1 we get the spaces 119897infin(Δ119898

) 119888(Δ119898) and 1198880(Δ119898

)

studied by Et and Colak [9] Taking V = 119898 = 1 we getthe spaces 119897

infin(Δ) 119888(Δ) and 119888

0(Δ) introduced and studied

by Kızmaz [8] For more details about difference sequencespaces see [15ndash19]

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

such that

(1) 119891(119909) = 0 if and only if 119909 = 0(2) 119891(119909 + 119910) le 119891(119909) + 119891(119910) for all 119909 119910 ge 0

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 413850 8 pageshttpdxdoiorg1011552015413850

2 Journal of Function Spaces

(3) 119891 is increasing(4) 119891 is continuous from the right at 0

It follows that 119891 must be continuous everywhere on [0infin)The modulus function may be bounded or unbounded Forexample if we take 119891(119909) = 119909(119909 + 1) then 119891(119909) is boundedIf 119891(119909) = 119909119901 0 lt 119901 lt 1 then the modulus function 119891(119909)is unbounded Subsequently modulus function has beendiscussed in [20] and references therein

In [21] Mursaleen introduced the idea of 120582-statisticalconvergence by extending the concept of [119881 120582] summabilityFurther Savas and Das [22] unified the two approachesand gave new concepts of 119868-statistical convergence 119868-119878

120582-

convergence and 119868-[119881 120582] convergence Later some pioneerworks have been extended in this direction by numerousauthors such as Belen and Mohiuddine [23] Das et al [24]Gurdal and Sarı [25] and references therein Quite recentlymany authors including Et et al [26] and Maddox [27] haveconstructed some sequence spaces by using modulus func-tion anddifference sequences and investigate their properties

Let 119883 and 119884 be two sequence spaces and 119860 = (119886119899119896) be an

infinite matrix of real or complex numbers 119886119899119896 where 119899 119896 isin

N Then we say that 119860 defines a matrix mapping from119883 into119884 if for every sequence 119909 = (119909

119896)infin

119896=0isin 119883 the sequence 119860119909 =

119860119899(119909)infin

119899=0 the 119860-transform of 119909 is in 119884 where

119860119899(119909) =

infin

sum

119896=0

119886119899119896119909119896

(119899 isin N) (3)

By (119883 119884) we denote the class of all matrices 119860 such that 119860 119883 rarr 119884Thus119860 isin (119883 119884) if and only if the series on the right-hand side of (3) converges for each 119899 isin N and every 119909 isin 119883

The matrix domain 119883119860

of an infinite matrix 119860 in asequence space119883 is defined by

119883119860= 119909 = (119909

119896) 119860119909 isin 119883 (4)

The approach constructing a new sequence space by meansof the matrix domain of a particular limitation method hasrecently been employed by several authors (see [28])

Let 120582 = (120582119899) be a nondecreasing sequence of positive

numbers such that 120582119899+1

le 120582119899+ 1 120582

1= 1 120582

119899rarr infin as

119899 rarr infin and 119868119899= [119899 minus 120582

119899+ 1 119899]

Let N be a nonempty set Then a family of sets 119868 sube 2N

(power set of N) is said to be an ideal if 119868 is additive that is119860 119861 isin 119868 rArr 119860cup119861 isin 119868 and119860 isin 119868119861 sube 119860 rArr 119861 isin 119868 A nonemptyfamily of sets m(119868) sube 2N is said to be filtered on N if and onlyifΦ notin m(119868) for 119860 119861 isin m(119868) we have 119860 cap 119861 isin m(119868) and for each119860 isin m(119868) and 119860 sube 119861 implying 119861 isin m(119868)

An ideal 119868 sube 2N is called nontrivial if 119868 = 2N A nontrivial

ideal 119868 sube 2N is called admissible if 119909 119909 isin N sube 119868 A

nontrivial ideal ismaximal if there cannot exist any nontrivialideal 119869 = 119868 containing 119868 as a subset For each ideal 119868 thereexist a filter m(119868) corresponding to 119868 that is m(119868) = 119870 sube N

119870119888

isin 119868 where119870119888 = N 119870

Definition 1 A sequence 119909 = (119909119896) in a 119899-normed space

(119883 sdot sdot) is said to be statistically convergent to some119871 isin 119883 if for each 120576 gt 0 the set 119860(120576) = 119896 isin N 119909

119896minus

119871 1199111 119911

119899minus1 ge 120576 has its natural density zero

Definition 2 (see [26]) A sequence 119909 = (119909119896) is said to be 120582119898

119883-

statistically convergent to the number 119871 if for every 120576 gt 0lim119899rarrinfin

(1120582119899)|119896 isin 119868

119899 Δ119898

119909119896minus 119871 ge 120576| = 0 In this case

one writes 119878120582(Δ119898

119883) minus lim119896rarrinfin

119909119896= 119871

Definition 3 (see [22]) A sequence 119909 = (119909119896) is said to

be 119868-[119881 120582] summable to 119871 if for any 120575 gt 0 119899 isin N

(1120582119899) sum119896isin119868119899

119909119896minus 119871 ge 120575 isin 119868 where 119868

119899= [119899 minus 120582

119899+ 1 119899]

Definition 4 (see [22]) A sequence 119909 = (119909119896) is said to be 119868-120582-

statistically convergent or 119868-119878120582convergent to 119871 if for every

120576 gt 0 and120575 gt 0 119899 isin N (1120582119899)|119896 isin 119868

119899 119909119896minus119871 ge 120575| isin 119868 In

this case one writes 119909119896rarr 119871(119868-119878

120582) or 119868-119878

120582minus lim119896rarrinfin

119909119896= 119871

The following well-known lemma is required for estab-lishing some important results in this paper

Lemma 5 Let F = (119891119896) be a sequence of modulus functions

and 0 lt 120575 lt 1Then for each119909 gt 120575 one has119891119896(119909) le 2119891

119896(1)119909120575

Throughout the paperNwill denote the set of all positiveintegers By 119878119899

119883we denote the space of all sequences defined

over 119899-normed space (119883 sdot sdot)Let 119868 sube 2N be an admissible ideal F = (119891

119896) a sequence

modulus functions 119901 = (119901119896) a bounded sequence of positive

(strictly) real numbers 119906 = (119906119896) a sequence of positive real

numbers 119860 = (119886119899119896) an infinite matrix and (119883 sdot sdot) a 119899-

normed spaceThen for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we

define the following sequence spaces

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]0

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 120575

isin 119868

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 120575

isin 119868

(5)

Journal of Function Spaces 3

for some 119871 gt 0 and

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]infin

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870

isin 119868

(6)

for some119870 gt 0If we takeF(119909) = 119891

119896(119909) = 119909 119906 = (119906

119896) = 1 for all 119896 isin N

V = 1 and 119860 = (119886119899119896) = 119868 (identity matrix) and 119899-normed

space is replaced by 2-normed space thenwe get the sequencespaces defined by Kumar et al [29]

The following inequality will be used throughout thepaper Let 119901 = (119901

119896) be a sequence of positive real numbers

with 0 lt 119901119896le sup

119896119901119896= 119867 and let 119863 = max1 2119867minus1 Then

for the factorable sequences (119886119896) and (119887

119896) in the complex

plane we have

1003816100381610038161003816119886119896+ 119887119896

1003816100381610038161003816

119901119896

le 119863(1003816100381610038161003816119886119896

1003816100381610038161003816

119901119896

+1003816100381610038161003816119887119896

1003816100381610038161003816

119901119896

) (7)

2 Main Results

The main purpose of this section is to study some topo-logical properties and some inclusion relations between thesequence spaces which we have defined above

Theorem 6 SupposeF = (119891119896) is a sequence of modulus func-

tions 119901 = (119901119896) is a bounded sequence of strictly positive real

numbers 119906 = (119906119896) is a sequence of positive real numbers and

119860 = (119886119899119896) is an infinite matrix Then the spaces 119881119868F[Δ

119898

V 120582

119860 119906 119901 sdot sdot]0 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot] and 119881119868

F[Δ119898

V

120582 119860 119906 119901 sdot sdot]infin

are linear spaces over the real field R

Proof The proof of the theorem is easy so we omit it

For the next result we shall define the following sequencespace

119881F [Δ119898

V 120582 119860 119906 119901 sdot sdot]infin

=

119909 = (119909119896) isin 119878119899

119883

sup119899

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt infin

(8)

Theorem 7 LetF = (119891119896) be a sequence of modulus functions

and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then the space 119881F[Δ119898V 120582 119860 119906 119901 sdot sdot]0 is aparanormed space with the paranorm

119892 (119909)

= sup119899

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

(9)

where 119866 = max(1119867) and 0 lt 119901119896le sup

119896119901119896= 119867

Proof Clearly 119892(119909) = 119892(minus119909) for 119909 isin 119881119868

F[Δ119898

V 120582 119860 119906 119901

sdot sdot]0 It is trivial that 119906

119896Δ119898

V 119909119896 = 0 for 119909 = 0 Since119891119896(0) = 0 we get 119892(119909) = 0 for 119909 = 0 Since 119901

119896119866 le 1 using

Minkowskirsquos inequality we have

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V (119909119896 + 119910119896) 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817

+1003817100381710038171003817119906119896Δ119898

V 119910119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

+ (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119910119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

(10)

Hence 119892(119909 + 119910) le 119892(119909) + 119892(119910) Finally to check the continu-ity of scalar multiplication let 120572 be any complex numbertherefore by definition

119892 (120572119909)

= sup119899

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le 119862119867119866

120572119892 (119909)

(11)

4 Journal of Function Spaces

where 119862120572is a positive integer such that 120572 le 119862

120572 Let 120572 rarr 0

for any fixed 119909 with 119892(119909) = 0 By definition for |120572| lt 1 wehave

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598

for 119899 gt 119873 (120598)

(12)

Also for 1 le 119899 le 119873 taking 120572 small enough since119891119896is conin-

uous we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598 (13)

Now (12) and (13) imply that 119892(120572119909) rarr 0 as 120572 rarr 0 Thiscompletes the proof

Theorem 8 Let 119906 = (119906119896) be a sequence of positive real

numbers Then for119898 ge 1 the following inclusions

(i) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]0

(ii) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]infin

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]infin

are strict

Proof We will prove the result for 119881119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0only The others can be proved similarly

Suppose 119909 isin 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0 by definition

for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)] ge 120575

isin 119868

(14)

By the property of modulus function we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]

le

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)

+119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

le 119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

+119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)] by (7)

(15)

Now for given 120575 gt 0 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)] ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

cup

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

(16)

for each 1199111 119911

119899minus1isin 119883 Since 119909 isin 119881

119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0 it follows that the sets on the right-hand side in

the above containment belong to 119868 Hence119909 isin 119881119868F[Δ119898minus1

V 120582 119860

119906 sdot sdot]0 To show that the inclusion is strict we give the

following exampleWe take 119891

119896(119909) = 119909 120582

119899= 119899 and 119860 = (119886

119899119896) = 119868 for all

119899 119896 isin N V = 1 and consider a sequence 119909 = (119909119896) = 119896

119898minus1then 119909 isin 119881

119868

F[Δ119898

V 120582 119860 119906 sdot sdot]0 but does not belong to119881119868

F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0for 119906 = (119906

119896) = 1 119896 isin N This

shows that the inclusion is strict

Theorem 9 Let F1015840 = (1198911015840119896) and F10158401015840 = (11989110158401015840

119896) be sequences of

modulus functions If lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 gt 0 then

119881119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Let lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 then there exists

a constant 119872 gt 0 such that 1198911015840119896(119905) ge 119872119891

10158401015840

119896(119905) for all 119905 ge 0

Therefore for each 1199111 119911

119899minus1isin 119883 we have

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge (119872)1198671

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(17)

Then for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have the

following relationship

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575

Journal of Function Spaces 5

sube

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575 (119872)119867

(18)

Since 119909 isin 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] it follows that theset on left-side of the above containment belongs to 119868 whichgives 119909 isin 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Theorem 10 Let F = (119891119896) F1015840 = (1198911015840

119896) and F10158401015840 = (11989110158401015840

119896) be

sequences of modulus functions Then

(i) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot]

(ii) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] cup 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901

sdot sdot] sub 119881119868

F1015840+F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof (i) Let 119909 = (119909119896) isin 119881

119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] forevery 120576 gt 0 and for some 119871 gt 0 such that

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896minus119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120576

isin 119868

(19)

for each 1199111 119911

119899minus1isin 119883 For given 120576 gt 0 we choose120575 isin (0 1)

such that 119891119896(119905) lt 120576 for all 0 lt 119905 lt 120575 On the other hand we

have

1

120582119899

sum

119896isin119868119899

119886119899119896[(119891119896∘ 1198911015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

=

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111119911119899minus1)]119901119896lt120575

119886119899119896[(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

+

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111 119911119899minus1)]119901119896ge120575

119886119899119896

sdot [(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

le (120576)119867

+max(1 (2119891119896(1)

120575

)

119867

)

1

120582119899

sdot sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(20)

by Lemma 5By using (19) we obtain 119909 isin 119881

119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot](ii)The result of this part is proved by using the following

inequality

1

120582119899

sum

119896isin119868119899

119886119899119896[(1198911015840

119896+ 11989110158401015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

le

119863

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

+

119863

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(21)

where sup119896119901119896= 119867 and119863 = max(1 2119867minus1)

Theorem 11 Let F = (119891119896) be a sequence of modulus func-

tions and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then 119881119868[Δ119898V 120582 119860 119906 119901 sdot sdot] sube 119881119868

F[Δ119898

V 120582 119860 119906

119901 sdot sdot]

Proof This can be proved by using the same techniques as inTheorem 10

Theorem 12 Let F = (119891119896) be a sequence of modulus func-

tions If lim sup119905rarrinfin

(119891119896(119905)119905) = 119872 gt 0 then 119881119868F[Δ

119898

V 120582 119860 119906

119901 sdot sdot] sube 119881119868

[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] andlim sup

119905rarrinfin(119891119896(119905)119905) = 119872 gt 0 then there exists a constant

119870 gt 0 such that 119891119896(119905) ge 119870119905 for all 119905 ge 0 Thus we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge (119870)1198671

120582119899

sum

119896isin119868119899

119886119899119896[(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

(22)

for each 1199111 119911

119899minus1isin 119883 This completes the proof

Theorem 13 If 0 lt 119901119896le 119902119896and (119902

119896119901119896) be bounded Then

119881119868

F[Δ119898

V 120582 119860 119906 119902 sdot sdot] sub 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof It is easy to prove so we omit the detail

3 Statistical Convergence

The notion of statistical convergence introduced by Fast [30]in 1951 and later developed by Fridy [31] Salat [32] and many

6 Journal of Function Spaces

others Furthermore Kostyrko et al [33] presented a veryinteresting generalization of statistical convergence called as119868-convergence Some recent developments in this regard canbe found in [34ndash37] and many others

In this section we define a new class of generalized statis-tical convergent sequences with the help of an ideal modulusfunctions and infinite matrix We also made an effort toestablish a strong connection between this convergence andthe sequence space 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot]

Definition 14 Let 119868 sube 119875(N) be a non-trivial ideal and120582 = (120582119899)

be a non-decreasing sequence A sequence 119909 = (119909119896) isin 119883

is said to be 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent to a number 119871

provided that for every 120576 gt 0 120575 gt 0 and 1199111 119911

119899minus1isin 119883 the

set

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575 isin 119868

(23)

In this case we write 119878Δ119898

V120582(119868 119906 119860) minus lim

119896rarrinfin119906119896Δ119898

V 119909119896 1199111

119911119899minus1 = 119871120588 119911

1 119911

119899minus1 Let 119878Δ

119898

V120582(119868 119860 119906 sdot sdot) denotes

the set of all 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent sequences in119883

Theorem 15 LetF = (119891119896) be a sequence of modulus functions

and 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin Then

119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119878Δ119898

V120582[119868 119860 119906 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] and120576 gt 0 be given Then for each 119911

1 119911

119899minus1isin 119883 we obtain

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

=

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

+

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111 119911119899minus1lt120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899

[119891119896(120576)]119901119896

ge sum

119896isin119868119899

min ([119891119896(120576)]ℎ

[119891119896(120576)]119867

)

ge 119870

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

(24)

where119870 = min([119891119896(120576)]ℎ

[119891119896(120576)]119867

) Then for every 120575 gt 0 and1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870120575

(25)

Since 119909119896rarr 119871(119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]) so that 119878Δ119898

V120582(119868 119906

119860) minus lim119896rarrinfin

119906119896Δ119898

V 119909119896 1199111 119911119899minus1 = 119871 1199111 119911119899minus1

Theorem 16 LetF = (119891119896) be a sequence ofmodulus functions

and 119901 = (119901119896) be a bounded sequence of strictly positive real

numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin then

119878Δ119898

V120582[119868 119860 119906 sdot sdot] sub 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof By using ([11 Theorem 35]) we can prove easily

Theorem 17 LetF = (119891119896) be a bounded sequence of modulus

functions and 119901 = (119901119896) be a bounded sequence of strictly

positive real numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le

sup119896119901119896= 119867 lt infin Then 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881119868F[Δ

119898

V 120582 119860

119906 119901 sdot sdot] if and only ifF = (119891119896) is a bounded

Proof This part can be obtained by combining Theorems 15and 16 Conversely supposeF = (119891

119896) be unbounded defined

by 119891119896(119896) = 119896 for all 119896 isin N We take a fixed set 119861 isin 119868 where 119868

is an admissible ideal and define 119909 = (119909119896) as follows

119909119896=

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 notin 119861

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 isin 119861

0 otherwise

(26)

For given 120576 gt 0 and for each 1199111 119911

119899minus1isin 119883 we have

lim119899rarrinfin

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

lt

[radic120582119899]

120582119899

997888rarr 0

(27)

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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

2 Journal of Function Spaces

(3) 119891 is increasing(4) 119891 is continuous from the right at 0

It follows that 119891 must be continuous everywhere on [0infin)The modulus function may be bounded or unbounded Forexample if we take 119891(119909) = 119909(119909 + 1) then 119891(119909) is boundedIf 119891(119909) = 119909119901 0 lt 119901 lt 1 then the modulus function 119891(119909)is unbounded Subsequently modulus function has beendiscussed in [20] and references therein

In [21] Mursaleen introduced the idea of 120582-statisticalconvergence by extending the concept of [119881 120582] summabilityFurther Savas and Das [22] unified the two approachesand gave new concepts of 119868-statistical convergence 119868-119878

120582-

convergence and 119868-[119881 120582] convergence Later some pioneerworks have been extended in this direction by numerousauthors such as Belen and Mohiuddine [23] Das et al [24]Gurdal and Sarı [25] and references therein Quite recentlymany authors including Et et al [26] and Maddox [27] haveconstructed some sequence spaces by using modulus func-tion anddifference sequences and investigate their properties

Let 119883 and 119884 be two sequence spaces and 119860 = (119886119899119896) be an

infinite matrix of real or complex numbers 119886119899119896 where 119899 119896 isin

N Then we say that 119860 defines a matrix mapping from119883 into119884 if for every sequence 119909 = (119909

119896)infin

119896=0isin 119883 the sequence 119860119909 =

119860119899(119909)infin

119899=0 the 119860-transform of 119909 is in 119884 where

119860119899(119909) =

infin

sum

119896=0

119886119899119896119909119896

(119899 isin N) (3)

By (119883 119884) we denote the class of all matrices 119860 such that 119860 119883 rarr 119884Thus119860 isin (119883 119884) if and only if the series on the right-hand side of (3) converges for each 119899 isin N and every 119909 isin 119883

The matrix domain 119883119860

of an infinite matrix 119860 in asequence space119883 is defined by

119883119860= 119909 = (119909

119896) 119860119909 isin 119883 (4)

The approach constructing a new sequence space by meansof the matrix domain of a particular limitation method hasrecently been employed by several authors (see [28])

Let 120582 = (120582119899) be a nondecreasing sequence of positive

numbers such that 120582119899+1

le 120582119899+ 1 120582

1= 1 120582

119899rarr infin as

119899 rarr infin and 119868119899= [119899 minus 120582

119899+ 1 119899]

Let N be a nonempty set Then a family of sets 119868 sube 2N

(power set of N) is said to be an ideal if 119868 is additive that is119860 119861 isin 119868 rArr 119860cup119861 isin 119868 and119860 isin 119868119861 sube 119860 rArr 119861 isin 119868 A nonemptyfamily of sets m(119868) sube 2N is said to be filtered on N if and onlyifΦ notin m(119868) for 119860 119861 isin m(119868) we have 119860 cap 119861 isin m(119868) and for each119860 isin m(119868) and 119860 sube 119861 implying 119861 isin m(119868)

An ideal 119868 sube 2N is called nontrivial if 119868 = 2N A nontrivial

ideal 119868 sube 2N is called admissible if 119909 119909 isin N sube 119868 A

nontrivial ideal ismaximal if there cannot exist any nontrivialideal 119869 = 119868 containing 119868 as a subset For each ideal 119868 thereexist a filter m(119868) corresponding to 119868 that is m(119868) = 119870 sube N

119870119888

isin 119868 where119870119888 = N 119870

Definition 1 A sequence 119909 = (119909119896) in a 119899-normed space

(119883 sdot sdot) is said to be statistically convergent to some119871 isin 119883 if for each 120576 gt 0 the set 119860(120576) = 119896 isin N 119909

119896minus

119871 1199111 119911

119899minus1 ge 120576 has its natural density zero

Definition 2 (see [26]) A sequence 119909 = (119909119896) is said to be 120582119898

119883-

statistically convergent to the number 119871 if for every 120576 gt 0lim119899rarrinfin

(1120582119899)|119896 isin 119868

119899 Δ119898

119909119896minus 119871 ge 120576| = 0 In this case

one writes 119878120582(Δ119898

119883) minus lim119896rarrinfin

119909119896= 119871

Definition 3 (see [22]) A sequence 119909 = (119909119896) is said to

be 119868-[119881 120582] summable to 119871 if for any 120575 gt 0 119899 isin N

(1120582119899) sum119896isin119868119899

119909119896minus 119871 ge 120575 isin 119868 where 119868

119899= [119899 minus 120582

119899+ 1 119899]

Definition 4 (see [22]) A sequence 119909 = (119909119896) is said to be 119868-120582-

statistically convergent or 119868-119878120582convergent to 119871 if for every

120576 gt 0 and120575 gt 0 119899 isin N (1120582119899)|119896 isin 119868

119899 119909119896minus119871 ge 120575| isin 119868 In

this case one writes 119909119896rarr 119871(119868-119878

120582) or 119868-119878

120582minus lim119896rarrinfin

119909119896= 119871

The following well-known lemma is required for estab-lishing some important results in this paper

Lemma 5 Let F = (119891119896) be a sequence of modulus functions

and 0 lt 120575 lt 1Then for each119909 gt 120575 one has119891119896(119909) le 2119891

119896(1)119909120575

Throughout the paperNwill denote the set of all positiveintegers By 119878119899

119883we denote the space of all sequences defined

over 119899-normed space (119883 sdot sdot)Let 119868 sube 2N be an admissible ideal F = (119891

119896) a sequence

modulus functions 119901 = (119901119896) a bounded sequence of positive

(strictly) real numbers 119906 = (119906119896) a sequence of positive real

numbers 119860 = (119886119899119896) an infinite matrix and (119883 sdot sdot) a 119899-

normed spaceThen for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we

define the following sequence spaces

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]0

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 120575

isin 119868

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 120575

isin 119868

(5)

Journal of Function Spaces 3

for some 119871 gt 0 and

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]infin

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870

isin 119868

(6)

for some119870 gt 0If we takeF(119909) = 119891

119896(119909) = 119909 119906 = (119906

119896) = 1 for all 119896 isin N

V = 1 and 119860 = (119886119899119896) = 119868 (identity matrix) and 119899-normed

space is replaced by 2-normed space thenwe get the sequencespaces defined by Kumar et al [29]

The following inequality will be used throughout thepaper Let 119901 = (119901

119896) be a sequence of positive real numbers

with 0 lt 119901119896le sup

119896119901119896= 119867 and let 119863 = max1 2119867minus1 Then

for the factorable sequences (119886119896) and (119887

119896) in the complex

plane we have

1003816100381610038161003816119886119896+ 119887119896

1003816100381610038161003816

119901119896

le 119863(1003816100381610038161003816119886119896

1003816100381610038161003816

119901119896

+1003816100381610038161003816119887119896

1003816100381610038161003816

119901119896

) (7)

2 Main Results

The main purpose of this section is to study some topo-logical properties and some inclusion relations between thesequence spaces which we have defined above

Theorem 6 SupposeF = (119891119896) is a sequence of modulus func-

tions 119901 = (119901119896) is a bounded sequence of strictly positive real

numbers 119906 = (119906119896) is a sequence of positive real numbers and

119860 = (119886119899119896) is an infinite matrix Then the spaces 119881119868F[Δ

119898

V 120582

119860 119906 119901 sdot sdot]0 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot] and 119881119868

F[Δ119898

V

120582 119860 119906 119901 sdot sdot]infin

are linear spaces over the real field R

Proof The proof of the theorem is easy so we omit it

For the next result we shall define the following sequencespace

119881F [Δ119898

V 120582 119860 119906 119901 sdot sdot]infin

=

119909 = (119909119896) isin 119878119899

119883

sup119899

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt infin

(8)

Theorem 7 LetF = (119891119896) be a sequence of modulus functions

and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then the space 119881F[Δ119898V 120582 119860 119906 119901 sdot sdot]0 is aparanormed space with the paranorm

119892 (119909)

= sup119899

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

(9)

where 119866 = max(1119867) and 0 lt 119901119896le sup

119896119901119896= 119867

Proof Clearly 119892(119909) = 119892(minus119909) for 119909 isin 119881119868

F[Δ119898

V 120582 119860 119906 119901

sdot sdot]0 It is trivial that 119906

119896Δ119898

V 119909119896 = 0 for 119909 = 0 Since119891119896(0) = 0 we get 119892(119909) = 0 for 119909 = 0 Since 119901

119896119866 le 1 using

Minkowskirsquos inequality we have

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V (119909119896 + 119910119896) 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817

+1003817100381710038171003817119906119896Δ119898

V 119910119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

+ (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119910119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

(10)

Hence 119892(119909 + 119910) le 119892(119909) + 119892(119910) Finally to check the continu-ity of scalar multiplication let 120572 be any complex numbertherefore by definition

119892 (120572119909)

= sup119899

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le 119862119867119866

120572119892 (119909)

(11)

4 Journal of Function Spaces

where 119862120572is a positive integer such that 120572 le 119862

120572 Let 120572 rarr 0

for any fixed 119909 with 119892(119909) = 0 By definition for |120572| lt 1 wehave

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598

for 119899 gt 119873 (120598)

(12)

Also for 1 le 119899 le 119873 taking 120572 small enough since119891119896is conin-

uous we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598 (13)

Now (12) and (13) imply that 119892(120572119909) rarr 0 as 120572 rarr 0 Thiscompletes the proof

Theorem 8 Let 119906 = (119906119896) be a sequence of positive real

numbers Then for119898 ge 1 the following inclusions

(i) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]0

(ii) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]infin

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]infin

are strict

Proof We will prove the result for 119881119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0only The others can be proved similarly

Suppose 119909 isin 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0 by definition

for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)] ge 120575

isin 119868

(14)

By the property of modulus function we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]

le

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)

+119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

le 119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

+119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)] by (7)

(15)

Now for given 120575 gt 0 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)] ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

cup

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

(16)

for each 1199111 119911

119899minus1isin 119883 Since 119909 isin 119881

119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0 it follows that the sets on the right-hand side in

the above containment belong to 119868 Hence119909 isin 119881119868F[Δ119898minus1

V 120582 119860

119906 sdot sdot]0 To show that the inclusion is strict we give the

following exampleWe take 119891

119896(119909) = 119909 120582

119899= 119899 and 119860 = (119886

119899119896) = 119868 for all

119899 119896 isin N V = 1 and consider a sequence 119909 = (119909119896) = 119896

119898minus1then 119909 isin 119881

119868

F[Δ119898

V 120582 119860 119906 sdot sdot]0 but does not belong to119881119868

F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0for 119906 = (119906

119896) = 1 119896 isin N This

shows that the inclusion is strict

Theorem 9 Let F1015840 = (1198911015840119896) and F10158401015840 = (11989110158401015840

119896) be sequences of

modulus functions If lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 gt 0 then

119881119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Let lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 then there exists

a constant 119872 gt 0 such that 1198911015840119896(119905) ge 119872119891

10158401015840

119896(119905) for all 119905 ge 0

Therefore for each 1199111 119911

119899minus1isin 119883 we have

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge (119872)1198671

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(17)

Then for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have the

following relationship

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575

Journal of Function Spaces 5

sube

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575 (119872)119867

(18)

Since 119909 isin 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] it follows that theset on left-side of the above containment belongs to 119868 whichgives 119909 isin 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Theorem 10 Let F = (119891119896) F1015840 = (1198911015840

119896) and F10158401015840 = (11989110158401015840

119896) be

sequences of modulus functions Then

(i) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot]

(ii) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] cup 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901

sdot sdot] sub 119881119868

F1015840+F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof (i) Let 119909 = (119909119896) isin 119881

119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] forevery 120576 gt 0 and for some 119871 gt 0 such that

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896minus119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120576

isin 119868

(19)

for each 1199111 119911

119899minus1isin 119883 For given 120576 gt 0 we choose120575 isin (0 1)

such that 119891119896(119905) lt 120576 for all 0 lt 119905 lt 120575 On the other hand we

have

1

120582119899

sum

119896isin119868119899

119886119899119896[(119891119896∘ 1198911015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

=

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111119911119899minus1)]119901119896lt120575

119886119899119896[(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

+

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111 119911119899minus1)]119901119896ge120575

119886119899119896

sdot [(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

le (120576)119867

+max(1 (2119891119896(1)

120575

)

119867

)

1

120582119899

sdot sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(20)

by Lemma 5By using (19) we obtain 119909 isin 119881

119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot](ii)The result of this part is proved by using the following

inequality

1

120582119899

sum

119896isin119868119899

119886119899119896[(1198911015840

119896+ 11989110158401015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

le

119863

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

+

119863

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(21)

where sup119896119901119896= 119867 and119863 = max(1 2119867minus1)

Theorem 11 Let F = (119891119896) be a sequence of modulus func-

tions and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then 119881119868[Δ119898V 120582 119860 119906 119901 sdot sdot] sube 119881119868

F[Δ119898

V 120582 119860 119906

119901 sdot sdot]

Proof This can be proved by using the same techniques as inTheorem 10

Theorem 12 Let F = (119891119896) be a sequence of modulus func-

tions If lim sup119905rarrinfin

(119891119896(119905)119905) = 119872 gt 0 then 119881119868F[Δ

119898

V 120582 119860 119906

119901 sdot sdot] sube 119881119868

[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] andlim sup

119905rarrinfin(119891119896(119905)119905) = 119872 gt 0 then there exists a constant

119870 gt 0 such that 119891119896(119905) ge 119870119905 for all 119905 ge 0 Thus we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge (119870)1198671

120582119899

sum

119896isin119868119899

119886119899119896[(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

(22)

for each 1199111 119911

119899minus1isin 119883 This completes the proof

Theorem 13 If 0 lt 119901119896le 119902119896and (119902

119896119901119896) be bounded Then

119881119868

F[Δ119898

V 120582 119860 119906 119902 sdot sdot] sub 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof It is easy to prove so we omit the detail

3 Statistical Convergence

The notion of statistical convergence introduced by Fast [30]in 1951 and later developed by Fridy [31] Salat [32] and many

6 Journal of Function Spaces

others Furthermore Kostyrko et al [33] presented a veryinteresting generalization of statistical convergence called as119868-convergence Some recent developments in this regard canbe found in [34ndash37] and many others

In this section we define a new class of generalized statis-tical convergent sequences with the help of an ideal modulusfunctions and infinite matrix We also made an effort toestablish a strong connection between this convergence andthe sequence space 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot]

Definition 14 Let 119868 sube 119875(N) be a non-trivial ideal and120582 = (120582119899)

be a non-decreasing sequence A sequence 119909 = (119909119896) isin 119883

is said to be 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent to a number 119871

provided that for every 120576 gt 0 120575 gt 0 and 1199111 119911

119899minus1isin 119883 the

set

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575 isin 119868

(23)

In this case we write 119878Δ119898

V120582(119868 119906 119860) minus lim

119896rarrinfin119906119896Δ119898

V 119909119896 1199111

119911119899minus1 = 119871120588 119911

1 119911

119899minus1 Let 119878Δ

119898

V120582(119868 119860 119906 sdot sdot) denotes

the set of all 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent sequences in119883

Theorem 15 LetF = (119891119896) be a sequence of modulus functions

and 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin Then

119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119878Δ119898

V120582[119868 119860 119906 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] and120576 gt 0 be given Then for each 119911

1 119911

119899minus1isin 119883 we obtain

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

=

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

+

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111 119911119899minus1lt120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899

[119891119896(120576)]119901119896

ge sum

119896isin119868119899

min ([119891119896(120576)]ℎ

[119891119896(120576)]119867

)

ge 119870

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

(24)

where119870 = min([119891119896(120576)]ℎ

[119891119896(120576)]119867

) Then for every 120575 gt 0 and1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870120575

(25)

Since 119909119896rarr 119871(119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]) so that 119878Δ119898

V120582(119868 119906

119860) minus lim119896rarrinfin

119906119896Δ119898

V 119909119896 1199111 119911119899minus1 = 119871 1199111 119911119899minus1

Theorem 16 LetF = (119891119896) be a sequence ofmodulus functions

and 119901 = (119901119896) be a bounded sequence of strictly positive real

numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin then

119878Δ119898

V120582[119868 119860 119906 sdot sdot] sub 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof By using ([11 Theorem 35]) we can prove easily

Theorem 17 LetF = (119891119896) be a bounded sequence of modulus

functions and 119901 = (119901119896) be a bounded sequence of strictly

positive real numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le

sup119896119901119896= 119867 lt infin Then 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881119868F[Δ

119898

V 120582 119860

119906 119901 sdot sdot] if and only ifF = (119891119896) is a bounded

Proof This part can be obtained by combining Theorems 15and 16 Conversely supposeF = (119891

119896) be unbounded defined

by 119891119896(119896) = 119896 for all 119896 isin N We take a fixed set 119861 isin 119868 where 119868

is an admissible ideal and define 119909 = (119909119896) as follows

119909119896=

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 notin 119861

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 isin 119861

0 otherwise

(26)

For given 120576 gt 0 and for each 1199111 119911

119899minus1isin 119883 we have

lim119899rarrinfin

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

lt

[radic120582119899]

120582119899

997888rarr 0

(27)

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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

Journal of Function Spaces 3

for some 119871 gt 0 and

119881119868

F [Δ119898

V 120582 119860 119906 119901 sdot sdot]infin

=

119909 = (119909119896) isin 119878119899

119883

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870

isin 119868

(6)

for some119870 gt 0If we takeF(119909) = 119891

119896(119909) = 119909 119906 = (119906

119896) = 1 for all 119896 isin N

V = 1 and 119860 = (119886119899119896) = 119868 (identity matrix) and 119899-normed

space is replaced by 2-normed space thenwe get the sequencespaces defined by Kumar et al [29]

The following inequality will be used throughout thepaper Let 119901 = (119901

119896) be a sequence of positive real numbers

with 0 lt 119901119896le sup

119896119901119896= 119867 and let 119863 = max1 2119867minus1 Then

for the factorable sequences (119886119896) and (119887

119896) in the complex

plane we have

1003816100381610038161003816119886119896+ 119887119896

1003816100381610038161003816

119901119896

le 119863(1003816100381610038161003816119886119896

1003816100381610038161003816

119901119896

+1003816100381610038161003816119887119896

1003816100381610038161003816

119901119896

) (7)

2 Main Results

The main purpose of this section is to study some topo-logical properties and some inclusion relations between thesequence spaces which we have defined above

Theorem 6 SupposeF = (119891119896) is a sequence of modulus func-

tions 119901 = (119901119896) is a bounded sequence of strictly positive real

numbers 119906 = (119906119896) is a sequence of positive real numbers and

119860 = (119886119899119896) is an infinite matrix Then the spaces 119881119868F[Δ

119898

V 120582

119860 119906 119901 sdot sdot]0 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot] and 119881119868

F[Δ119898

V

120582 119860 119906 119901 sdot sdot]infin

are linear spaces over the real field R

Proof The proof of the theorem is easy so we omit it

For the next result we shall define the following sequencespace

119881F [Δ119898

V 120582 119860 119906 119901 sdot sdot]infin

=

119909 = (119909119896) isin 119878119899

119883

sup119899

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt infin

(8)

Theorem 7 LetF = (119891119896) be a sequence of modulus functions

and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then the space 119881F[Δ119898V 120582 119860 119906 119901 sdot sdot]0 is aparanormed space with the paranorm

119892 (119909)

= sup119899

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

(9)

where 119866 = max(1119867) and 0 lt 119901119896le sup

119896119901119896= 119867

Proof Clearly 119892(119909) = 119892(minus119909) for 119909 isin 119881119868

F[Δ119898

V 120582 119860 119906 119901

sdot sdot]0 It is trivial that 119906

119896Δ119898

V 119909119896 = 0 for 119909 = 0 Since119891119896(0) = 0 we get 119892(119909) = 0 for 119909 = 0 Since 119901

119896119866 le 1 using

Minkowskirsquos inequality we have

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V (119909119896 + 119910119896) 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817

+1003817100381710038171003817119906119896Δ119898

V 119910119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

+ (

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119910119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

(10)

Hence 119892(119909 + 119910) le 119892(119909) + 119892(119910) Finally to check the continu-ity of scalar multiplication let 120572 be any complex numbertherefore by definition

119892 (120572119909)

= sup119899

(

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

)

1119866

le 119862119867119866

120572119892 (119909)

(11)

4 Journal of Function Spaces

where 119862120572is a positive integer such that 120572 le 119862

120572 Let 120572 rarr 0

for any fixed 119909 with 119892(119909) = 0 By definition for |120572| lt 1 wehave

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598

for 119899 gt 119873 (120598)

(12)

Also for 1 le 119899 le 119873 taking 120572 small enough since119891119896is conin-

uous we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598 (13)

Now (12) and (13) imply that 119892(120572119909) rarr 0 as 120572 rarr 0 Thiscompletes the proof

Theorem 8 Let 119906 = (119906119896) be a sequence of positive real

numbers Then for119898 ge 1 the following inclusions

(i) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]0

(ii) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]infin

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]infin

are strict

Proof We will prove the result for 119881119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0only The others can be proved similarly

Suppose 119909 isin 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0 by definition

for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)] ge 120575

isin 119868

(14)

By the property of modulus function we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]

le

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)

+119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

le 119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

+119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)] by (7)

(15)

Now for given 120575 gt 0 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)] ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

cup

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

(16)

for each 1199111 119911

119899minus1isin 119883 Since 119909 isin 119881

119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0 it follows that the sets on the right-hand side in

the above containment belong to 119868 Hence119909 isin 119881119868F[Δ119898minus1

V 120582 119860

119906 sdot sdot]0 To show that the inclusion is strict we give the

following exampleWe take 119891

119896(119909) = 119909 120582

119899= 119899 and 119860 = (119886

119899119896) = 119868 for all

119899 119896 isin N V = 1 and consider a sequence 119909 = (119909119896) = 119896

119898minus1then 119909 isin 119881

119868

F[Δ119898

V 120582 119860 119906 sdot sdot]0 but does not belong to119881119868

F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0for 119906 = (119906

119896) = 1 119896 isin N This

shows that the inclusion is strict

Theorem 9 Let F1015840 = (1198911015840119896) and F10158401015840 = (11989110158401015840

119896) be sequences of

modulus functions If lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 gt 0 then

119881119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Let lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 then there exists

a constant 119872 gt 0 such that 1198911015840119896(119905) ge 119872119891

10158401015840

119896(119905) for all 119905 ge 0

Therefore for each 1199111 119911

119899minus1isin 119883 we have

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge (119872)1198671

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(17)

Then for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have the

following relationship

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575

Journal of Function Spaces 5

sube

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575 (119872)119867

(18)

Since 119909 isin 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] it follows that theset on left-side of the above containment belongs to 119868 whichgives 119909 isin 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Theorem 10 Let F = (119891119896) F1015840 = (1198911015840

119896) and F10158401015840 = (11989110158401015840

119896) be

sequences of modulus functions Then

(i) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot]

(ii) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] cup 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901

sdot sdot] sub 119881119868

F1015840+F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof (i) Let 119909 = (119909119896) isin 119881

119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] forevery 120576 gt 0 and for some 119871 gt 0 such that

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896minus119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120576

isin 119868

(19)

for each 1199111 119911

119899minus1isin 119883 For given 120576 gt 0 we choose120575 isin (0 1)

such that 119891119896(119905) lt 120576 for all 0 lt 119905 lt 120575 On the other hand we

have

1

120582119899

sum

119896isin119868119899

119886119899119896[(119891119896∘ 1198911015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

=

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111119911119899minus1)]119901119896lt120575

119886119899119896[(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

+

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111 119911119899minus1)]119901119896ge120575

119886119899119896

sdot [(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

le (120576)119867

+max(1 (2119891119896(1)

120575

)

119867

)

1

120582119899

sdot sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(20)

by Lemma 5By using (19) we obtain 119909 isin 119881

119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot](ii)The result of this part is proved by using the following

inequality

1

120582119899

sum

119896isin119868119899

119886119899119896[(1198911015840

119896+ 11989110158401015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

le

119863

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

+

119863

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(21)

where sup119896119901119896= 119867 and119863 = max(1 2119867minus1)

Theorem 11 Let F = (119891119896) be a sequence of modulus func-

tions and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then 119881119868[Δ119898V 120582 119860 119906 119901 sdot sdot] sube 119881119868

F[Δ119898

V 120582 119860 119906

119901 sdot sdot]

Proof This can be proved by using the same techniques as inTheorem 10

Theorem 12 Let F = (119891119896) be a sequence of modulus func-

tions If lim sup119905rarrinfin

(119891119896(119905)119905) = 119872 gt 0 then 119881119868F[Δ

119898

V 120582 119860 119906

119901 sdot sdot] sube 119881119868

[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] andlim sup

119905rarrinfin(119891119896(119905)119905) = 119872 gt 0 then there exists a constant

119870 gt 0 such that 119891119896(119905) ge 119870119905 for all 119905 ge 0 Thus we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge (119870)1198671

120582119899

sum

119896isin119868119899

119886119899119896[(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

(22)

for each 1199111 119911

119899minus1isin 119883 This completes the proof

Theorem 13 If 0 lt 119901119896le 119902119896and (119902

119896119901119896) be bounded Then

119881119868

F[Δ119898

V 120582 119860 119906 119902 sdot sdot] sub 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof It is easy to prove so we omit the detail

3 Statistical Convergence

The notion of statistical convergence introduced by Fast [30]in 1951 and later developed by Fridy [31] Salat [32] and many

6 Journal of Function Spaces

others Furthermore Kostyrko et al [33] presented a veryinteresting generalization of statistical convergence called as119868-convergence Some recent developments in this regard canbe found in [34ndash37] and many others

In this section we define a new class of generalized statis-tical convergent sequences with the help of an ideal modulusfunctions and infinite matrix We also made an effort toestablish a strong connection between this convergence andthe sequence space 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot]

Definition 14 Let 119868 sube 119875(N) be a non-trivial ideal and120582 = (120582119899)

be a non-decreasing sequence A sequence 119909 = (119909119896) isin 119883

is said to be 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent to a number 119871

provided that for every 120576 gt 0 120575 gt 0 and 1199111 119911

119899minus1isin 119883 the

set

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575 isin 119868

(23)

In this case we write 119878Δ119898

V120582(119868 119906 119860) minus lim

119896rarrinfin119906119896Δ119898

V 119909119896 1199111

119911119899minus1 = 119871120588 119911

1 119911

119899minus1 Let 119878Δ

119898

V120582(119868 119860 119906 sdot sdot) denotes

the set of all 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent sequences in119883

Theorem 15 LetF = (119891119896) be a sequence of modulus functions

and 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin Then

119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119878Δ119898

V120582[119868 119860 119906 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] and120576 gt 0 be given Then for each 119911

1 119911

119899minus1isin 119883 we obtain

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

=

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

+

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111 119911119899minus1lt120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899

[119891119896(120576)]119901119896

ge sum

119896isin119868119899

min ([119891119896(120576)]ℎ

[119891119896(120576)]119867

)

ge 119870

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

(24)

where119870 = min([119891119896(120576)]ℎ

[119891119896(120576)]119867

) Then for every 120575 gt 0 and1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870120575

(25)

Since 119909119896rarr 119871(119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]) so that 119878Δ119898

V120582(119868 119906

119860) minus lim119896rarrinfin

119906119896Δ119898

V 119909119896 1199111 119911119899minus1 = 119871 1199111 119911119899minus1

Theorem 16 LetF = (119891119896) be a sequence ofmodulus functions

and 119901 = (119901119896) be a bounded sequence of strictly positive real

numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin then

119878Δ119898

V120582[119868 119860 119906 sdot sdot] sub 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof By using ([11 Theorem 35]) we can prove easily

Theorem 17 LetF = (119891119896) be a bounded sequence of modulus

functions and 119901 = (119901119896) be a bounded sequence of strictly

positive real numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le

sup119896119901119896= 119867 lt infin Then 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881119868F[Δ

119898

V 120582 119860

119906 119901 sdot sdot] if and only ifF = (119891119896) is a bounded

Proof This part can be obtained by combining Theorems 15and 16 Conversely supposeF = (119891

119896) be unbounded defined

by 119891119896(119896) = 119896 for all 119896 isin N We take a fixed set 119861 isin 119868 where 119868

is an admissible ideal and define 119909 = (119909119896) as follows

119909119896=

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 notin 119861

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 isin 119861

0 otherwise

(26)

For given 120576 gt 0 and for each 1199111 119911

119899minus1isin 119883 we have

lim119899rarrinfin

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

lt

[radic120582119899]

120582119899

997888rarr 0

(27)

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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 Journal of Function Spaces

where 119862120572is a positive integer such that 120572 le 119862

120572 Let 120572 rarr 0

for any fixed 119909 with 119892(119909) = 0 By definition for |120572| lt 1 wehave

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598

for 119899 gt 119873 (120598)

(12)

Also for 1 le 119899 le 119873 taking 120572 small enough since119891119896is conin-

uous we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 120572119909119896 1199111 119911119899minus11003817100381710038171003817)]119901119896

lt 120598 (13)

Now (12) and (13) imply that 119892(120572119909) rarr 0 as 120572 rarr 0 Thiscompletes the proof

Theorem 8 Let 119906 = (119906119896) be a sequence of positive real

numbers Then for119898 ge 1 the following inclusions

(i) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]0

(ii) 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]infin

sub 119881119868

F[Δ119898

V 120582 119860 119906

sdot sdot]infin

are strict

Proof We will prove the result for 119881119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0only The others can be proved similarly

Suppose 119909 isin 119881119868F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0 by definition

for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)] ge 120575

isin 119868

(14)

By the property of modulus function we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)]

le

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)

+119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

le 119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

+119863

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)] by (7)

(15)

Now for given 120575 gt 0 we have

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 1199111 119911119899minus11003817100381710038171003817)] ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

cup

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(

10038171003817100381710038171003817119906119896Δ119898minus1

V 119909119896+1 1199111 119911

119899minus1

10038171003817100381710038171003817)]

ge

120575

2119863

(16)

for each 1199111 119911

119899minus1isin 119883 Since 119909 isin 119881

119868

F[Δ119898minus1

V 120582 119860 119906

sdot sdot]0 it follows that the sets on the right-hand side in

the above containment belong to 119868 Hence119909 isin 119881119868F[Δ119898minus1

V 120582 119860

119906 sdot sdot]0 To show that the inclusion is strict we give the

following exampleWe take 119891

119896(119909) = 119909 120582

119899= 119899 and 119860 = (119886

119899119896) = 119868 for all

119899 119896 isin N V = 1 and consider a sequence 119909 = (119909119896) = 119896

119898minus1then 119909 isin 119881

119868

F[Δ119898

V 120582 119860 119906 sdot sdot]0 but does not belong to119881119868

F[Δ119898minus1

V 120582 119860 119906 sdot sdot]0for 119906 = (119906

119896) = 1 119896 isin N This

shows that the inclusion is strict

Theorem 9 Let F1015840 = (1198911015840119896) and F10158401015840 = (11989110158401015840

119896) be sequences of

modulus functions If lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 gt 0 then

119881119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Let lim sup119905rarrinfin

(1198911015840

119896(119905)11989110158401015840

119896(119905)) = 119875 then there exists

a constant 119872 gt 0 such that 1198911015840119896(119905) ge 119872119891

10158401015840

119896(119905) for all 119905 ge 0

Therefore for each 1199111 119911

119899minus1isin 119883 we have

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge (119872)1198671

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(17)

Then for every 120575 gt 0 and 1199111 119911

119899minus1isin 119883 we have the

following relationship

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575

Journal of Function Spaces 5

sube

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575 (119872)119867

(18)

Since 119909 isin 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] it follows that theset on left-side of the above containment belongs to 119868 whichgives 119909 isin 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Theorem 10 Let F = (119891119896) F1015840 = (1198911015840

119896) and F10158401015840 = (11989110158401015840

119896) be

sequences of modulus functions Then

(i) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot]

(ii) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] cup 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901

sdot sdot] sub 119881119868

F1015840+F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof (i) Let 119909 = (119909119896) isin 119881

119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] forevery 120576 gt 0 and for some 119871 gt 0 such that

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896minus119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120576

isin 119868

(19)

for each 1199111 119911

119899minus1isin 119883 For given 120576 gt 0 we choose120575 isin (0 1)

such that 119891119896(119905) lt 120576 for all 0 lt 119905 lt 120575 On the other hand we

have

1

120582119899

sum

119896isin119868119899

119886119899119896[(119891119896∘ 1198911015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

=

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111119911119899minus1)]119901119896lt120575

119886119899119896[(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

+

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111 119911119899minus1)]119901119896ge120575

119886119899119896

sdot [(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

le (120576)119867

+max(1 (2119891119896(1)

120575

)

119867

)

1

120582119899

sdot sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(20)

by Lemma 5By using (19) we obtain 119909 isin 119881

119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot](ii)The result of this part is proved by using the following

inequality

1

120582119899

sum

119896isin119868119899

119886119899119896[(1198911015840

119896+ 11989110158401015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

le

119863

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

+

119863

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(21)

where sup119896119901119896= 119867 and119863 = max(1 2119867minus1)

Theorem 11 Let F = (119891119896) be a sequence of modulus func-

tions and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then 119881119868[Δ119898V 120582 119860 119906 119901 sdot sdot] sube 119881119868

F[Δ119898

V 120582 119860 119906

119901 sdot sdot]

Proof This can be proved by using the same techniques as inTheorem 10

Theorem 12 Let F = (119891119896) be a sequence of modulus func-

tions If lim sup119905rarrinfin

(119891119896(119905)119905) = 119872 gt 0 then 119881119868F[Δ

119898

V 120582 119860 119906

119901 sdot sdot] sube 119881119868

[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] andlim sup

119905rarrinfin(119891119896(119905)119905) = 119872 gt 0 then there exists a constant

119870 gt 0 such that 119891119896(119905) ge 119870119905 for all 119905 ge 0 Thus we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge (119870)1198671

120582119899

sum

119896isin119868119899

119886119899119896[(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

(22)

for each 1199111 119911

119899minus1isin 119883 This completes the proof

Theorem 13 If 0 lt 119901119896le 119902119896and (119902

119896119901119896) be bounded Then

119881119868

F[Δ119898

V 120582 119860 119906 119902 sdot sdot] sub 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof It is easy to prove so we omit the detail

3 Statistical Convergence

The notion of statistical convergence introduced by Fast [30]in 1951 and later developed by Fridy [31] Salat [32] and many

6 Journal of Function Spaces

others Furthermore Kostyrko et al [33] presented a veryinteresting generalization of statistical convergence called as119868-convergence Some recent developments in this regard canbe found in [34ndash37] and many others

In this section we define a new class of generalized statis-tical convergent sequences with the help of an ideal modulusfunctions and infinite matrix We also made an effort toestablish a strong connection between this convergence andthe sequence space 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot]

Definition 14 Let 119868 sube 119875(N) be a non-trivial ideal and120582 = (120582119899)

be a non-decreasing sequence A sequence 119909 = (119909119896) isin 119883

is said to be 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent to a number 119871

provided that for every 120576 gt 0 120575 gt 0 and 1199111 119911

119899minus1isin 119883 the

set

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575 isin 119868

(23)

In this case we write 119878Δ119898

V120582(119868 119906 119860) minus lim

119896rarrinfin119906119896Δ119898

V 119909119896 1199111

119911119899minus1 = 119871120588 119911

1 119911

119899minus1 Let 119878Δ

119898

V120582(119868 119860 119906 sdot sdot) denotes

the set of all 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent sequences in119883

Theorem 15 LetF = (119891119896) be a sequence of modulus functions

and 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin Then

119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119878Δ119898

V120582[119868 119860 119906 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] and120576 gt 0 be given Then for each 119911

1 119911

119899minus1isin 119883 we obtain

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

=

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

+

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111 119911119899minus1lt120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899

[119891119896(120576)]119901119896

ge sum

119896isin119868119899

min ([119891119896(120576)]ℎ

[119891119896(120576)]119867

)

ge 119870

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

(24)

where119870 = min([119891119896(120576)]ℎ

[119891119896(120576)]119867

) Then for every 120575 gt 0 and1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870120575

(25)

Since 119909119896rarr 119871(119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]) so that 119878Δ119898

V120582(119868 119906

119860) minus lim119896rarrinfin

119906119896Δ119898

V 119909119896 1199111 119911119899minus1 = 119871 1199111 119911119899minus1

Theorem 16 LetF = (119891119896) be a sequence ofmodulus functions

and 119901 = (119901119896) be a bounded sequence of strictly positive real

numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin then

119878Δ119898

V120582[119868 119860 119906 sdot sdot] sub 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof By using ([11 Theorem 35]) we can prove easily

Theorem 17 LetF = (119891119896) be a bounded sequence of modulus

functions and 119901 = (119901119896) be a bounded sequence of strictly

positive real numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le

sup119896119901119896= 119867 lt infin Then 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881119868F[Δ

119898

V 120582 119860

119906 119901 sdot sdot] if and only ifF = (119891119896) is a bounded

Proof This part can be obtained by combining Theorems 15and 16 Conversely supposeF = (119891

119896) be unbounded defined

by 119891119896(119896) = 119896 for all 119896 isin N We take a fixed set 119861 isin 119868 where 119868

is an admissible ideal and define 119909 = (119909119896) as follows

119909119896=

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 notin 119861

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 isin 119861

0 otherwise

(26)

For given 120576 gt 0 and for each 1199111 119911

119899minus1isin 119883 we have

lim119899rarrinfin

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

lt

[radic120582119899]

120582119899

997888rarr 0

(27)

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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

Journal of Function Spaces 5

sube

119899 isin N

1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120575 (119872)119867

(18)

Since 119909 isin 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] it follows that theset on left-side of the above containment belongs to 119868 whichgives 119909 isin 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Theorem 10 Let F = (119891119896) F1015840 = (1198911015840

119896) and F10158401015840 = (11989110158401015840

119896) be

sequences of modulus functions Then

(i) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119881119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot]

(ii) 119881119868F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] cup 119881119868

F10158401015840[Δ119898

V 120582 119860 119906 119901

sdot sdot] sub 119881119868

F1015840+F10158401015840[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof (i) Let 119909 = (119909119896) isin 119881

119868

F1015840[Δ119898

V 120582 119860 119906 119901 sdot sdot] forevery 120576 gt 0 and for some 119871 gt 0 such that

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896minus119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

ge 120576

isin 119868

(19)

for each 1199111 119911

119899minus1isin 119883 For given 120576 gt 0 we choose120575 isin (0 1)

such that 119891119896(119905) lt 120576 for all 0 lt 119905 lt 120575 On the other hand we

have

1

120582119899

sum

119896isin119868119899

119886119899119896[(119891119896∘ 1198911015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

=

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111119911119899minus1)]119901119896lt120575

119886119899119896[(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

+

1

120582119899

sum

119896isin119868119899[1198911015840

119896(119906119896Δ

119898

V 119909119896minus1198711199111 119911119899minus1)]119901119896ge120575

119886119899119896

sdot [(119891119896∘ 1198911015840

119896)

sdot (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]

119901119896

le (120576)119867

+max(1 (2119891119896(1)

120575

)

119867

)

1

120582119899

sdot sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(20)

by Lemma 5By using (19) we obtain 119909 isin 119881

119868

F∘F1015840[Δ119898

V 120582 119860 119906 119901

sdot sdot](ii)The result of this part is proved by using the following

inequality

1

120582119899

sum

119896isin119868119899

119886119899119896[(1198911015840

119896+ 11989110158401015840

119896) (1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

le

119863

120582119899

sum

119896isin119868119899

119886119899119896[1198911015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

+

119863

120582119899

sum

119896isin119868119899

119886119899119896[11989110158401015840

119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]

119901119896

(21)

where sup119896119901119896= 119867 and119863 = max(1 2119867minus1)

Theorem 11 Let F = (119891119896) be a sequence of modulus func-

tions and 119901 = (119901119896) a bounded sequence of strictly positive real

numbers Then 119881119868[Δ119898V 120582 119860 119906 119901 sdot sdot] sube 119881119868

F[Δ119898

V 120582 119860 119906

119901 sdot sdot]

Proof This can be proved by using the same techniques as inTheorem 10

Theorem 12 Let F = (119891119896) be a sequence of modulus func-

tions If lim sup119905rarrinfin

(119891119896(119905)119905) = 119872 gt 0 then 119881119868F[Δ

119898

V 120582 119860 119906

119901 sdot sdot] sube 119881119868

[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] andlim sup

119905rarrinfin(119891119896(119905)119905) = 119872 gt 0 then there exists a constant

119870 gt 0 such that 119891119896(119905) ge 119870119905 for all 119905 ge 0 Thus we have

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge (119870)1198671

120582119899

sum

119896isin119868119899

119886119899119896[(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

(22)

for each 1199111 119911

119899minus1isin 119883 This completes the proof

Theorem 13 If 0 lt 119901119896le 119902119896and (119902

119896119901119896) be bounded Then

119881119868

F[Δ119898

V 120582 119860 119906 119902 sdot sdot] sub 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof It is easy to prove so we omit the detail

3 Statistical Convergence

The notion of statistical convergence introduced by Fast [30]in 1951 and later developed by Fridy [31] Salat [32] and many

6 Journal of Function Spaces

others Furthermore Kostyrko et al [33] presented a veryinteresting generalization of statistical convergence called as119868-convergence Some recent developments in this regard canbe found in [34ndash37] and many others

In this section we define a new class of generalized statis-tical convergent sequences with the help of an ideal modulusfunctions and infinite matrix We also made an effort toestablish a strong connection between this convergence andthe sequence space 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot]

Definition 14 Let 119868 sube 119875(N) be a non-trivial ideal and120582 = (120582119899)

be a non-decreasing sequence A sequence 119909 = (119909119896) isin 119883

is said to be 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent to a number 119871

provided that for every 120576 gt 0 120575 gt 0 and 1199111 119911

119899minus1isin 119883 the

set

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575 isin 119868

(23)

In this case we write 119878Δ119898

V120582(119868 119906 119860) minus lim

119896rarrinfin119906119896Δ119898

V 119909119896 1199111

119911119899minus1 = 119871120588 119911

1 119911

119899minus1 Let 119878Δ

119898

V120582(119868 119860 119906 sdot sdot) denotes

the set of all 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent sequences in119883

Theorem 15 LetF = (119891119896) be a sequence of modulus functions

and 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin Then

119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119878Δ119898

V120582[119868 119860 119906 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] and120576 gt 0 be given Then for each 119911

1 119911

119899minus1isin 119883 we obtain

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

=

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

+

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111 119911119899minus1lt120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899

[119891119896(120576)]119901119896

ge sum

119896isin119868119899

min ([119891119896(120576)]ℎ

[119891119896(120576)]119867

)

ge 119870

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

(24)

where119870 = min([119891119896(120576)]ℎ

[119891119896(120576)]119867

) Then for every 120575 gt 0 and1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870120575

(25)

Since 119909119896rarr 119871(119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]) so that 119878Δ119898

V120582(119868 119906

119860) minus lim119896rarrinfin

119906119896Δ119898

V 119909119896 1199111 119911119899minus1 = 119871 1199111 119911119899minus1

Theorem 16 LetF = (119891119896) be a sequence ofmodulus functions

and 119901 = (119901119896) be a bounded sequence of strictly positive real

numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin then

119878Δ119898

V120582[119868 119860 119906 sdot sdot] sub 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof By using ([11 Theorem 35]) we can prove easily

Theorem 17 LetF = (119891119896) be a bounded sequence of modulus

functions and 119901 = (119901119896) be a bounded sequence of strictly

positive real numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le

sup119896119901119896= 119867 lt infin Then 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881119868F[Δ

119898

V 120582 119860

119906 119901 sdot sdot] if and only ifF = (119891119896) is a bounded

Proof This part can be obtained by combining Theorems 15and 16 Conversely supposeF = (119891

119896) be unbounded defined

by 119891119896(119896) = 119896 for all 119896 isin N We take a fixed set 119861 isin 119868 where 119868

is an admissible ideal and define 119909 = (119909119896) as follows

119909119896=

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 notin 119861

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 isin 119861

0 otherwise

(26)

For given 120576 gt 0 and for each 1199111 119911

119899minus1isin 119883 we have

lim119899rarrinfin

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

lt

[radic120582119899]

120582119899

997888rarr 0

(27)

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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

6 Journal of Function Spaces

others Furthermore Kostyrko et al [33] presented a veryinteresting generalization of statistical convergence called as119868-convergence Some recent developments in this regard canbe found in [34ndash37] and many others

In this section we define a new class of generalized statis-tical convergent sequences with the help of an ideal modulusfunctions and infinite matrix We also made an effort toestablish a strong connection between this convergence andthe sequence space 119881119868F[Δ

119898

V 120582 119860 119906 119901 sdot sdot]

Definition 14 Let 119868 sube 119875(N) be a non-trivial ideal and120582 = (120582119899)

be a non-decreasing sequence A sequence 119909 = (119909119896) isin 119883

is said to be 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent to a number 119871

provided that for every 120576 gt 0 120575 gt 0 and 1199111 119911

119899minus1isin 119883 the

set

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575 isin 119868

(23)

In this case we write 119878Δ119898

V120582(119868 119906 119860) minus lim

119896rarrinfin119906119896Δ119898

V 119909119896 1199111

119911119899minus1 = 119871120588 119911

1 119911

119899minus1 Let 119878Δ

119898

V120582(119868 119860 119906 sdot sdot) denotes

the set of all 119878Δ119898

V120582(119868 119906 119860 sdot sdot)-convergent sequences in119883

Theorem 15 LetF = (119891119896) be a sequence of modulus functions

and 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin Then

119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] sub 119878Δ119898

V120582[119868 119860 119906 sdot sdot]

Proof Suppose 119909 = (119909119896) isin 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] and120576 gt 0 be given Then for each 119911

1 119911

119899minus1isin 119883 we obtain

1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

=

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

+

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111 119911119899minus1lt120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899119906119896Δ119898

V 119909119896minus1198711199111119911119899minus1ge120576

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871

1199111 119911

119899minus1

1003817100381710038171003817)]119901119896

ge

1

120582119899

sum

119896isin119868119899

[119891119896(120576)]119901119896

ge sum

119896isin119868119899

min ([119891119896(120576)]ℎ

[119891119896(120576)]119867

)

ge 119870

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

(24)

where119870 = min([119891119896(120576)]ℎ

[119891119896(120576)]119867

) Then for every 120575 gt 0 and1199111 119911

119899minus1isin 119883 we have

119899 isin N 1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

ge 120575

sube

119899 isin N 1

120582119899

sum

119896isin119868119899

119886119899119896[119891119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 119871 1199111 119911119899minus11003817100381710038171003817)]119901119896

ge 119870120575

(25)

Since 119909119896rarr 119871(119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]) so that 119878Δ119898

V120582(119868 119906

119860) minus lim119896rarrinfin

119906119896Δ119898

V 119909119896 1199111 119911119899minus1 = 119871 1199111 119911119899minus1

Theorem 16 LetF = (119891119896) be a sequence ofmodulus functions

and 119901 = (119901119896) be a bounded sequence of strictly positive real

numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le sup

119896119901119896= 119867 lt infin then

119878Δ119898

V120582[119868 119860 119906 sdot sdot] sub 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot]

Proof By using ([11 Theorem 35]) we can prove easily

Theorem 17 LetF = (119891119896) be a bounded sequence of modulus

functions and 119901 = (119901119896) be a bounded sequence of strictly

positive real numbers If 0 lt inf119896119901119896= ℎ le 119901

119896le

sup119896119901119896= 119867 lt infin Then 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881119868F[Δ

119898

V 120582 119860

119906 119901 sdot sdot] if and only ifF = (119891119896) is a bounded

Proof This part can be obtained by combining Theorems 15and 16 Conversely supposeF = (119891

119896) be unbounded defined

by 119891119896(119896) = 119896 for all 119896 isin N We take a fixed set 119861 isin 119868 where 119868

is an admissible ideal and define 119909 = (119909119896) as follows

119909119896=

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 notin 119861

119896119898+1

for 119899 minus [radic120582119899] + 1 le 119896 le 119899 119899 isin 119861

0 otherwise

(26)

For given 120576 gt 0 and for each 1199111 119911

119899minus1isin 119883 we have

lim119899rarrinfin

1

120582119899

1003816100381610038161003816119896 isin 119868

119899 119886119899119896(1003817100381710038171003817119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus11003817100381710038171003817) ge 120576

1003816100381610038161003816

lt

[radic120582119899]

120582119899

997888rarr 0

(27)

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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

Journal of Function Spaces 7

for 119899 notin 119861 Hence for 120575 gt 0 there exists a positive integer 1198990

such that (1120582119899)|119896 isin 119868

119899 119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge

120576| lt 120575 for 119899 notin 119861 and 119899 ge 1198990 Now we have (1120582

119899)|119896 isin 119868

119899

119886119899119896119906119896Δ119898

V 119909119896 minus 0 1199111 119911119899minus1 ge 120576| ge 120575 sub 119861 cup (1 2 1198990 minus

1) Since 119868 be an admissible ideal it follows that 119878Δ119898

V120582(119868 119906 119860)minus

lim119896rarrinfin

119906119896119909119896 1199111 119911

119899minus1 rarr 0 for each 119911

1 119911

119899minus1isin 119883

On the other hand if we take 119901 = (119901119896) = 1 for all 119896 isin N

then 119909119896notin 119881119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] This contradicts thefact 119878Δ

119898

V120582[119868 119860 119906 sdot sdot] = 119881

119868

F[Δ119898

V 120582 119860 119906 119901 sdot sdot] soour supposition is wrong

Conflict of Interests

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

Acknowledgments

This project was funded by the Deanship of ScientificResearch (DSR) at King Abdulaziz University Jeddah undergrant no (69-130-35-RG) The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] S Gahler ldquoLineare 2-normierte Raumerdquo MathematischeNachrichten vol 28 no 1-2 pp 1ndash43 1964

[2] A Misiak ldquon-inner product spacesrdquo Mathematische Nachrich-ten vol 140 pp 299ndash319 1989

[3] H Gunawan ldquoOn n-inner product n-norms and the Cauchy-Schwartz inequalityrdquo Scientiae Mathematicae Japonicae vol 5pp 47ndash54 2001

[4] H Gunawan ldquoThe space of 119901-summable sequences and its nat-ural 119899-normrdquo Bulletin of the Australian Mathematical Societyvol 64 no 1 pp 137ndash147 2001

[5] H Gunawan and M Mashadi ldquoOn 119899-normed spacesrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol27 no 10 pp 631ndash639 2001

[6] K Raj S K Sharma and A K Sharma ldquoSome differencesequence spaces in n-normed spaces defined by Musielak-Orlicz functionrdquo Armenian Journal of Mathematics vol 3 pp127ndash141 2010

[7] A Sahiner M Gurdal S Saltan and H Gunawan ldquoIdealconvergence in 2minusnormed spacesrdquo Taiwanese Journal of Math-ematics vol 11 no 5 pp 1477ndash1484 2007

[8] H Kızmaz ldquoOn certain sequence spacesrdquoCanadianMathemat-ical Bulletin vol 24 no 2 pp 169ndash176 1981

[9] M Et and R Colak ldquoOn some generalized difference sequencespacesrdquo Soochow Journal of Mathematics vol 21 no 4 pp 377ndash386 1995

[10] C A Bektas M Et and R Colak ldquoGeneralized differencesequence spaces and their dual spacesrdquo Journal of MathematicalAnalysis and Applications vol 292 no 2 pp 423ndash432 2004

[11] M Et andA Esi ldquoOnKothe-Toeplitz duals of generalized differ-ence sequence spacesrdquo Bulletin of the Malaysian MathematicalSciences Society vol 23 no 1 pp 25ndash32 2000

[12] B C Tripathy and A Esi ldquoA new type of difference sequencespacesrdquo The International Journal of Science amp Technology vol1 pp 11ndash14 2006

[13] A Esi B C Tripathy and B Sarma ldquoOn some new typegeneralized difference sequence spacesrdquo Mathematica Slovacavol 57 no 5 pp 475ndash482 2007

[14] B C Tripathy A Esi and B Tripathy ldquoOn a new type of gen-eralized difference Cesaro sequence spacesrdquo Soochow Journal ofMathematics vol 31 no 3 pp 333ndash340 2005

[15] T Bilgin ldquoSome new difference sequences spaces defined by anOrlicz functionrdquo Filomat no 17 pp 1ndash8 2003

[16] M Et ldquoStrongly almost summable difference sequences of order119898 defined by a modulusrdquo Studia Scientiarum MathematicarumHungarica vol 40 no 4 pp 463ndash476 2003

[17] S A Mohiuddine K Raj and A Alotaibi ldquoOn some classes ofdouble difference sequences of interval numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 516956 8 pages 2014

[18] S A Mohiuddine K Raj and A Alotaibi ldquoSome paranormeddouble difference sequence spaces for Orlicz functions andbounded-regular matricesrdquo Abstract and Applied Analysis vol2014 Article ID 419064 10 pages 2014

[19] S A Mohiuddine K Raj and A Alotaibi ldquoGeneralized spacesof double sequences for Orlicz functions and bounded-regularmatrices over n-normed spacesrdquo Journal of Inequalities andApplications vol 2014 article 332 2014

[20] Y Altin and M Et ldquoGeneralized difference sequence spacesdefined by a modulus function in a locally convex spacerdquoSoochow Journal of Mathematics vol 31 no 2 pp 233ndash2432005

[21] M Mursaleen ldquo120582-statistical convergencerdquo Mathematica Slo-vaca vol 50 pp 111ndash115 2000

[22] E Savas and P Das ldquoA generalized statistical convergence viaidealsrdquo Applied Mathematics Letters vol 24 no 6 pp 826ndash8302011

[23] C Belen and S A Mohiuddine ldquoGeneralized weighted statis-tical convergence and applicationrdquo Applied Mathematics andComputation vol 219 no 18 pp 9821ndash9826 2013

[24] P Das E Savas and S K Ghosal ldquoOn generalizations of cer-tain summability methods using idealsrdquo Applied MathematicsLetters vol 24 no 9 pp 1509ndash1514 2011

[25] M Gurdal and H Sarı ldquoExtremal A-statistical limit points viaidealsrdquo Journal of the EgyptianMathematical Society vol 22 no1 pp 55ndash58 2014

[26] M Et Y Altin andH Altinok ldquoOn some generalized differencesequence spaces defined by amodulus functionrdquo Filomat no 17pp 23ndash33 2003

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

[28] E Savas and M Mursaleen ldquoMatrix transformations in somesequence spacesrdquo Istanbul Universitesi Fen Fakultesi MatematikDergisi vol 52 pp 1ndash5 1993

[29] S Kumar V Kumar and S S Bhatia ldquoGeneralized sequencespaces in 2-normed spaces defined by ideal and a modulusfunctionrdquo Analele Stiintifice ale Universitatii 2014

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

[31] J A Fridy ldquoOn statistical convergencerdquo Analysis vol 5 no 4pp 301ndash313 1985

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

[33] P Kostyrko T Salat and W Wilczynski ldquoI- convergencerdquo RealAnalysis Exchange vol 26 pp 669ndash686 2000

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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

8 Journal of Function Spaces

[34] M Et A Alotaibi and S AMohiuddine ldquoOn (998779119898 119868)-statisticalconvergence of order 120572rdquoThe Scientific World Journal vol 2014Article ID 535419 5 pages 2014

[35] F Gezer and S Karakus ldquoI and Ilowast convergent functionsequencesrdquo Mathematical Communications vol 10 pp 71ndash802005

[36] B Hazarika and S A Mohiuddine ldquoIdeal convergence ofrandom variablesrdquo Journal of Function Spaces and Applicationsvol 2013 Article ID 148249 7 pages 2013

[37] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence inprobabilistic normed spacesrdquoMathematica Slovaca vol 62 no1 pp 49ndash62 2012

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