Research Article Spaces of Ideal Convergent …downloads.hindawi.com/journals/tswj/2014/134534.pdfSpaces of Ideal Convergent Sequences M.Mursaleen 1 andSunilK.Sharma 2 Department of

Research ArticleSpaces of Ideal Convergent Sequences

M Mursaleen1 and Sunil K Sharma2

1 Department of Mathematics Aligarh Muslim University Aligarh 202002 India2Department of Mathematics Model Institute of Engineering amp Technology Kot Bhalwal JampK 181122 India

Correspondence should be addressed to M Mursaleen mursaleenmgmailcom

Received 20 August 2013 Accepted 26 November 2013 Published 28 January 2014

Academic Editors F Basar and J Xu

Copyright copy 2014 M Mursaleen and S K Sharma This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In the present paper we introduce some sequence spaces using ideal convergence and Musielak-Orlicz function M = (119872119896) Wealso examine some topological properties of the resulting sequence spaces

1 Introduction and Preliminaries

The notion of ideal convergence was first introduced byKostyrko et al [1] as a generalization of statistical convergencewhich was further studied in topological spaces by Das et alsee [2] More applications of ideals can be seen in [2 3]We continue in this direction and introduce 119868-convergenceof generalized sequences with respect to Musielak-Orliczfunction

A familyI sub 2

119883 of subsets of a nonempty set119883 is said tobe an ideal in119883 if

(1) 120601 isin I(2) 119860 119861 isin I imply 119860 cup 119861 isin I(3) 119860 isin I 119861 sub 119860 imply 119861 isin I

while an admissible ideal I of 119883 further satisfies 119909 isin Ifor each 119909 isin 119883 see [1] A sequence (119909119899)119899isinN in 119883 is said to be119868-convergent to 119909 isin 119883 If for each 120598 gt 0 the set 119860(120598) = 119899 isinN 119909119899minus119909 ge 120598 belongs toI see [1] For more details aboutideal convergent sequence spaces see [4ndash10] and referencestherein

Mursaleen and Noman [11] introduced the notion of 120582-convergent and 120582-bounded sequences as follows

Let 120582 = (120582119896)infin

119896=1be a strictly increasing sequence of

positive real numbers tending to infinity that is

0 lt 1205820 lt 1205821 lt sdot sdot sdot 120582119896 997888rarr infin as 119896 997888rarr infin (1)

The sequence 119909 = (119909119896) isin 119908 is 120582-convergent to the number 119871called the 120582-limit of 119909 if Λ119898(119909) rarr 119871 as119898 rarr infin where

Λ119898 (119909) =

1

120582119898

119898

sum

119896=1

(120582119896 minus 120582119896minus1) 119909119896 (2)

The sequence 119909=(119909119896)isin119908 is 120582-bounded if sup119898|Λ119898(119909)| lt

infin It is well known [11] that if lim119898 119909119898 = 119886 in the ordinarysense of convergence then

lim119898(

1

120582119898

(

119898

sum

119896=1

(120582119896 minus 120582119896minus1)1003816100381610038161003816

119909119896 minus 1198861003816100381610038161003816

)) = 0 (3)

This implies that

lim119898

1003816100381610038161003816

Λ119898 (119909) minus 1198861003816100381610038161003816

= lim119898

1003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898

119898

sum

119896=1

(120582119896 minus 120582119896minus1) (119909119896 minus 119886)

1003816100381610038161003816100381610038161003816100381610038161003816

= 0

(4)

which yields that lim119898Λ119898(119909) = 119886 and hence 119909 = (119909119896) isin 119908 is120582-convergent to 119886

Let 119883 be a linear metric space A function 119901 119883 rarr R iscalled paranorm if

(1) 119901(119909) ge 0 for all 119909 isin 119883(2) 119901(minus119909) = 119901(119909) for all 119909 isin 119883(3) 119901(119909 + 119910) le 119901(119909) + 119901(119910) for all 119909 119910 isin 119883

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 134534 6 pageshttpdxdoiorg1011552014134534

2 The Scientific World Journal

(4) if (120582119899) is a sequence of scalars with 120582119899 rarr 120582 as 119899 rarrinfin and (119909119899) is a sequence of vectors with 119901(119909119899minus119909) rarr0 as 119899 rarr infin then 119901(120582119899119909119899 minus 120582119909) rarr 0 as 119899 rarr infin

A paranorm 119901 for which 119901(119909) = 0 implies that 119909 = 0

is called total paranorm and the pair (119883 119901) is called a totalparanormed space It is well known that the metric of anylinear metric space is given by some total paranorm (see[12Theorem 1042 P-183]) For more details about sequencespaces see [13ndash15] and references therein

An Orlicz function119872 is a function which is continuousnondecreasing and convex with 119872(0) = 0 119872(119909) gt 0 for119909 gt 0 and119872(119909) rarr infin as 119909 rarr infin

Lindenstrauss and Tzafriri [16] used the idea of Orliczfunction to define the following sequence space Let 119908 be thespace of all real or complex sequences 119909 = (119909119896) Then

ℓ119872 = 119909 isin 119908

infin

sum

119896=1

119872(

1003816100381610038161003816

119909119896

1003816100381610038161003816

120588

) lt infin (5)

which is called an Orlicz sequence space The space ℓ119872 is aBanach space with the norm

119909 = inf 120588 gt 0 infin

sum

119896=1

119872(

1003816100381610038161003816

119909119896

1003816100381610038161003816

120588

) le 1 (6)

It is shown in [16] that every Orlicz sequence space ℓ119872contains a subspace isomorphic to ℓ119901 (119901 ge 1) The Δ 2-condition is equivalent to 119872(119871119909) le 119896119871119872(119909) for all valuesof 119909 ge 0 and for 119871 gt 1

A sequence M = (119872119896) of Orlicz function is called aMusielak-Orlicz function see [17 18] A sequenceN = (119873119896)

defined by

119873119896 (V) = sup |V| 119906 minus (119872119896) 119906 ge 0 119896 = 1 2 (7)

is called the complementary function of a Musielak-Orliczfunction M For a given Musielak-Orlicz function M theMusielak-Orlicz sequence space 119905M and its subspace ℎM aredefined as follows

119905M = 119909 isin 119908 119868M (119888119909) lt infin for some 119888 gt 0

ℎM = 119909 isin 119908 119868M (119888119909) lt infin for all 119888 gt 0 (8)

where 119868M is a convex modular defined by

119868M (119909) =

infin

sum

119896=1

119872119896 (119909119896) 119909 = (119909119896) isin 119905M (9)

We consider 119905M equipped with the Luxemburg norm

119909 = inf 119896 gt 0 119868M (119909

119896

) le 1 (10)

or equipped with the Orlicz norm

119909

0= inf 1

119896

(1 + 119868M (119896119909)) 119896 gt 0 (11)

Let M = (119872119896) be a Musielak-Orlicz function and let119901 = (119901119896) be a bounded sequence of positive real numbersWe define the following sequence spaces

119888

119868(M Λ 119901)

= 119909 = (119909119896) isin 119908 119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

= 0

for some 119871 and 120588 gt 0

119888

119868

0(M Λ 119901)

= 119909 = (119909119896) isin 119908 119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

for some 120588 gt 0

119897infin (M Λ 119901) = 119909 = (119909119896) isin 119908 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

lt infin


(12)

We can write

119898

119868(M Λ 119901) = 119888

119868(M Λ 119901) cap 119897infin (M Λ 119901)

119898

119868

0(M Λ 119901) = 119888

119868

0(M Λ 119901) cap 119897infin (M Λ 119901)

(13)

If we take 119901 = (119901119896) = 1 for all 119896 isin N we have

119888

119868(M Λ)

= 119909 = (119909119896) isin 119908 119868 minus lim119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

) = 0

for some 119871 and 120588 gt 0

119888

119868

0(M Λ) = 119909 = (119909119896) isin 119908 119868 minus lim

119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

) = 0


119897infin (M Λ) = 119909 = (119909119896) isin 119908 sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

) lt infin


(14)

The Scientific World Journal 3

The following inequality will be used throughout thepaper If 0 le 119901119896 le sup119901119896 = 119867119863 = max(1 2119867minus1) then

1003816100381610038161003816

119886119896 + 119887119896

1003816100381610038161003816

119901119896le 119863

1003816100381610038161003816

119886119896

1003816100381610038161003816

119901119896+

1003816100381610038161003816

119861119896

1003816100381610038161003816

119901119896 (15)

for all 119896 and 119886119896 119887119896 isin C Also |119886|119901119896 le max(1 |119886|119867) for all119886 isin C

The main aim of this paper is to study some ideal conver-gent sequence spaces defined by a Musielak-Orlicz functionM = (119872119896) We also make an effort to study some topologicalproperties and prove some inclusion relations between thesespaces

2 Main Results

Theorem 1 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen the spaces 119888119868(M Λ 119901) 119888119868

0(M Λ 119901) 119898119868(M Λ 119901) and

119898

119868

0(M Λ 119901) are linear

Proof Let 119909 119910 isin 119888119868(M Λ 119901) and let 120572 120573 be scalars Thenthere exist positive numbers 1205881 and 1205882 such that

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

= 0 for some 1198711 isin C

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896


(16)

For a given 120598 gt 0 we have

1198631 = 119896 isin N 119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

1198632 = 119896 isin N 119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896

(17)

Let 1205883 = max2|120572|1205881 2|120573|1205882 Since M = (119872119896) is non-decreasing convex function so by using inequality (15) wehave

lim119896

119872119896(

|Λ 119896((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))|

1205883

)

119901119896

le lim119896

119872119896(

|120572|

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205883

+

1003816100381610038161003816

120573

1003816100381610038161003816

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205883

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

+ lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896

(18)

Now by (17) we have

119896 isin N lim119896

119872119896(

1003816100381610038161003816

Λ 119896 ((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))1003816100381610038161003816

1205883

)

119901119896

gt 120598 sub 1198631 cup 1198632

(19)

Therefore 120572119909 + 120573119910 isin 119888

119868(M Λ 119901) Hence 119888119868(M Λ 119901)

is a linear space Similarly we can prove that 1198881198680(M Λ 119901)

119898

119868(M Λ 119901) and119898119868

0(M Λ 119901) are linear spaces

Theorem 2 Let M = (119872119896) be a Musielak-Orlicz functionThen

119888

119868

0(M Λ 119901) sub 119888

119868(M Λ 119901) sub 119897infin (M Λ 119901) (20)

Proof Let 119909 isin 119888119868(M Λ 119901) Then there exist 119871 isin C and 120588 gt 0such that

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

= 0 (21)

We have

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

2120588

)

119901119896

le

1

2

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

+119872119896

1

2

(

|119871|

120588

)

119901119896

(22)

Taking supremum over 119896 on both sides we get 119909 isin

119897infin(M Λ 119901) The inclusion 1198881198680(M Λ 119901) sub 119888

119868(M Λ 119901) is

obvious Thus

119888

119868

0(M Λ 119901) sub 119888

119868(M Λ 119901) sub 119897infin (M Λ 119901) (23)

This completes the proof of the theorem

Theorem 3 LetM = (119872119896) be aMusielak-Orlicz function andlet 119901 = (119901119896) be a bounded sequence of positive real numbersThen 119897infin(M Λ 119901) is a paranormed space with paranormdefined by

119892 (119909) = inf 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

le 1 (24)

Proof It is clear that 119892(119909) = 119892(minus119909) Since119872119896(0) = 0 we get119892(0) = 0 Let us take 119909 119910 isin 119897infin (M Λ 119901) Let

119861 (119909) = 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

le 1

119861 (119910) = 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

120588

)

119901119896

le 1

(25)


Let 1205881 isin 119861(119909) and 1205882 isin 119861(119910) If 120588 = 1205881 + 1205882 then we have

sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909 + 119910)1003816100381610038161003816

120588

)

le (

1205881

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

+ (

1205882

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

(26)

Thus sup119896119872119896(|Λ(119909 + 119910)|(1205881 + 1205882))

119901119896le 1 and

119892 (119909 + 119910) le inf (1205881 + 1205882) gt 0 1205881 isin 119861 (119909) 1205882 isin 119861 (119910)

le inf 1205881 gt 0 1205881 isin 119861 (119909)

+ inf 1205882 gt 0 1205882 isin 119861 (119910)

= 119892 (119909) + 119892 (119910)

(27)

Let 120590119904 rarr 120590 where 120590 120590119904 isin C and let 119892(119909119904 minus 119909) rarr 0 as119904 rarr infin We have to show that 119892(120590119904119909119904minus120590119909) rarr 0 as 119904 rarr infinLet

119861 (119909

119904) = 120588119904 gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

120588119904

)

119901119896

le 1

119861 (119909

119904minus 119909) = 120588

1015840

119904gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

119901119896

le 1

(28)

If 120588119904 isin 119861(119909119904) and 1205881015840

119904isin 119861(119909

119904minus 119909) then we observe that

119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le 119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909

119904)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

+

1003816100381610038161003816

(120590119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

120588119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

)

120588119904

)

+

|120590| 120588

1015840

119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

(29)

From the above inequality it follows that

119872119896(

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

119901119896

le 1 (30)

and consequently

119892 (120590

119904119909

119904minus 120590119909) le inf (120588119904

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

+ 120588

1015840

119904|120590|) gt 0

120588119904 isin 119861 (119909119904) 120588

1015840

119904isin 119861 (119909

119904minus 119909)

le (

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

) gt 0 inf 120588 gt 0 120588119904 isin 119861 (119909119904)

+ (|120590|) gt 0 inf (1205881015840119904)

119901119899119867

120588

1015840

119904isin 119861 (119909

119904minus 119909)

997888rarr 0 as 119904 997888rarr infin

(31)

This completes the proof

Theorem 4 Let M1015840 = (1198721015840119896) and M10158401015840 = (11987210158401015840

119896) be Musielak-

Orlicz functions that satisfy the Δ 2-condition Then(i) 119885(M10158401015840 Λ 119901) sube 119885(M1015840 ∘M10158401015840 Λ 119901)(ii) 119885(M1015840 Λ 119901) cap 119885(M10158401015840 Λ 119901) sube 119885(M1015840 +M10158401015840 Λ 119901) for119885 = 119888

119868 119888

119868

0 119898

119868 119898

119868

0

Proof (i) Let 119909 isin 1198881198680(M10158401015840 Λ 119901) Then there exists 120588 gt 0 such

that

119868 minus lim119896

M10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (32)

Let 120598 gt 0 and choose 120575with 0 lt 120575 lt 1 such that1198721015840119896(119905) lt 120598

for 0 le 119905 le 120575 Write 119910119896 = 11987210158401015840

119896(|Λ 119896(119909)|120588)

119901119896 and consider

lim0le119910119896le120575

119896isinN

119872

1015840

119896(119910119896) = lim

119910119896le120575

119896isinN

119872

1015840

119896(119910119896) + lim

119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) (33)

SinceM = (119872119896) satisfies Δ 2-condition we have

lim119910119896le120575

119896isinN

119872

1015840

119896(119910119896) le 119872

1015840

119896(2) lim119910119896le120575

119896isinN

(119910119896) (34)

For 119910119896 gt 120575 we have

119910119896 lt

119910119896

120575

lt 1 +

119910119896

120575

(35)

SinceM1015840 = (1198721015840119896) is nondecreasing and convex it follows

that

119872

1015840

119896(119910119896) lt 119872

1015840

119896(1 +

119910119896

120575

) lt

1

2

119872

1015840

119896(2) +

1

2

119872

1015840

119896(2119910119896)

120575

(36)

SinceM1015840 = (1198721015840119896) satisfies Δ 2-condition we have

119872

1015840

119896(119910119896) lt

1

2

119870

119910119896

120575

119872

1015840

119896(2) +

1

2

119870

119910119896

120575

119872

1015840

119896(2) = 119870

119910119896

120575

119872

1015840

119896(2)

(37)

Hence

lim119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) le max (1 119870120575minus11198721015840

119896(2)) lim

119910119896le120575

119896isinN

(119910119896) (38)


From (32) (34) and (38) we have 119909 = (119909119896) isin 119888

119868

0(M1015840 ∘

M10158401015840 Λ 119901)Thus 1198881198680(M10158401015840 Λ 119901) sube 119888119868

0(M1015840∘M10158401015840 Λ 119901) Similarly

we can prove the other cases(ii) Let 119909 isin 119888119868

0(M1015840 Λ 119901)cap119888119868

0(M10158401015840 Λ 119901)Then there exists

120588 gt 0 such that

119868 minus lim119896

119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

119868 minus lim119896

119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

(39)

The rest of the proof follows from the following equality

lim119896isinN(119872

1015840

119896+119872

10158401015840

119896)(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= lim119896isinN119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

+ lim119896isinN119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(40)

Corollary 5 Let M = (119872119896) be a Musielak-Orlicz functionwhich satisfies Δ 2-condition Then 119885(119901 Λ) sube 119885(M Λ 119901)

holds for 119885 = 119888119868 1198881198680 119898

119868 and1198981198680

Proof Theproof follows fromTheorem 3by putting11987210158401015840119896(119909) =

119909 and1198721015840119896(119909) = 119872119896(119909) forall119909 isin [0infin)

Theorem 6 The spaces 1198881198680(M Λ 119901) and 119898119868

0(M Λ 119901) are

solid

Proof We will prove for the space 1198881198680(M Λ 119901Λ) Let 119909 isin

119888

119868

0(M Λ 119901) Then there exists 120588 gt 0 such that

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (41)

Let (120572119896) be a sequence of scalars with |120572119896| le 1 forall119896 isin NThen the result follows from the following inequality

lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (120572119909)1003816100381610038161003816

120588

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(42)

and this completes the proof Similarly we can prove for thespace119898119868

0(M Λ 119901)

Corollary 7 The spaces 1198881198680(M Λ 119901) and 119898119868

0(M Λ 119901) are

monotone

Proof It is easy to prove so we omit the details

Theorem 8 The spaces 119888119868(M Λ 119901) and 1198881198680(M Λ 119901) are seq-

uence algebra

Proof Let 119909 119910 isin 1198881198680(M Λ 119901) Then

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

119901119896

= 0 for some 1205881 gt 0

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

119901119896

= 0 for some 1205882 gt 0

(43)

Let 120588 = 1205881 + 1205882 Then we can show that

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909 sdot 119910)1003816100381610038161003816

120588

)

119901119896

= 0 (44)

Thus (119909 sdot 119910) isin 119888

119868

0(M Λ 119901) Hence 119888119868

0(M Λ 119901) is a

sequence algebra Similarly we can prove that 119888119868(M Λ 119901) isa sequence algebra

Conflict of Interests

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

References

[1] P Kostyrko T Salat and W Wilczynski ldquo119868-convergencerdquo RealAnalysis Exchange vol 26 no 2 pp 669ndash685 2000

[2] P Das P Kostyrko W Wilczynski and P Malik ldquo119868 and 119868lowast-convergence of double sequencesrdquo Mathematica Slovaca vol58 no 5 pp 605ndash620 2008

[3] P Das and P Malik ldquoOn the statistical and 119868 variation of doublesequencesrdquo Real Analysis Exchange vol 33 no 2 pp 351ndash3632008

[4] V Kumar ldquoOn 119868 and 119868lowast-convergence of double sequencesrdquoMathematical Communications vol 12 no 2 pp 171ndash181 2007

[5] M Mursaleen and A Alotaibi ldquoOn 119868-convergence in random2-normed spacesrdquoMathematica Slovaca vol 61 no 6 pp 933ndash940 2011

[6] M Mursaleen S A Mohiuddine and O H H Edely ldquoOn theideal convergence of double sequences in intuitionistic fuzzynormed spacesrdquo Computers amp Mathematics with Applicationsvol 59 no 2 pp 603ndash611 2010

[7] M Mursaleen and S A Mohiuddine ldquoOn ideal convergence ofdouble sequences in probabilistic normed spacesrdquo Mathemati-cal Reports vol 12(62) no 4 pp 359ndash371 2010

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

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

[10] B C Tripathy and B Hazarika ldquoSome 119868-convergent sequencespaces defined by Orlicz functionsrdquo Acta Mathematicae Appli-catae Sinica vol 27 no 1 pp 149ndash154 2011

[11] M Mursaleen and A K Noman ldquoOn some new sequencespaces of non-absolute type related to the spaces ℓ119901 and ℓinfin IrdquoFilomat vol 25 no 2 pp 33ndash51 2011

[12] A Wilansky Summability through Functional Analysis vol 85ofNorth-HollandMathematics Studies North-Holland Publish-ing Amsterdam The Netherlands 1984


[13] K Raj and S K Sharma ldquoSome sequence spaces in 2-normedspaces defined by Musielak-Orlicz functionrdquo Acta UniversitatisSapientiae vol 3 no 1 pp 97ndash109 2011

[14] K Raj and S K Sharma ldquoSome generalized difference doublesequence spaces defined by a sequence of Orlicz-functionsrdquoCubo vol 14 no 3 pp 167ndash190 2012

[15] K Raj and S K Sharma ldquoSome multiplier sequence spacesdefined by a Musielak-Orlicz function in 119899-normed spacesrdquoNew Zealand Journal of Mathematics vol 42 pp 45ndash56 2012

[16] J Lindenstrauss and L Tzafriri ldquoOn Orlicz sequence spacesrdquoIsrael Journal of Mathematics vol 10 pp 379ndash390 1971

[17] L Maligranda Orlicz Spaces and Interpolation vol 5 of Semi-nars in Mathematics Polish Academy of Science 1989

[18] J Musielak Orlicz Spaces and Modular Spaces vol 1034 ofLecture Notes in Mathematics Springer 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(4) if (120582119899) is a sequence of scalars with 120582119899 rarr 120582 as 119899 rarrinfin and (119909119899) is a sequence of vectors with 119901(119909119899minus119909) rarr0 as 119899 rarr infin then 119901(120582119899119909119899 minus 120582119909) rarr 0 as 119899 rarr infin

A paranorm 119901 for which 119901(119909) = 0 implies that 119909 = 0

is called total paranorm and the pair (119883 119901) is called a totalparanormed space It is well known that the metric of anylinear metric space is given by some total paranorm (see[12Theorem 1042 P-183]) For more details about sequencespaces see [13ndash15] and references therein

An Orlicz function119872 is a function which is continuousnondecreasing and convex with 119872(0) = 0 119872(119909) gt 0 for119909 gt 0 and119872(119909) rarr infin as 119909 rarr infin

Lindenstrauss and Tzafriri [16] used the idea of Orliczfunction to define the following sequence space Let 119908 be thespace of all real or complex sequences 119909 = (119909119896) Then

ℓ119872 = 119909 isin 119908

infin

sum

119896=1

119872(

1003816100381610038161003816

119909119896

1003816100381610038161003816

120588

) lt infin (5)

which is called an Orlicz sequence space The space ℓ119872 is aBanach space with the norm

119909 = inf 120588 gt 0 infin

sum

119896=1

119872(

1003816100381610038161003816

119909119896

1003816100381610038161003816

120588

) le 1 (6)

It is shown in [16] that every Orlicz sequence space ℓ119872contains a subspace isomorphic to ℓ119901 (119901 ge 1) The Δ 2-condition is equivalent to 119872(119871119909) le 119896119871119872(119909) for all valuesof 119909 ge 0 and for 119871 gt 1

A sequence M = (119872119896) of Orlicz function is called aMusielak-Orlicz function see [17 18] A sequenceN = (119873119896)

defined by

119873119896 (V) = sup |V| 119906 minus (119872119896) 119906 ge 0 119896 = 1 2 (7)

is called the complementary function of a Musielak-Orliczfunction M For a given Musielak-Orlicz function M theMusielak-Orlicz sequence space 119905M and its subspace ℎM aredefined as follows

119905M = 119909 isin 119908 119868M (119888119909) lt infin for some 119888 gt 0

ℎM = 119909 isin 119908 119868M (119888119909) lt infin for all 119888 gt 0 (8)

where 119868M is a convex modular defined by

119868M (119909) =

infin

sum

119896=1

119872119896 (119909119896) 119909 = (119909119896) isin 119905M (9)

We consider 119905M equipped with the Luxemburg norm

119909 = inf 119896 gt 0 119868M (119909

119896

) le 1 (10)

or equipped with the Orlicz norm

119909

0= inf 1

119896

(1 + 119868M (119896119909)) 119896 gt 0 (11)

Let M = (119872119896) be a Musielak-Orlicz function and let119901 = (119901119896) be a bounded sequence of positive real numbersWe define the following sequence spaces

119888

119868(M Λ 119901)

= 119909 = (119909119896) isin 119908 119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

= 0

for some 119871 and 120588 gt 0

119888

119868

0(M Λ 119901)

= 119909 = (119909119896) isin 119908 119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0


119897infin (M Λ 119901) = 119909 = (119909119896) isin 119908 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

lt infin


(12)

We can write

119898

119868(M Λ 119901) = 119888

119868(M Λ 119901) cap 119897infin (M Λ 119901)

119898

119868

0(M Λ 119901) = 119888

119868

0(M Λ 119901) cap 119897infin (M Λ 119901)

(13)

If we take 119901 = (119901119896) = 1 for all 119896 isin N we have

119888

119868(M Λ)

= 119909 = (119909119896) isin 119908 119868 minus lim119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

) = 0

for some 119871 and 120588 gt 0

119888

119868

0(M Λ) = 119909 = (119909119896) isin 119908 119868 minus lim

119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

) = 0


119897infin (M Λ) = 119909 = (119909119896) isin 119908 sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

) lt infin


(14)



1003816100381610038161003816

119886119896 + 119887119896

1003816100381610038161003816

119901119896le 119863

1003816100381610038161003816

119886119896

1003816100381610038161003816

119901119896+

1003816100381610038161003816

119861119896

1003816100381610038161003816

119901119896 (15)



2 Main Results


0(M Λ 119901) 119898119868(M Λ 119901) and

119898

119868



119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896


(16)


1198631 = 119896 isin N 119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

1198632 = 119896 isin N 119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896

(17)


lim119896

119872119896(

|Λ 119896((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))|

1205883

)

119901119896

le lim119896

119872119896(

|120572|

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205883

+

1003816100381610038161003816

120573

1003816100381610038161003816

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205883

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

+ lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896

(18)

Now by (17) we have

119896 isin N lim119896

119872119896(

1003816100381610038161003816

Λ 119896 ((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))1003816100381610038161003816

1205883

)

119901119896

gt 120598 sub 1198631 cup 1198632

(19)

Therefore 120572119909 + 120573119910 isin 119888

119868(M Λ 119901) Hence 119888119868(M Λ 119901)


119898

119868(M Λ 119901) and119898119868



119888

119868

0(M Λ 119901) sub 119888

119868(M Λ 119901) sub 119897infin (M Λ 119901) (20)


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

= 0 (21)

We have

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

2120588

)

119901119896

le

1

2

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

+119872119896

1

2

(

|119871|

120588

)

119901119896

(22)



119868(M Λ 119901) is

obvious Thus

119888

119868

0(M Λ 119901) sub 119888

119868(M Λ 119901) sub 119897infin (M Λ 119901) (23)



119892 (119909) = inf 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

le 1 (24)


119861 (119909) = 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

le 1

119861 (119910) = 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

120588

)

119901119896

le 1

(25)



sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909 + 119910)1003816100381610038161003816

120588

)

le (

1205881

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

+ (

1205882

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

(26)

Thus sup119896119872119896(|Λ(119909 + 119910)|(1205881 + 1205882))

119901119896le 1 and


le inf 1205881 gt 0 1205881 isin 119861 (119909)

+ inf 1205882 gt 0 1205882 isin 119861 (119910)

= 119892 (119909) + 119892 (119910)

(27)


119861 (119909

119904) = 120588119904 gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

120588119904

)

119901119896

le 1

119861 (119909

119904minus 119909) = 120588

1015840

119904gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

119901119896

le 1

(28)

If 120588119904 isin 119861(119909119904) and 1205881015840

119904isin 119861(119909


119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le 119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909

119904)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

+

1003816100381610038161003816

(120590119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

120588119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

)

120588119904

)

+

|120590| 120588

1015840

119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

(29)


119872119896(

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

119901119896

le 1 (30)

and consequently

119892 (120590

119904119909

119904minus 120590119909) le inf (120588119904

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

+ 120588

1015840

119904|120590|) gt 0

120588119904 isin 119861 (119909119904) 120588

1015840

119904isin 119861 (119909

119904minus 119909)

le (

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

) gt 0 inf 120588 gt 0 120588119904 isin 119861 (119909119904)

+ (|120590|) gt 0 inf (1205881015840119904)

119901119899119867

120588

1015840

119904isin 119861 (119909

119904minus 119909)


(31)





119868 119888

119868

0 119898

119868 119898

119868

0


that

119868 minus lim119896

M10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (32)


for 0 le 119905 le 120575 Write 119910119896 = 11987210158401015840

119896(|Λ 119896(119909)|120588)


lim0le119910119896le120575

119896isinN

119872

1015840

119896(119910119896) = lim

119910119896le120575

119896isinN

119872

1015840

119896(119910119896) + lim

119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) (33)


lim119910119896le120575

119896isinN

119872

1015840

119896(119910119896) le 119872

1015840

119896(2) lim119910119896le120575

119896isinN

(119910119896) (34)

For 119910119896 gt 120575 we have

119910119896 lt

119910119896

120575

lt 1 +

119910119896

120575

(35)


that

119872

1015840

119896(119910119896) lt 119872

1015840

119896(1 +

119910119896

120575

) lt

1

2

119872

1015840

119896(2) +

1

2

119872

1015840

119896(2119910119896)

120575

(36)


119872

1015840

119896(119910119896) lt

1

2

119870

119910119896

120575

119872

1015840

119896(2) +

1

2

119870

119910119896

120575

119872

1015840

119896(2) = 119870

119910119896

120575

119872

1015840

119896(2)

(37)

Hence

lim119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) le max (1 119870120575minus11198721015840

119896(2)) lim

119910119896le120575

119896isinN

(119910119896) (38)



119868

0(M1015840 ∘

M10158401015840 Λ 119901)Thus 1198881198680(M10158401015840 Λ 119901) sube 119888119868

0(M1015840∘M10158401015840 Λ 119901) Similarly


0(M1015840 Λ 119901)cap119888119868



119868 minus lim119896

119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

119868 minus lim119896

119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

(39)


lim119896isinN(119872

1015840

119896+119872

10158401015840

119896)(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= lim119896isinN119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

+ lim119896isinN119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(40)


holds for 119885 = 119888119868 1198881198680 119898

119868 and1198981198680




0(M Λ 119901) are

solid


119888

119868


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (41)


lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (120572119909)1003816100381610038161003816

120588

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(42)


0(M Λ 119901)


0(M Λ 119901) are

monotone



uence algebra


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

119901119896

= 0 for some 1205881 gt 0

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

119901119896

= 0 for some 1205882 gt 0

(43)


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909 sdot 119910)1003816100381610038161003816

120588

)

119901119896

= 0 (44)

Thus (119909 sdot 119910) isin 119888

119868

0(M Λ 119901) Hence 119888119868

0(M Λ 119901) is a




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003816100381610038161003816

119886119896 + 119887119896

1003816100381610038161003816

119901119896le 119863

1003816100381610038161003816

119886119896

1003816100381610038161003816

119901119896+

1003816100381610038161003816

119861119896

1003816100381610038161003816

119901119896 (15)



2 Main Results


0(M Λ 119901) 119898119868(M Λ 119901) and

119898

119868



119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896


(16)


1198631 = 119896 isin N 119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

1198632 = 119896 isin N 119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896

(17)


lim119896

119872119896(

|Λ 119896((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))|

1205883

)

119901119896

le lim119896

119872119896(

|120572|

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205883

+

1003816100381610038161003816

120573

1003816100381610038161003816

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205883

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711

1003816100381610038161003816

1205881

)

119901119896

+ lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910) minus 1198712

1003816100381610038161003816

1205882

)

119901119896

(18)

Now by (17) we have

119896 isin N lim119896

119872119896(

1003816100381610038161003816

Λ 119896 ((120572119909 + 120573119910) minus (1205721198711 + 1205731198712))1003816100381610038161003816

1205883

)

119901119896

gt 120598 sub 1198631 cup 1198632

(19)

Therefore 120572119909 + 120573119910 isin 119888

119868(M Λ 119901) Hence 119888119868(M Λ 119901)


119898

119868(M Λ 119901) and119898119868



119888

119868

0(M Λ 119901) sub 119888

119868(M Λ 119901) sub 119897infin (M Λ 119901) (20)


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

= 0 (21)

We have

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

2120588

)

119901119896

le

1

2

119872119896(

1003816100381610038161003816

Λ 119896 (119909) minus 1198711003816100381610038161003816

120588

)

119901119896

+119872119896

1

2

(

|119871|

120588

)

119901119896

(22)



119868(M Λ 119901) is

obvious Thus

119888

119868

0(M Λ 119901) sub 119888

119868(M Λ 119901) sub 119897infin (M Λ 119901) (23)



119892 (119909) = inf 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

le 1 (24)


119861 (119909) = 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

le 1

119861 (119910) = 120588 gt 0 sup119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

120588

)

119901119896

le 1

(25)



sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909 + 119910)1003816100381610038161003816

120588

)

le (

1205881

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

+ (

1205882

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

(26)

Thus sup119896119872119896(|Λ(119909 + 119910)|(1205881 + 1205882))

119901119896le 1 and


le inf 1205881 gt 0 1205881 isin 119861 (119909)

+ inf 1205882 gt 0 1205882 isin 119861 (119910)

= 119892 (119909) + 119892 (119910)

(27)


119861 (119909

119904) = 120588119904 gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

120588119904

)

119901119896

le 1

119861 (119909

119904minus 119909) = 120588

1015840

119904gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

119901119896

le 1

(28)

If 120588119904 isin 119861(119909119904) and 1205881015840

119904isin 119861(119909


119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le 119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909

119904)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

+

1003816100381610038161003816

(120590119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

120588119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

)

120588119904

)

+

|120590| 120588

1015840

119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

(29)


119872119896(

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

119901119896

le 1 (30)

and consequently

119892 (120590

119904119909

119904minus 120590119909) le inf (120588119904

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

+ 120588

1015840

119904|120590|) gt 0

120588119904 isin 119861 (119909119904) 120588

1015840

119904isin 119861 (119909

119904minus 119909)

le (

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

) gt 0 inf 120588 gt 0 120588119904 isin 119861 (119909119904)

+ (|120590|) gt 0 inf (1205881015840119904)

119901119899119867

120588

1015840

119904isin 119861 (119909

119904minus 119909)


(31)





119868 119888

119868

0 119898

119868 119898

119868

0


that

119868 minus lim119896

M10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (32)


for 0 le 119905 le 120575 Write 119910119896 = 11987210158401015840

119896(|Λ 119896(119909)|120588)


lim0le119910119896le120575

119896isinN

119872

1015840

119896(119910119896) = lim

119910119896le120575

119896isinN

119872

1015840

119896(119910119896) + lim

119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) (33)


lim119910119896le120575

119896isinN

119872

1015840

119896(119910119896) le 119872

1015840

119896(2) lim119910119896le120575

119896isinN

(119910119896) (34)

For 119910119896 gt 120575 we have

119910119896 lt

119910119896

120575

lt 1 +

119910119896

120575

(35)


that

119872

1015840

119896(119910119896) lt 119872

1015840

119896(1 +

119910119896

120575

) lt

1

2

119872

1015840

119896(2) +

1

2

119872

1015840

119896(2119910119896)

120575

(36)


119872

1015840

119896(119910119896) lt

1

2

119870

119910119896

120575

119872

1015840

119896(2) +

1

2

119870

119910119896

120575

119872

1015840

119896(2) = 119870

119910119896

120575

119872

1015840

119896(2)

(37)

Hence

lim119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) le max (1 119870120575minus11198721015840

119896(2)) lim

119910119896le120575

119896isinN

(119910119896) (38)



119868

0(M1015840 ∘

M10158401015840 Λ 119901)Thus 1198881198680(M10158401015840 Λ 119901) sube 119888119868

0(M1015840∘M10158401015840 Λ 119901) Similarly


0(M1015840 Λ 119901)cap119888119868



119868 minus lim119896

119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

119868 minus lim119896

119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

(39)


lim119896isinN(119872

1015840

119896+119872

10158401015840

119896)(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= lim119896isinN119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

+ lim119896isinN119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(40)


holds for 119885 = 119888119868 1198881198680 119898

119868 and1198981198680




0(M Λ 119901) are

solid


119888

119868


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (41)


lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (120572119909)1003816100381610038161003816

120588

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(42)


0(M Λ 119901)


0(M Λ 119901) are

monotone



uence algebra


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

119901119896

= 0 for some 1205881 gt 0

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

119901119896

= 0 for some 1205882 gt 0

(43)


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909 sdot 119910)1003816100381610038161003816

120588

)

119901119896

= 0 (44)

Thus (119909 sdot 119910) isin 119888

119868

0(M Λ 119901) Hence 119888119868

0(M Λ 119901) is a




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909 + 119910)1003816100381610038161003816

120588

)

le (

1205881

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

+ (

1205882

1205881 + 1205882

) sup119896

119872119896 (

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

(26)

Thus sup119896119872119896(|Λ(119909 + 119910)|(1205881 + 1205882))

119901119896le 1 and


le inf 1205881 gt 0 1205881 isin 119861 (119909)

+ inf 1205882 gt 0 1205882 isin 119861 (119910)

= 119892 (119909) + 119892 (119910)

(27)


119861 (119909

119904) = 120588119904 gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

120588119904

)

119901119896

le 1

119861 (119909

119904minus 119909) = 120588

1015840

119904gt 0 sup

119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

119901119896

le 1

(28)

If 120588119904 isin 119861(119909119904) and 1205881015840

119904isin 119861(119909


119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le 119872119896 (

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909

119904)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

+

1003816100381610038161003816

(120590119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

le

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

120588119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

(

1003816100381610038161003816

Λ 119896 (119909119904)

1003816100381610038161003816

)

120588119904

)

+

|120590| 120588

1015840

119904

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

119872119896 (

1003816100381610038161003816

Λ 119896 (119909119904minus 119909)

1003816100381610038161003816

120588

1015840119904

)

(29)


119872119896(

1003816100381610038161003816

Λ 119896 (120590119904119909

119904minus 120590119909)

1003816100381610038161003816

120588119904 |120590119904minus 120590| + 120588

1015840119904|120590|

)

119901119896

le 1 (30)

and consequently

119892 (120590

119904119909

119904minus 120590119909) le inf (120588119904

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

+ 120588

1015840

119904|120590|) gt 0

120588119904 isin 119861 (119909119904) 120588

1015840

119904isin 119861 (119909

119904minus 119909)

le (

1003816100381610038161003816

120590

119904minus 120590

1003816100381610038161003816

) gt 0 inf 120588 gt 0 120588119904 isin 119861 (119909119904)

+ (|120590|) gt 0 inf (1205881015840119904)

119901119899119867

120588

1015840

119904isin 119861 (119909

119904minus 119909)


(31)





119868 119888

119868

0 119898

119868 119898

119868

0


that

119868 minus lim119896

M10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (32)


for 0 le 119905 le 120575 Write 119910119896 = 11987210158401015840

119896(|Λ 119896(119909)|120588)


lim0le119910119896le120575

119896isinN

119872

1015840

119896(119910119896) = lim

119910119896le120575

119896isinN

119872

1015840

119896(119910119896) + lim

119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) (33)


lim119910119896le120575

119896isinN

119872

1015840

119896(119910119896) le 119872

1015840

119896(2) lim119910119896le120575

119896isinN

(119910119896) (34)

For 119910119896 gt 120575 we have

119910119896 lt

119910119896

120575

lt 1 +

119910119896

120575

(35)


that

119872

1015840

119896(119910119896) lt 119872

1015840

119896(1 +

119910119896

120575

) lt

1

2

119872

1015840

119896(2) +

1

2

119872

1015840

119896(2119910119896)

120575

(36)


119872

1015840

119896(119910119896) lt

1

2

119870

119910119896

120575

119872

1015840

119896(2) +

1

2

119870

119910119896

120575

119872

1015840

119896(2) = 119870

119910119896

120575

119872

1015840

119896(2)

(37)

Hence

lim119910119896gt120575

119896isinN

119872

1015840

119896(119910119896) le max (1 119870120575minus11198721015840

119896(2)) lim

119910119896le120575

119896isinN

(119910119896) (38)



119868

0(M1015840 ∘

M10158401015840 Λ 119901)Thus 1198881198680(M10158401015840 Λ 119901) sube 119888119868

0(M1015840∘M10158401015840 Λ 119901) Similarly


0(M1015840 Λ 119901)cap119888119868



119868 minus lim119896

119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

119868 minus lim119896

119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

(39)


lim119896isinN(119872

1015840

119896+119872

10158401015840

119896)(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= lim119896isinN119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

+ lim119896isinN119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(40)


holds for 119885 = 119888119868 1198881198680 119898

119868 and1198981198680




0(M Λ 119901) are

solid


119888

119868


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (41)


lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (120572119909)1003816100381610038161003816

120588

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(42)


0(M Λ 119901)


0(M Λ 119901) are

monotone



uence algebra


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

119901119896

= 0 for some 1205881 gt 0

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

119901119896

= 0 for some 1205882 gt 0

(43)


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909 sdot 119910)1003816100381610038161003816

120588

)

119901119896

= 0 (44)

Thus (119909 sdot 119910) isin 119888

119868

0(M Λ 119901) Hence 119888119868

0(M Λ 119901) is a




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868

0(M1015840 ∘

M10158401015840 Λ 119901)Thus 1198881198680(M10158401015840 Λ 119901) sube 119888119868

0(M1015840∘M10158401015840 Λ 119901) Similarly


0(M1015840 Λ 119901)cap119888119868



119868 minus lim119896

119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

119868 minus lim119896

119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0

(39)


lim119896isinN(119872

1015840

119896+119872

10158401015840

119896)(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= lim119896isinN119872

1015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

+ lim119896isinN119872

10158401015840

119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(40)


holds for 119885 = 119888119868 1198881198680 119898

119868 and1198981198680




0(M Λ 119901) are

solid


119888

119868


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

= 0 (41)


lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (120572119909)1003816100381610038161003816

120588

)

119901119896

le lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

120588

)

119901119896

(42)


0(M Λ 119901)


0(M Λ 119901) are

monotone



uence algebra


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909)1003816100381610038161003816

1205881

)

119901119896

= 0 for some 1205881 gt 0

119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119910)1003816100381610038161003816

1205882

)

119901119896

= 0 for some 1205882 gt 0

(43)


119868 minus lim119896

119872119896(

1003816100381610038161003816

Λ 119896 (119909 sdot 119910)1003816100381610038161003816

120588

)

119901119896

= 0 (44)

Thus (119909 sdot 119910) isin 119888

119868

0(M Λ 119901) Hence 119888119868

0(M Λ 119901) is a




References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Spaces of Ideal Convergent …downloads.hindawi.com/journals/tswj/2014/134534.pdfSpaces of Ideal Convergent Sequences M.Mursaleen 1 andSunilK.Sharma 2 Department of