Research ArticleAlmost Conservative Four-Dimensional Matrices through de laValleacutee-Poussin Mean

S A Mohiuddine and Abdullah Alotaibi

Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to S A Mohiuddine mohiuddinegmailcom

Received 1 November 2013 Accepted 17 December 2013 Published 2 January 2014

Academic Editor Mohammad Mursaleen

Copyright copy 2014 S A Mohiuddine and A Alotaibi 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

The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de laVallee-Poussin mean for double sequences We also define and characterize the generalized regularly almost conservative andalmost coercive four-dimensional matrices Further we characterize the infinite matrices which transform the sequence belongingto the space of absolutely convergent double series into the space of generalized almost convergence

1 Introduction and Preliminaries

Let 119897infinbe the Banach space of real bounded sequences 119909 = 119909

119899

with the usual norm 119909 = sup |119909119899| There exist continuous

linear functionals on 119897infin

called Banach limits [1] It is wellknown that any Banach limit of 119909 lies between lim inf 119909 andlim sup119909 The idea of almost convergence of Lorentz [2] isnarrowly connected with the limits of S Banach that is asequence 119909

119899isin 119897infinis almost convergent to 119897 if all of its Banach

limits are equal As an application of almost convergenceMohiuddine [3] obtained some approximation theorems forsequence of positive linear operator through this notion Fordouble sequence the notion of almost convergence was firstintroduced by Moricz and Rhoades [4] The authors of [5]introduced the notion of Banach limit for double sequenceand characterized the spaces of almost and strong almostconvergence for double sequences through some sublinearfunctionals For more details on these concepts one can referto [6ndash12]

We say that a double sequence 119909 = (119909119895119896

119895 119896 = 0 12 ) of real or complex numbers is bounded if

119909 = sup119895119896

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt infin (1)

denoted byLinfin the space of all bounded sequence (119909

119895119896)

A double sequence 119909 = (119909119895119896) of reals is called convergent

to some number 119871 in Pringsheimrsquos sense (briefly 119875-convergent

to 119871) [13] if for every 120598 gt 0 there exists 119873 isin N such that|119909119895119896minus 119871| lt 120598 whenever 119895 119896 ge 119873 where N = 1 2 3 If a double sequence 119909 = (119909

119895119896) in L

infinand 119909 is also 119875-

convergent to 119871 then we say that it is boundedly 119875-convergentto 119871 (briefly 119861119875-convergent to 119871)

A double sequence 119909 = (119909119895119896) is said to converge regu-

larly to 119871 (briefly 119877-convergent to 119871) if 119909 is converges inPringsheimrsquos sense and the limits 119909119895 = lim

119896119909119895119896(119895 isin N) and

119909119896

= lim119895119909119895119896(119896 isin N) exist Note that in this case the limits

lim119895lim119896119909119895119896and lim

119896lim119895119909119895119896exist and are equal to the119875-limit

of 119909Throughout this paper by C

119875 C119861119875 and C

119877 we denote

the space of all 119875-convergent 119861119875-convergent and 119877-convergent double sequences respectively Also the linearspace of all continuous linear functionals on C

119877is denoted

byC1015840119877

Let 119861 = (119887119901119902119895119896

119895 119896 = 0 1 2 ) be a four-dimensionalinfinite matrix of real numbers for all 119901 119902 = 0 1 2 and1198781a space of double sequences Let 119878

2be a double sequences

space converging with respect to a convergence rule ] isin

119875 119861119875 119877 Define

119878119861]1

=

119909 = (119909119895119896) 119861119909 = (119861

119901119902(119909))

= ] minussum119895119896

119887119901119902119895119896

119909119895119896

exists and 119861119909 isin 1198781

(2)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 412974 6 pageshttpdxdoiorg1011552014412974

2 Abstract and Applied Analysis

Then we say that a four-dimensionalmatrix119861maps the space1198782into the space 119878

1if 1198782sub 119878119861]1

and is denoted by (1198782 1198781)

Moricz and Rhoades [4] extended the notion of almostconvergence from single to double sequence and character-ized some matrix classes involving this concept A doublesequence 119909 = (119909

119895119896) of real numbers is said to be almost

convergent to a number 119871 if

lim119901119902rarrinfin

sup119898119899gt0

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119901119902

119898+119901minus1

sum

119895=119898

119899+119902minus1

sum

119896=119899

119909119895119896minus 119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (3)

For more details on double sequences and 4-dimensionalmatrices one can refer to [14ndash20]

Using the notion of almost convergence for singlesequence King [21] introduced a slightly more general classof matrices than the conservative and regular matrices thatis almost conservative and almost regular matrices andpresented its characterization In [22] Schaefer presentedsome interesting characterization for almost convergenceThe Steinhaus-type theorem for the concepts of almostregular and almost coercivematrices was proved by Basar andSolak [23] In this paper we generalize the concept of almostconvergence for double sequences with the help of doublegeneralized de la Vallee-Poussinmean and called it (Λ)almostconvergence Using this concept we define the notions ofregularly (Λ)almost conservative and (Λ)almost coercivefour-dimensional matrices and obtain their necessity andsufficient conditions Further we introduce the space L

1

of all absolutely convergent double series and characterizethe matrix class (L

1FΛ) where F

Λdenotes the space of

(Λ)almost convergence for double sequences

2 Main Results

Definition 1 Let 120582 = (120582119898 119898 = 0 1 2 ) and 120583 = (120583

119899 119899 =

0 1 2 ) be two nondecreasing sequences of positive realswith each tending to infin such that 120582

119898+1le 120582119898+ 1 120582

1= 0

120583119899+1

le 120583119899+ 1 120583

1= 0 and

I119898119899

(119909) =1

120582119898120583119899

sum

119895isin119869119898

sum

119896isin119868119899

119909119895119896 (4)

is called the double generalized de la Vallee-Poussin meanwhere 119869

119898= [119898minus120582

119898+1119898] and 119868

119899= [119899minus120583

119899+1 119899] We denote

the set of all 120582 and 120583 type sequences by using the symbol Λ

Definition 2 A double sequence 119909 = (119909119895119896) of reals is said

to be (Λ)almost convergent (briefly FΛ-convergent) to some

number 119871 if 119909 isin F120582120583

where

FΛ= 119909 = (119909

119895119896) 119875- lim119898119899rarrinfin

Ω119898119899119904119905

(119909)

= 119871 exists uniformly in 119904 119905 119871 = FΛ- lim119909

Ω119898119899119904119905

(119909) =1

120582119898120583119899

sum

119895isin119869119898

sum

119896isin119868119899

119909119895+119904119896+119905

(5)

denoted by FΛ the space of all (Λ)almost convergent

sequences (119909119895119896) Note thatC

119861119875sub FΛsub Linfin

Remark 3 If we take 120582119898= 119898 and 120583

119899= 119899 then the notion of

(Λ)almost convergence reduced to almost convergence dueto Moricz and Rhoades [4]

Definition 4 A four-dimensional matrix 119861 = (119887119901119902119895119896

) issaid to be regularly (Λ)almost conservative if it maps every119877-convergent double sequence into F

Λ-convergent double

sequence that is 119861 isin (C119877FΛ) In addition ifF

Λ- lim119860119909 =

119877- lim119909 then 119861 is regularly (Λ)almost regular

Definition 5 A matrix 119861 = (119887119901119902119895119896

) is said to be (Λ)almostcoercive if it maps every 119861119875-convergent double sequence(119909119895119896) intoF

Λ-convergent double sequence briefly a matrix

119861 in (C119861119875FΛ)

Theorem 6 A matrix 119861 = (119887119901119902119895119896

) is regularly (Λ)almostconservative if and only if

(CR1) 119861 = sup

119901119902sum119895119896|119887119901119902119895119896

| lt infin

(CR2) lim119898119899rarrinfin

120572(119898 119899 119904 119905 119895 119896) = 119906119895119896 for each 119895 119896 (uni-

formly in 119904 119905)(CR3) lim119898119899rarrinfin

sum119896|120572(119898 119899 119904 119905 119895 119896)| = 119906

1198950 for each 119895

(uniformly in 119904 119905)(CR4) lim119898119899rarrinfin

sum119895|120572(119898 119899 119904 119905 119895 119896)| = 119906

0119896 for each 119896

(uniformly in 119904 119905)(CR5) lim119898119899rarrinfin

sum119895119896120572(119898 119899 119904 119905 119895 119896) = 119906 (uniformly in 119904

119905)

where

120573 (119898 119899 119904 119905 119895 119896) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

(6)

In this case theFΛ-limit of 119861119909 is

ℓ119906 +

infin

sum

119895=0

(ℓ119895minus ℓ) 119906

1198950+

infin

sum

119896=0

(ℎ119896minus ℓ) 119906

0119896

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ) 119906

119895119896

(7)

where ℓ = 119877- lim119909

Proof Necessity Suppose that 119861 is regularly (Λ)almost con-servative matrix Fix 119904 119905 isin Z the set of integers Let

Ω119898119899119904119905

(119909) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119898

120588119901+119904119902+119905

(119909) (8)

where

120588119901119902

(119909) =

infin

sum

119895=0

infin

sum

119896=0

119887119901119902119895119896

119909119895119896 (9)

It is clear that

120588119901119902

isin C1015840

119877 119901 119902 = 0 1 2 (10)

Abstract and Applied Analysis 3

Hence Ω119898119899119904119905

isin C1015840119877for 119898 119899 isin N Since 119861 is regularly

(Λ)almost conservative we have

119875- lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω (119909) (say) (11)

uniformly in 119904 119905 It follows that (Ω119898119899119904119905

(119909)) is bounded for119909 isin C

119877and fixed 119904 119905 Hence Ω

119898119899119904119905(119909) is bounded by the

uniform boundedness principleFor each 119894 V isin Z+ define the sequence 119910 = 119910(119898 119899 119904 119905)

by

119910119895119896

=

sgn( sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) if 0 le 119895 le 119894 0 le 119896 le V

0 if V lt 119896 119894 lt 119895

(12)

Then a double sequence 119910 isin C119877 119910 = 1 and

1003816100381610038161003816Ω119898119899119904119905 (119910)1003816100381610038161003816 =

1

120582119898120583119899

119894

sum

119895=0

V

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(13)

Hence1003816100381610038161003816Ω119898119899119904119905 (119910)

1003816100381610038161003816 le1003817100381710038171003817Ω119898119899119904119905

100381710038171003817100381710038171003817100381710038171199101003817100381710038171003817 =

1003817100381710038171003817Ω1198981198991199041199051003817100381710038171003817 (14)

Therefore

1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003817100381710038171003817Ω119898119899119904119905

1003817100381710038171003817 (15)

so that condition (CR1) follows

The sequences 119865(119887119888) = (119891(119887119888)

119895119896) 119865(119887) = (119891

(119887)

119895119896) 119866(119888) = (119892

(119888)

119895119896)

and 119866 = (119892119895119896) are defined by

119891(119902119903)

119895119896=

1 if (119895 119896) = (119887 119888)

0 otherwise

119891(119887)

119895119896=

1 if 119895 = 119887

0 otherwise

119892(119888)

119895119896=

1 if 119896 = 119888

0 otherwise

119892119895119896

= 1 forall119895 119896

(16)

Since 119865(119895119896) 119865(119895) 119866(119896) 119866 isin C119877 the 119875-limit of Ω

119898119899119904119905(119865(119895119896)

)Ω119898119899119904119905

(119865(119895)

) Ω119898119899119904119905

(119866(119896)

) and Ω119898119899119904119905

(119866) must exist uni-formly in 119904 119905 Hence the conditions (CR

2)ndash(CR

5) must hold

respectively

Sufficiency Suppose that the conditions (CR1)ndash(CR

5) hold

and a double sequence 119909 = (119909119895119896) isin C119877 Fix 119904 119905 isin Z Then

Ω119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

1003816100381610038161003816Ω119898119899119904119905 (119909)1003816100381610038161003816 le

1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817119909119895119896

10038171003817100381710038171003817

(17)

Therefore by (CR1) we have |Ω

119898119899119904119905(119909)| le 119862

119904119905119909 where

119862119904119905

is a constant independent of 119898 119899 Hence Ω119898119899119904119905

isin C1015840119877

and the sequence (Ω119898119899119904119905

) is bounded for each 119904 119905 isin Z+It follow from the conditions (CR

2) (CR

3) (CR

4) and (CR

5)

that the 119875-limit of Ω119898119899119904119905

(119865(119895119896)

) Ω119898119899119904119905

(119865(119895)

) Ω119898119899119904119905

(119866(119896)

)andΩ

119898119899119904119905(119866) exist for all 119895 119896 119904 and 119905 Since119866 119865(119895)119866(119896) and

119865(119895119896) is a fundamental set inC

119877(see [24]) it follows that

lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω119904119905(119909) (18)

exists and Ω119904119905isin C1015840119877 ThereforeΩ

119904119905has the form

Ω119904119905(119909) = ℓΩ

119904119905(119866) +

infin

sum

119895=0

(ℓ119895minus ℓ)Ω

119904119905(119865(119895)

)

+

infin

sum

119896=0

(ℎ119896minus ℓ)Ω

119904119905(119866(119896)

)

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896+ ℓ)Ω

119904119905(119865(119895119896)

)

(19)

But Ω119904119905(119865(119895119896)

) = 119906119895119896 Ω119904119905(119865(119895)

) = 1199061198950 Ω119904119905(119866(119896)

) = 1199060119896

andΩ119904119905(119866) = 119906 by the conditions (CR

2)ndash(CR

5) respectively

Hence

lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω (119909) (20)

exists for each 119909 isin C119877and 119904 119905 = 0 1 2 with

Ω (119909) = ℓ119906 +

infin

sum

119895=0

(ℓ119895minus ℓ) 119906

1198950+

infin

sum

119896=0

(ℎ119896minus ℓ) 119906

0119896

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ) 119906

119895119896

(21)

SinceΩ119898119899119904119905

(119909) isin C1015840119877for each119898 119899 119904 and 119905 it has the form

Ω119898119899119904119905

(119909) = ℓΩ119898119899119904119905

(119866) +

infin

sum

119895=0

(ℓ119895minus ℓ)Ω

119898119899119904119905(119865(119895)

)

+

infin

sum

119896=0

(ℎ119896minus ℓ)Ω

119898119899119904119905(119866(119896)

)

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ)Ω

119898119899119904119905(119865(119895119896)

)

(22)

It is easy to see from (21) and (22) that the convergence of(Ω119898119899119904119905

(119909)) to Ω(119909) is uniform in 119904 119905 since Ω119898119899119904119905

(119866) rarr

119906 Ω119898119899119904119905

(119865(119895)

) rarr 1199061198950 Ω119898119899119904119905

(119866(119896)

) rarr 1199060119896 and

Ω119898119899119904119905

(119865(119895119896)

) rarr 119906119895119896(119898 119899 rarr infin) uniformly in 119904 119905

Therefore 119861 is regularly (Λ)almost conservative

Let us recall the following lemma which is proved byMursaleen and Mohiuddine [25]

4 Abstract and Applied Analysis

Lemma 7 Let 119860(119904 119905) = (119886119898119899119895119896

(119904 119905)) 119904 119905 = 0 1 2 be asequence of infinite matrices such that

(i) 119860(119904 119905) lt 119867 lt +infin for all 119904 119905 and

(ii) for each 119895 119896 lim119898119899

119886119898119899119895119896

(119904 119905) = 0 uniformly in 119904 119905

Then

lim119898119899

sum

119895

sum

119896

119886119898119899119895119896

(119904 119905) 119909119895119896

= 0 uniformly in 119904 119905 for each 119909 isin Linfin

(23)

if and only if

lim119898119899

sum

119895

sum

119896

10038161003816100381610038161003816119886119898119899119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (24)

Theorem8 Amatrix119861 = (119887119901119902119895119896

) is (Λ)almost coercive if andonly if

(AC1) 119861 = sup

119901119902sum119895119896|119887119901119902119895119896

| lt infin

(AC2) lim119898119899rarrinfin

120572(119898 119899 119904 119905 119895 119896) = 119906119895119896 for each 119895 119896 (uni-

formly in 119904 119905)

(AC3) lim119898119899rarrinfin

suminfin

119895=1suminfin

119896=1(1120582119898120583119899)| sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

minus119906119895119896| = 0 uniformly in 119904 119905

In this case the FΛ-limit of 119861119909 is suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

for every(119909119895119896) isin L

infin

Proof Sufficiency Assume that conditions (AC1)ndash(AC

3)

hold For any positive integers 119869 119870

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816119906119895119896

10038161003816100381610038161003816=

119869

sum

119895=1

119870

sum

119896=1

lim119898119899rarrinfin

1

120582119898120583119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

= lim119898119899rarrinfin

1

120582119898120583119899

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le lim sup119898119899

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816119887119901+119904119902+119905119895119896

10038161003816100381610038161003816

le 119861

(25)

This shows that suminfin

119895=1suminfin

119896=1|119906119895119896| converges and that

suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

is defined for every double sequence119909 = (119909

119895119896) isin L

infin

Let (119909119895119896) be any arbitrary bounded double sequence For

every positive integers119898 11989910038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

(1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896)119909119895119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

le sup119904119905

[

[

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

le 119909 sup119904119905

[

[

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

(26)

Letting 119901 119902 rarr infin and using condition (AC3) we get

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

997888rarr

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896 (27)

Hence 119861119909 isin FΛwithF

Λ- lim119861119909 = sum

infin

119895=1suminfin

119896=1119906119895119896119909119895119896

Necessity Let 119861 be (Λ)almost coercive matrix This impliesthat a four-dimensional matrix 119861 is (Λ)almost conservativethen we have conditions (AC

1) and (AC

2) from Theorem 6

Now we have to show that (AC3) holds

Suppose that for some 119904 119905 we have

lim sup119898119899

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119873 gt 0

(28)

Since 119861 is finite therefore119873 is also finite We observe thatsince suminfin

119895=1suminfin

119896=1|119906119895119896| lt +infin and 119861 is (Λ)almost coercive

the matrix 119860 = (119886119901119902119895119896

) where 119886119901119902119895119896

= 119887119901119902119895119896

minus 119906119895119896

isalso (Λ)almost coercive matrix By an argument similar tothat ofTheorem 21 in [26] for single sequences one can find119909 isin L

infinfor which 119860119909 notin F

Λ This contradiction implies the

necessity of (AC3)

Now we use Lemma 7 to show that this convergence isuniform in 119904 119905 Let

ℎ119898119899119895119896

(119904 119905) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896] (29)

and let119867(119904 119905) be thematrix (ℎ119898119899119895119896

(119904 119905)) It is easy to see that119867(119904 119905) le 2119861 for every 119904 119905 and from condition (AC

2)

lim119898119899

ℎ119898119899119895119896

(119904 119905) = 0 for each 119895 119896 uniformly in 119904 119905 (30)

For any 119909 isin Linfin

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

= FΛ- lim119861119909 minus

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(31)

Abstract and Applied Analysis 5

and the limit exists uniformly in 119904 119905 since 119861119909 isin FΛ

Moreover this limit is zero since10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le119909sum

infin

119895=1suminfin

119896=1

10038161003816100381610038161003816sum119901isin119869119898

sum119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896]10038161003816100381610038161003816

120582119898120583119899

(32)

Hence

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816ℎ119898119898119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (33)

This shows that matrix 119861 = (119887119901119902119895119896

) satisfies condition (AC3)

In the following theorem we characterize the four-dimensional matrices of type (L

1FΛ) where

L1=

119909 = (119909119895119896)

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt infin

(34)

the space of all absolutely convergent double series

Theorem 9 A matrix 119861 isin (L1FΛ) if and only if it satisfies

the following conditions

(i) sup119898119899119904119905119895119896

|(1120582119898120583119899) sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

| lt infin

and the condition (CR2) of Theorem 6 holds

Proof Sufficiency Suppose that conditions (i) and (CR2)

hold For any double sequence 119909 = (119909119895119896) isin L

1 we see that

lim119898119899rarrinfin

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

=

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(35)

uniformly in 119904 119905 and it also converges absolutely Further-more

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (36)

converges absolutely for each119898 119899 119904 and 119905 Given 120598 gt 0 thereexist 119895

∘= 119895∘(120598) and 119896

∘= 119896∘(120598) such that

sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt 120598 (37)

By the condition (CR2) we can find119898

∘ 119899∘isin N such that

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin (38)

for all 119898 gt 119898∘and 119899 gt 119899

∘ uniformly in 119904 119905 Now by using

the conditions (37) (38) and (CR2) we get

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816

(39)

for all119898 gt 119898∘ 119899 gt 119899

∘and uniformly in 119904 119905 Hence (37) holds

Necessity Suppose that 119861 isin (L1FΛ) The condition (CR

2)

follows from the fact that 119864 isin L1 where 119864 = (119890

(119895119896)

) with119890(119895119896)

= 1 for all 119895 119896 To verify the condition (i) we define acontinuous linear functional 119871

119898119899119904119905(119909) onL

1by

119871119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (40)

Now

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(41)

and hence

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(42)

For any fixed 119895 119896 isin N we define a double sequence 119909 = (119909119894ℓ)

by

119909119894ℓ

=

sgn( 1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) for (119894 ℓ) = (119895 119896)

0 for (119894 ℓ) = (119895 119896)

(43)

Then 1199091= 1 and

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(44)

so that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 ge sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

6 Abstract and Applied Analysis

It follows from (42) and (45) that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 = sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(46)

Since 119861 isin (L1FΛ) we have

sup119898119899119904119905

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816

= sup119898119899119904119905

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(47)

Hence by the uniform boundedness principle we obtain

sup119898119899119904119905

1003817100381710038171003817119871119898119899119904119905 (119909)1003817100381710038171003817 = sup119898119899119904119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(48)

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no(130-100-D1434) The authors therefore acknowledge withthanks DSR technical and financial support

References

[1] S BanachTheorie des Operations Lineaires 1932[2] G G Lorentz ldquoA contribution to the theory of divergent

sequencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948[3] S A Mohiuddine ldquoAn application of almost convergence in

approximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[4] F Moricz and B E Rhoades ldquoAlmost convergence of doublesequences and strong regularity of summability matricesrdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 104 no 2 pp 283ndash294 1988

[5] M Mursaleen and S A Mohiuddine ldquoBanach limit and somenew spaces of double sequencesrdquo Turkish Journal of Mathemat-ics vol 36 no 1 pp 121ndash130 2012

[6] M Basarir ldquoOn the strong almost convergence of doublesequencesrdquo PeriodicaMathematica Hungarica vol 30 no 3 pp177ndash181 1995

[7] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[8] F Cunjalo ldquoAlmost convergence of double subsequencesrdquoFilomat vol 22 no 2 pp 87ndash93 2008

[9] K Kayaduman and C Cakan ldquoThe cesaro core of doublesequencesrdquo Abstract and Applied Analysis vol 2011 Article ID950364 9 pages 2011

[10] Mursaleen ldquoAlmost strongly regular matrices and a core the-orem for double sequencesrdquo Journal of Mathematical Analysisand Applications vol 293 no 2 pp 523ndash531 2004

[11] Mursaleen and O H H Edely ldquoAlmost convergence and acore theorem for double sequencesrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 532ndash540 2004

[12] M Zeltser M Mursaleen and S A Mohiuddine ldquoOn almostconservative matrix methods for double sequence spacesrdquoPublicationes Mathematicae Debrecen vol 75 no 3-4 pp 387ndash399 2009

[13] A Pringsheim ldquoZur theorie der zweifach unendlichen zahlen-folgenrdquo Mathematische Annalen vol 53 no 3 pp 289ndash3211900

[14] B Altay and F Basar ldquoSome new spaces of double sequencesrdquoJournal of Mathematical Analysis and Applications vol 309 no1 pp 70ndash90 2005

[15] H J Hamilton ldquoTransformations of multiple sequencesrdquo DukeMathematical Journal vol 2 no 1 pp 29ndash60 1936

[16] S A Mohiuddine and A Alotaibi ldquoSome spaces of doublesequences obtained through invariant mean and related con-ceptsrdquo Abstract and Applied Analysis vol 2013 Article ID507950 11 pages 2013

[17] Mursaleen and S A Mohiuddine ldquoDouble 120590-multiplicativematricesrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 991ndash996 2007

[18] M Mursaleen and S A Mohiuddine ldquoOn 120590-conservative andboundedly 120590-conservative four-dimensional matricesrdquo Com-puters amp Mathematics with Applications vol 59 no 2 pp 880ndash885 2010

[19] P Schaefer ldquoInfinite matrices and invariant meansrdquo Proceedingsof the AmericanMathematical Society vol 36 pp 104ndash110 1972

[20] GM Robison ldquoDivergent double sequences and seriesrdquo Trans-actions of the American Mathematical Society vol 28 no 1 pp50ndash73 1926

[21] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966

[22] P Schaefer ldquoMatrix transformations of almost convergentsequencesrdquo Mathematische Zeitschrift vol 112 pp 321ndash3251969

[23] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991

[24] F Moricz ldquoExtensions of the spaces 119888 and 1198880from single to

double sequencesrdquoActaMathematicaHungarica vol 57 no 1-2pp 129ndash136 1991

[25] MMursaleen and S AMohiuddine ldquoRegularly 120590-conservativeand 120590-coercive four dimensional matricesrdquoComputers ampMath-ematics with Applications vol 56 no 6 pp 1580ndash1586 2008

[26] C Eizen and G Laush ldquoInfinite matrices and almost conver-gencerdquoMathematica Japonica vol 14 pp 137ndash143 1969

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 Abstract and Applied Analysis

Then we say that a four-dimensionalmatrix119861maps the space1198782into the space 119878

1if 1198782sub 119878119861]1

and is denoted by (1198782 1198781)

Moricz and Rhoades [4] extended the notion of almostconvergence from single to double sequence and character-ized some matrix classes involving this concept A doublesequence 119909 = (119909

119895119896) of real numbers is said to be almost

convergent to a number 119871 if

lim119901119902rarrinfin

sup119898119899gt0

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119901119902

119898+119901minus1

sum

119895=119898

119899+119902minus1

sum

119896=119899

119909119895119896minus 119871

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (3)

For more details on double sequences and 4-dimensionalmatrices one can refer to [14ndash20]

Using the notion of almost convergence for singlesequence King [21] introduced a slightly more general classof matrices than the conservative and regular matrices thatis almost conservative and almost regular matrices andpresented its characterization In [22] Schaefer presentedsome interesting characterization for almost convergenceThe Steinhaus-type theorem for the concepts of almostregular and almost coercivematrices was proved by Basar andSolak [23] In this paper we generalize the concept of almostconvergence for double sequences with the help of doublegeneralized de la Vallee-Poussinmean and called it (Λ)almostconvergence Using this concept we define the notions ofregularly (Λ)almost conservative and (Λ)almost coercivefour-dimensional matrices and obtain their necessity andsufficient conditions Further we introduce the space L

1

of all absolutely convergent double series and characterizethe matrix class (L

1FΛ) where F

Λdenotes the space of

(Λ)almost convergence for double sequences

2 Main Results

Definition 1 Let 120582 = (120582119898 119898 = 0 1 2 ) and 120583 = (120583

119899 119899 =

0 1 2 ) be two nondecreasing sequences of positive realswith each tending to infin such that 120582

119898+1le 120582119898+ 1 120582

1= 0

120583119899+1

le 120583119899+ 1 120583

1= 0 and

I119898119899

(119909) =1

120582119898120583119899

sum

119895isin119869119898

sum

119896isin119868119899

119909119895119896 (4)

is called the double generalized de la Vallee-Poussin meanwhere 119869

119898= [119898minus120582

119898+1119898] and 119868

119899= [119899minus120583

119899+1 119899] We denote

the set of all 120582 and 120583 type sequences by using the symbol Λ

Definition 2 A double sequence 119909 = (119909119895119896) of reals is said

to be (Λ)almost convergent (briefly FΛ-convergent) to some

number 119871 if 119909 isin F120582120583

where

FΛ= 119909 = (119909

119895119896) 119875- lim119898119899rarrinfin

Ω119898119899119904119905

(119909)

= 119871 exists uniformly in 119904 119905 119871 = FΛ- lim119909

Ω119898119899119904119905

(119909) =1

120582119898120583119899

sum

119895isin119869119898

sum

119896isin119868119899

119909119895+119904119896+119905

(5)

denoted by FΛ the space of all (Λ)almost convergent

sequences (119909119895119896) Note thatC

119861119875sub FΛsub Linfin

Remark 3 If we take 120582119898= 119898 and 120583

119899= 119899 then the notion of

(Λ)almost convergence reduced to almost convergence dueto Moricz and Rhoades [4]

Definition 4 A four-dimensional matrix 119861 = (119887119901119902119895119896

) issaid to be regularly (Λ)almost conservative if it maps every119877-convergent double sequence into F

Λ-convergent double

sequence that is 119861 isin (C119877FΛ) In addition ifF

Λ- lim119860119909 =

119877- lim119909 then 119861 is regularly (Λ)almost regular

Definition 5 A matrix 119861 = (119887119901119902119895119896

) is said to be (Λ)almostcoercive if it maps every 119861119875-convergent double sequence(119909119895119896) intoF

Λ-convergent double sequence briefly a matrix

119861 in (C119861119875FΛ)

Theorem 6 A matrix 119861 = (119887119901119902119895119896

) is regularly (Λ)almostconservative if and only if

(CR1) 119861 = sup

119901119902sum119895119896|119887119901119902119895119896

| lt infin

(CR2) lim119898119899rarrinfin

120572(119898 119899 119904 119905 119895 119896) = 119906119895119896 for each 119895 119896 (uni-

formly in 119904 119905)(CR3) lim119898119899rarrinfin

sum119896|120572(119898 119899 119904 119905 119895 119896)| = 119906

1198950 for each 119895

(uniformly in 119904 119905)(CR4) lim119898119899rarrinfin

sum119895|120572(119898 119899 119904 119905 119895 119896)| = 119906

0119896 for each 119896

(uniformly in 119904 119905)(CR5) lim119898119899rarrinfin

sum119895119896120572(119898 119899 119904 119905 119895 119896) = 119906 (uniformly in 119904

119905)

where

120573 (119898 119899 119904 119905 119895 119896) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

(6)

In this case theFΛ-limit of 119861119909 is

ℓ119906 +

infin

sum

119895=0

(ℓ119895minus ℓ) 119906

1198950+

infin

sum

119896=0

(ℎ119896minus ℓ) 119906

0119896

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ) 119906

119895119896

(7)

where ℓ = 119877- lim119909

Proof Necessity Suppose that 119861 is regularly (Λ)almost con-servative matrix Fix 119904 119905 isin Z the set of integers Let

Ω119898119899119904119905

(119909) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119898

120588119901+119904119902+119905

(119909) (8)

where

120588119901119902

(119909) =

infin

sum

119895=0

infin

sum

119896=0

119887119901119902119895119896

119909119895119896 (9)

It is clear that

120588119901119902

isin C1015840

119877 119901 119902 = 0 1 2 (10)

Abstract and Applied Analysis 3

Hence Ω119898119899119904119905

isin C1015840119877for 119898 119899 isin N Since 119861 is regularly

(Λ)almost conservative we have

119875- lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω (119909) (say) (11)

uniformly in 119904 119905 It follows that (Ω119898119899119904119905

(119909)) is bounded for119909 isin C

119877and fixed 119904 119905 Hence Ω

119898119899119904119905(119909) is bounded by the

uniform boundedness principleFor each 119894 V isin Z+ define the sequence 119910 = 119910(119898 119899 119904 119905)

by

119910119895119896

=

sgn( sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) if 0 le 119895 le 119894 0 le 119896 le V

0 if V lt 119896 119894 lt 119895

(12)

Then a double sequence 119910 isin C119877 119910 = 1 and

1003816100381610038161003816Ω119898119899119904119905 (119910)1003816100381610038161003816 =

1

120582119898120583119899

119894

sum

119895=0

V

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(13)

Hence1003816100381610038161003816Ω119898119899119904119905 (119910)

1003816100381610038161003816 le1003817100381710038171003817Ω119898119899119904119905

100381710038171003817100381710038171003817100381710038171199101003817100381710038171003817 =

1003817100381710038171003817Ω1198981198991199041199051003817100381710038171003817 (14)

Therefore

1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003817100381710038171003817Ω119898119899119904119905

1003817100381710038171003817 (15)

so that condition (CR1) follows

The sequences 119865(119887119888) = (119891(119887119888)

119895119896) 119865(119887) = (119891

(119887)

119895119896) 119866(119888) = (119892

(119888)

119895119896)

and 119866 = (119892119895119896) are defined by

119891(119902119903)

119895119896=

1 if (119895 119896) = (119887 119888)

0 otherwise

119891(119887)

119895119896=

1 if 119895 = 119887

0 otherwise

119892(119888)

119895119896=

1 if 119896 = 119888

0 otherwise

119892119895119896

= 1 forall119895 119896

(16)

Since 119865(119895119896) 119865(119895) 119866(119896) 119866 isin C119877 the 119875-limit of Ω

119898119899119904119905(119865(119895119896)

)Ω119898119899119904119905

(119865(119895)

) Ω119898119899119904119905

(119866(119896)

) and Ω119898119899119904119905

(119866) must exist uni-formly in 119904 119905 Hence the conditions (CR

2)ndash(CR

5) must hold

respectively

Sufficiency Suppose that the conditions (CR1)ndash(CR

5) hold

and a double sequence 119909 = (119909119895119896) isin C119877 Fix 119904 119905 isin Z Then

Ω119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

1003816100381610038161003816Ω119898119899119904119905 (119909)1003816100381610038161003816 le

1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817119909119895119896

10038171003817100381710038171003817

(17)

Therefore by (CR1) we have |Ω

119898119899119904119905(119909)| le 119862

119904119905119909 where

119862119904119905

is a constant independent of 119898 119899 Hence Ω119898119899119904119905

isin C1015840119877

and the sequence (Ω119898119899119904119905

) is bounded for each 119904 119905 isin Z+It follow from the conditions (CR

2) (CR

3) (CR

4) and (CR

5)

that the 119875-limit of Ω119898119899119904119905

(119865(119895119896)

) Ω119898119899119904119905

(119865(119895)

) Ω119898119899119904119905

(119866(119896)

)andΩ

119898119899119904119905(119866) exist for all 119895 119896 119904 and 119905 Since119866 119865(119895)119866(119896) and

119865(119895119896) is a fundamental set inC

119877(see [24]) it follows that

lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω119904119905(119909) (18)

exists and Ω119904119905isin C1015840119877 ThereforeΩ

119904119905has the form

Ω119904119905(119909) = ℓΩ

119904119905(119866) +

infin

sum

119895=0

(ℓ119895minus ℓ)Ω

119904119905(119865(119895)

)

+

infin

sum

119896=0

(ℎ119896minus ℓ)Ω

119904119905(119866(119896)

)

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896+ ℓ)Ω

119904119905(119865(119895119896)

)

(19)

But Ω119904119905(119865(119895119896)

) = 119906119895119896 Ω119904119905(119865(119895)

) = 1199061198950 Ω119904119905(119866(119896)

) = 1199060119896

andΩ119904119905(119866) = 119906 by the conditions (CR

2)ndash(CR

5) respectively

Hence

lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω (119909) (20)

exists for each 119909 isin C119877and 119904 119905 = 0 1 2 with

Ω (119909) = ℓ119906 +

infin

sum

119895=0

(ℓ119895minus ℓ) 119906

1198950+

infin

sum

119896=0

(ℎ119896minus ℓ) 119906

0119896

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ) 119906

119895119896

(21)

SinceΩ119898119899119904119905

(119909) isin C1015840119877for each119898 119899 119904 and 119905 it has the form

Ω119898119899119904119905

(119909) = ℓΩ119898119899119904119905

(119866) +

infin

sum

119895=0

(ℓ119895minus ℓ)Ω

119898119899119904119905(119865(119895)

)

+

infin

sum

119896=0

(ℎ119896minus ℓ)Ω

119898119899119904119905(119866(119896)

)

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ)Ω

119898119899119904119905(119865(119895119896)

)

(22)

It is easy to see from (21) and (22) that the convergence of(Ω119898119899119904119905

(119909)) to Ω(119909) is uniform in 119904 119905 since Ω119898119899119904119905

(119866) rarr

119906 Ω119898119899119904119905

(119865(119895)

) rarr 1199061198950 Ω119898119899119904119905

(119866(119896)

) rarr 1199060119896 and

Ω119898119899119904119905

(119865(119895119896)

) rarr 119906119895119896(119898 119899 rarr infin) uniformly in 119904 119905

Therefore 119861 is regularly (Λ)almost conservative

Let us recall the following lemma which is proved byMursaleen and Mohiuddine [25]

4 Abstract and Applied Analysis

Lemma 7 Let 119860(119904 119905) = (119886119898119899119895119896

(119904 119905)) 119904 119905 = 0 1 2 be asequence of infinite matrices such that

(i) 119860(119904 119905) lt 119867 lt +infin for all 119904 119905 and

(ii) for each 119895 119896 lim119898119899

119886119898119899119895119896

(119904 119905) = 0 uniformly in 119904 119905

Then

lim119898119899

sum

119895

sum

119896

119886119898119899119895119896

(119904 119905) 119909119895119896

= 0 uniformly in 119904 119905 for each 119909 isin Linfin

(23)

if and only if

lim119898119899

sum

119895

sum

119896

10038161003816100381610038161003816119886119898119899119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (24)

Theorem8 Amatrix119861 = (119887119901119902119895119896

) is (Λ)almost coercive if andonly if

(AC1) 119861 = sup

119901119902sum119895119896|119887119901119902119895119896

| lt infin

(AC2) lim119898119899rarrinfin

120572(119898 119899 119904 119905 119895 119896) = 119906119895119896 for each 119895 119896 (uni-

formly in 119904 119905)

(AC3) lim119898119899rarrinfin

suminfin

119895=1suminfin

119896=1(1120582119898120583119899)| sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

minus119906119895119896| = 0 uniformly in 119904 119905

In this case the FΛ-limit of 119861119909 is suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

for every(119909119895119896) isin L

infin

Proof Sufficiency Assume that conditions (AC1)ndash(AC

3)

hold For any positive integers 119869 119870

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816119906119895119896

10038161003816100381610038161003816=

119869

sum

119895=1

119870

sum

119896=1

lim119898119899rarrinfin

1

120582119898120583119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

= lim119898119899rarrinfin

1

120582119898120583119899

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le lim sup119898119899

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816119887119901+119904119902+119905119895119896

10038161003816100381610038161003816

le 119861

(25)

This shows that suminfin

119895=1suminfin

119896=1|119906119895119896| converges and that

suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

is defined for every double sequence119909 = (119909

119895119896) isin L

infin

Let (119909119895119896) be any arbitrary bounded double sequence For

every positive integers119898 11989910038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

(1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896)119909119895119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

le sup119904119905

[

[

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

le 119909 sup119904119905

[

[

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

(26)

Letting 119901 119902 rarr infin and using condition (AC3) we get

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

997888rarr

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896 (27)

Hence 119861119909 isin FΛwithF

Λ- lim119861119909 = sum

infin

119895=1suminfin

119896=1119906119895119896119909119895119896

Necessity Let 119861 be (Λ)almost coercive matrix This impliesthat a four-dimensional matrix 119861 is (Λ)almost conservativethen we have conditions (AC

1) and (AC

2) from Theorem 6

Now we have to show that (AC3) holds

Suppose that for some 119904 119905 we have

lim sup119898119899

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119873 gt 0

(28)

Since 119861 is finite therefore119873 is also finite We observe thatsince suminfin

119895=1suminfin

119896=1|119906119895119896| lt +infin and 119861 is (Λ)almost coercive

the matrix 119860 = (119886119901119902119895119896

) where 119886119901119902119895119896

= 119887119901119902119895119896

minus 119906119895119896

isalso (Λ)almost coercive matrix By an argument similar tothat ofTheorem 21 in [26] for single sequences one can find119909 isin L

infinfor which 119860119909 notin F

Λ This contradiction implies the

necessity of (AC3)

Now we use Lemma 7 to show that this convergence isuniform in 119904 119905 Let

ℎ119898119899119895119896

(119904 119905) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896] (29)

and let119867(119904 119905) be thematrix (ℎ119898119899119895119896

(119904 119905)) It is easy to see that119867(119904 119905) le 2119861 for every 119904 119905 and from condition (AC

2)

lim119898119899

ℎ119898119899119895119896

(119904 119905) = 0 for each 119895 119896 uniformly in 119904 119905 (30)

For any 119909 isin Linfin

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

= FΛ- lim119861119909 minus

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(31)

Abstract and Applied Analysis 5

and the limit exists uniformly in 119904 119905 since 119861119909 isin FΛ

Moreover this limit is zero since10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le119909sum

infin

119895=1suminfin

119896=1

10038161003816100381610038161003816sum119901isin119869119898

sum119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896]10038161003816100381610038161003816

120582119898120583119899

(32)

Hence

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816ℎ119898119898119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (33)

This shows that matrix 119861 = (119887119901119902119895119896

) satisfies condition (AC3)

In the following theorem we characterize the four-dimensional matrices of type (L

1FΛ) where

L1=

119909 = (119909119895119896)

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt infin

(34)

the space of all absolutely convergent double series

Theorem 9 A matrix 119861 isin (L1FΛ) if and only if it satisfies

the following conditions

(i) sup119898119899119904119905119895119896

|(1120582119898120583119899) sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

| lt infin

and the condition (CR2) of Theorem 6 holds

Proof Sufficiency Suppose that conditions (i) and (CR2)

hold For any double sequence 119909 = (119909119895119896) isin L

1 we see that

lim119898119899rarrinfin

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

=

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(35)

uniformly in 119904 119905 and it also converges absolutely Further-more

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (36)

converges absolutely for each119898 119899 119904 and 119905 Given 120598 gt 0 thereexist 119895

∘= 119895∘(120598) and 119896

∘= 119896∘(120598) such that

sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt 120598 (37)

By the condition (CR2) we can find119898

∘ 119899∘isin N such that

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin (38)

for all 119898 gt 119898∘and 119899 gt 119899

∘ uniformly in 119904 119905 Now by using

the conditions (37) (38) and (CR2) we get

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816

(39)

for all119898 gt 119898∘ 119899 gt 119899

∘and uniformly in 119904 119905 Hence (37) holds

Necessity Suppose that 119861 isin (L1FΛ) The condition (CR

2)

follows from the fact that 119864 isin L1 where 119864 = (119890

(119895119896)

) with119890(119895119896)

= 1 for all 119895 119896 To verify the condition (i) we define acontinuous linear functional 119871

119898119899119904119905(119909) onL

1by

119871119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (40)

Now

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(41)

and hence

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(42)

For any fixed 119895 119896 isin N we define a double sequence 119909 = (119909119894ℓ)

by

119909119894ℓ

=

sgn( 1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) for (119894 ℓ) = (119895 119896)

0 for (119894 ℓ) = (119895 119896)

(43)

Then 1199091= 1 and

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(44)

so that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 ge sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

6 Abstract and Applied Analysis

It follows from (42) and (45) that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 = sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(46)

Since 119861 isin (L1FΛ) we have

sup119898119899119904119905

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816

= sup119898119899119904119905

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(47)

Hence by the uniform boundedness principle we obtain

sup119898119899119904119905

1003817100381710038171003817119871119898119899119904119905 (119909)1003817100381710038171003817 = sup119898119899119904119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(48)

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no(130-100-D1434) The authors therefore acknowledge withthanks DSR technical and financial support

References

[1] S BanachTheorie des Operations Lineaires 1932[2] G G Lorentz ldquoA contribution to the theory of divergent

sequencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948[3] S A Mohiuddine ldquoAn application of almost convergence in

approximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[4] F Moricz and B E Rhoades ldquoAlmost convergence of doublesequences and strong regularity of summability matricesrdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 104 no 2 pp 283ndash294 1988

[5] M Mursaleen and S A Mohiuddine ldquoBanach limit and somenew spaces of double sequencesrdquo Turkish Journal of Mathemat-ics vol 36 no 1 pp 121ndash130 2012

[6] M Basarir ldquoOn the strong almost convergence of doublesequencesrdquo PeriodicaMathematica Hungarica vol 30 no 3 pp177ndash181 1995

[7] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[8] F Cunjalo ldquoAlmost convergence of double subsequencesrdquoFilomat vol 22 no 2 pp 87ndash93 2008

[9] K Kayaduman and C Cakan ldquoThe cesaro core of doublesequencesrdquo Abstract and Applied Analysis vol 2011 Article ID950364 9 pages 2011

[10] Mursaleen ldquoAlmost strongly regular matrices and a core the-orem for double sequencesrdquo Journal of Mathematical Analysisand Applications vol 293 no 2 pp 523ndash531 2004

[11] Mursaleen and O H H Edely ldquoAlmost convergence and acore theorem for double sequencesrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 532ndash540 2004

[12] M Zeltser M Mursaleen and S A Mohiuddine ldquoOn almostconservative matrix methods for double sequence spacesrdquoPublicationes Mathematicae Debrecen vol 75 no 3-4 pp 387ndash399 2009

[13] A Pringsheim ldquoZur theorie der zweifach unendlichen zahlen-folgenrdquo Mathematische Annalen vol 53 no 3 pp 289ndash3211900

[14] B Altay and F Basar ldquoSome new spaces of double sequencesrdquoJournal of Mathematical Analysis and Applications vol 309 no1 pp 70ndash90 2005

[15] H J Hamilton ldquoTransformations of multiple sequencesrdquo DukeMathematical Journal vol 2 no 1 pp 29ndash60 1936

[16] S A Mohiuddine and A Alotaibi ldquoSome spaces of doublesequences obtained through invariant mean and related con-ceptsrdquo Abstract and Applied Analysis vol 2013 Article ID507950 11 pages 2013

[17] Mursaleen and S A Mohiuddine ldquoDouble 120590-multiplicativematricesrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 991ndash996 2007

[18] M Mursaleen and S A Mohiuddine ldquoOn 120590-conservative andboundedly 120590-conservative four-dimensional matricesrdquo Com-puters amp Mathematics with Applications vol 59 no 2 pp 880ndash885 2010

[19] P Schaefer ldquoInfinite matrices and invariant meansrdquo Proceedingsof the AmericanMathematical Society vol 36 pp 104ndash110 1972

[20] GM Robison ldquoDivergent double sequences and seriesrdquo Trans-actions of the American Mathematical Society vol 28 no 1 pp50ndash73 1926

[21] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966

[22] P Schaefer ldquoMatrix transformations of almost convergentsequencesrdquo Mathematische Zeitschrift vol 112 pp 321ndash3251969

[23] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991

[24] F Moricz ldquoExtensions of the spaces 119888 and 1198880from single to

double sequencesrdquoActaMathematicaHungarica vol 57 no 1-2pp 129ndash136 1991

[25] MMursaleen and S AMohiuddine ldquoRegularly 120590-conservativeand 120590-coercive four dimensional matricesrdquoComputers ampMath-ematics with Applications vol 56 no 6 pp 1580ndash1586 2008

[26] C Eizen and G Laush ldquoInfinite matrices and almost conver-gencerdquoMathematica Japonica vol 14 pp 137ndash143 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Hence Ω119898119899119904119905

isin C1015840119877for 119898 119899 isin N Since 119861 is regularly

(Λ)almost conservative we have

119875- lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω (119909) (say) (11)

uniformly in 119904 119905 It follows that (Ω119898119899119904119905

(119909)) is bounded for119909 isin C

119877and fixed 119904 119905 Hence Ω

119898119899119904119905(119909) is bounded by the

uniform boundedness principleFor each 119894 V isin Z+ define the sequence 119910 = 119910(119898 119899 119904 119905)

by

119910119895119896

=

sgn( sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) if 0 le 119895 le 119894 0 le 119896 le V

0 if V lt 119896 119894 lt 119895

(12)

Then a double sequence 119910 isin C119877 119910 = 1 and

1003816100381610038161003816Ω119898119899119904119905 (119910)1003816100381610038161003816 =

1

120582119898120583119899

119894

sum

119895=0

V

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(13)

Hence1003816100381610038161003816Ω119898119899119904119905 (119910)

1003816100381610038161003816 le1003817100381710038171003817Ω119898119899119904119905

100381710038171003817100381710038171003817100381710038171199101003817100381710038171003817 =

1003817100381710038171003817Ω1198981198991199041199051003817100381710038171003817 (14)

Therefore

1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le1003817100381710038171003817Ω119898119899119904119905

1003817100381710038171003817 (15)

so that condition (CR1) follows

The sequences 119865(119887119888) = (119891(119887119888)

119895119896) 119865(119887) = (119891

(119887)

119895119896) 119866(119888) = (119892

(119888)

119895119896)

and 119866 = (119892119895119896) are defined by

119891(119902119903)

119895119896=

1 if (119895 119896) = (119887 119888)

0 otherwise

119891(119887)

119895119896=

1 if 119895 = 119887

0 otherwise

119892(119888)

119895119896=

1 if 119896 = 119888

0 otherwise

119892119895119896

= 1 forall119895 119896

(16)

Since 119865(119895119896) 119865(119895) 119866(119896) 119866 isin C119877 the 119875-limit of Ω

119898119899119904119905(119865(119895119896)

)Ω119898119899119904119905

(119865(119895)

) Ω119898119899119904119905

(119866(119896)

) and Ω119898119899119904119905

(119866) must exist uni-formly in 119904 119905 Hence the conditions (CR

2)ndash(CR

5) must hold

respectively

Sufficiency Suppose that the conditions (CR1)ndash(CR

5) hold

and a double sequence 119909 = (119909119895119896) isin C119877 Fix 119904 119905 isin Z Then

Ω119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

1003816100381610038161003816Ω119898119899119904119905 (119909)1003816100381610038161003816 le

1

120582119898120583119899

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038171003817100381710038171003817119909119895119896

10038171003817100381710038171003817

(17)

Therefore by (CR1) we have |Ω

119898119899119904119905(119909)| le 119862

119904119905119909 where

119862119904119905

is a constant independent of 119898 119899 Hence Ω119898119899119904119905

isin C1015840119877

and the sequence (Ω119898119899119904119905

) is bounded for each 119904 119905 isin Z+It follow from the conditions (CR

2) (CR

3) (CR

4) and (CR

5)

that the 119875-limit of Ω119898119899119904119905

(119865(119895119896)

) Ω119898119899119904119905

(119865(119895)

) Ω119898119899119904119905

(119866(119896)

)andΩ

119898119899119904119905(119866) exist for all 119895 119896 119904 and 119905 Since119866 119865(119895)119866(119896) and

119865(119895119896) is a fundamental set inC

119877(see [24]) it follows that

lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω119904119905(119909) (18)

exists and Ω119904119905isin C1015840119877 ThereforeΩ

119904119905has the form

Ω119904119905(119909) = ℓΩ

119904119905(119866) +

infin

sum

119895=0

(ℓ119895minus ℓ)Ω

119904119905(119865(119895)

)

+

infin

sum

119896=0

(ℎ119896minus ℓ)Ω

119904119905(119866(119896)

)

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896+ ℓ)Ω

119904119905(119865(119895119896)

)

(19)

But Ω119904119905(119865(119895119896)

) = 119906119895119896 Ω119904119905(119865(119895)

) = 1199061198950 Ω119904119905(119866(119896)

) = 1199060119896

andΩ119904119905(119866) = 119906 by the conditions (CR

2)ndash(CR

5) respectively

Hence

lim119898119899rarrinfin

Ω119898119899119904119905

(119909) = Ω (119909) (20)

exists for each 119909 isin C119877and 119904 119905 = 0 1 2 with

Ω (119909) = ℓ119906 +

infin

sum

119895=0

(ℓ119895minus ℓ) 119906

1198950+

infin

sum

119896=0

(ℎ119896minus ℓ) 119906

0119896

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ) 119906

119895119896

(21)

SinceΩ119898119899119904119905

(119909) isin C1015840119877for each119898 119899 119904 and 119905 it has the form

Ω119898119899119904119905

(119909) = ℓΩ119898119899119904119905

(119866) +

infin

sum

119895=0

(ℓ119895minus ℓ)Ω

119898119899119904119905(119865(119895)

)

+

infin

sum

119896=0

(ℎ119896minus ℓ)Ω

119898119899119904119905(119866(119896)

)

+

infin

sum

119895=0

infin

sum

119896=0

(119909119895119896minus ℓ119895minus ℎ119896minus ℓ)Ω

119898119899119904119905(119865(119895119896)

)

(22)

It is easy to see from (21) and (22) that the convergence of(Ω119898119899119904119905

(119909)) to Ω(119909) is uniform in 119904 119905 since Ω119898119899119904119905

(119866) rarr

119906 Ω119898119899119904119905

(119865(119895)

) rarr 1199061198950 Ω119898119899119904119905

(119866(119896)

) rarr 1199060119896 and

Ω119898119899119904119905

(119865(119895119896)

) rarr 119906119895119896(119898 119899 rarr infin) uniformly in 119904 119905

Therefore 119861 is regularly (Λ)almost conservative

Let us recall the following lemma which is proved byMursaleen and Mohiuddine [25]

4 Abstract and Applied Analysis

Lemma 7 Let 119860(119904 119905) = (119886119898119899119895119896

(119904 119905)) 119904 119905 = 0 1 2 be asequence of infinite matrices such that

(i) 119860(119904 119905) lt 119867 lt +infin for all 119904 119905 and

(ii) for each 119895 119896 lim119898119899

119886119898119899119895119896

(119904 119905) = 0 uniformly in 119904 119905

Then

lim119898119899

sum

119895

sum

119896

119886119898119899119895119896

(119904 119905) 119909119895119896

= 0 uniformly in 119904 119905 for each 119909 isin Linfin

(23)

if and only if

lim119898119899

sum

119895

sum

119896

10038161003816100381610038161003816119886119898119899119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (24)

Theorem8 Amatrix119861 = (119887119901119902119895119896

) is (Λ)almost coercive if andonly if

(AC1) 119861 = sup

119901119902sum119895119896|119887119901119902119895119896

| lt infin

(AC2) lim119898119899rarrinfin

120572(119898 119899 119904 119905 119895 119896) = 119906119895119896 for each 119895 119896 (uni-

formly in 119904 119905)

(AC3) lim119898119899rarrinfin

suminfin

119895=1suminfin

119896=1(1120582119898120583119899)| sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

minus119906119895119896| = 0 uniformly in 119904 119905

In this case the FΛ-limit of 119861119909 is suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

for every(119909119895119896) isin L

infin

Proof Sufficiency Assume that conditions (AC1)ndash(AC

3)

hold For any positive integers 119869 119870

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816119906119895119896

10038161003816100381610038161003816=

119869

sum

119895=1

119870

sum

119896=1

lim119898119899rarrinfin

1

120582119898120583119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

= lim119898119899rarrinfin

1

120582119898120583119899

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le lim sup119898119899

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816119887119901+119904119902+119905119895119896

10038161003816100381610038161003816

le 119861

(25)

This shows that suminfin

119895=1suminfin

119896=1|119906119895119896| converges and that

suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

is defined for every double sequence119909 = (119909

119895119896) isin L

infin

Let (119909119895119896) be any arbitrary bounded double sequence For

every positive integers119898 11989910038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

(1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896)119909119895119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

le sup119904119905

[

[

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

le 119909 sup119904119905

[

[

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

(26)

Letting 119901 119902 rarr infin and using condition (AC3) we get

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

997888rarr

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896 (27)

Hence 119861119909 isin FΛwithF

Λ- lim119861119909 = sum

infin

119895=1suminfin

119896=1119906119895119896119909119895119896

Necessity Let 119861 be (Λ)almost coercive matrix This impliesthat a four-dimensional matrix 119861 is (Λ)almost conservativethen we have conditions (AC

1) and (AC

2) from Theorem 6

Now we have to show that (AC3) holds

Suppose that for some 119904 119905 we have

lim sup119898119899

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119873 gt 0

(28)

Since 119861 is finite therefore119873 is also finite We observe thatsince suminfin

119895=1suminfin

119896=1|119906119895119896| lt +infin and 119861 is (Λ)almost coercive

the matrix 119860 = (119886119901119902119895119896

) where 119886119901119902119895119896

= 119887119901119902119895119896

minus 119906119895119896

isalso (Λ)almost coercive matrix By an argument similar tothat ofTheorem 21 in [26] for single sequences one can find119909 isin L

infinfor which 119860119909 notin F

Λ This contradiction implies the

necessity of (AC3)

Now we use Lemma 7 to show that this convergence isuniform in 119904 119905 Let

ℎ119898119899119895119896

(119904 119905) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896] (29)

and let119867(119904 119905) be thematrix (ℎ119898119899119895119896

(119904 119905)) It is easy to see that119867(119904 119905) le 2119861 for every 119904 119905 and from condition (AC

2)

lim119898119899

ℎ119898119899119895119896

(119904 119905) = 0 for each 119895 119896 uniformly in 119904 119905 (30)

For any 119909 isin Linfin

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

= FΛ- lim119861119909 minus

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(31)

Abstract and Applied Analysis 5

and the limit exists uniformly in 119904 119905 since 119861119909 isin FΛ

Moreover this limit is zero since10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le119909sum

infin

119895=1suminfin

119896=1

10038161003816100381610038161003816sum119901isin119869119898

sum119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896]10038161003816100381610038161003816

120582119898120583119899

(32)

Hence

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816ℎ119898119898119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (33)

This shows that matrix 119861 = (119887119901119902119895119896

) satisfies condition (AC3)

In the following theorem we characterize the four-dimensional matrices of type (L

1FΛ) where

L1=

119909 = (119909119895119896)

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt infin

(34)

the space of all absolutely convergent double series

Theorem 9 A matrix 119861 isin (L1FΛ) if and only if it satisfies

the following conditions

(i) sup119898119899119904119905119895119896

|(1120582119898120583119899) sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

| lt infin

and the condition (CR2) of Theorem 6 holds

Proof Sufficiency Suppose that conditions (i) and (CR2)

hold For any double sequence 119909 = (119909119895119896) isin L

1 we see that

lim119898119899rarrinfin

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

=

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(35)

uniformly in 119904 119905 and it also converges absolutely Further-more

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (36)

converges absolutely for each119898 119899 119904 and 119905 Given 120598 gt 0 thereexist 119895

∘= 119895∘(120598) and 119896

∘= 119896∘(120598) such that

sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt 120598 (37)

By the condition (CR2) we can find119898

∘ 119899∘isin N such that

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin (38)

for all 119898 gt 119898∘and 119899 gt 119899

∘ uniformly in 119904 119905 Now by using

the conditions (37) (38) and (CR2) we get

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816

(39)

for all119898 gt 119898∘ 119899 gt 119899

∘and uniformly in 119904 119905 Hence (37) holds

Necessity Suppose that 119861 isin (L1FΛ) The condition (CR

2)

follows from the fact that 119864 isin L1 where 119864 = (119890

(119895119896)

) with119890(119895119896)

= 1 for all 119895 119896 To verify the condition (i) we define acontinuous linear functional 119871

119898119899119904119905(119909) onL

1by

119871119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (40)

Now

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(41)

and hence

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(42)

For any fixed 119895 119896 isin N we define a double sequence 119909 = (119909119894ℓ)

by

119909119894ℓ

=

sgn( 1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) for (119894 ℓ) = (119895 119896)

0 for (119894 ℓ) = (119895 119896)

(43)

Then 1199091= 1 and

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(44)

so that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 ge sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

6 Abstract and Applied Analysis

It follows from (42) and (45) that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 = sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(46)

Since 119861 isin (L1FΛ) we have

sup119898119899119904119905

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816

= sup119898119899119904119905

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(47)

Hence by the uniform boundedness principle we obtain

sup119898119899119904119905

1003817100381710038171003817119871119898119899119904119905 (119909)1003817100381710038171003817 = sup119898119899119904119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(48)

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no(130-100-D1434) The authors therefore acknowledge withthanks DSR technical and financial support

References

[1] S BanachTheorie des Operations Lineaires 1932[2] G G Lorentz ldquoA contribution to the theory of divergent

sequencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948[3] S A Mohiuddine ldquoAn application of almost convergence in

approximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[4] F Moricz and B E Rhoades ldquoAlmost convergence of doublesequences and strong regularity of summability matricesrdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 104 no 2 pp 283ndash294 1988

[5] M Mursaleen and S A Mohiuddine ldquoBanach limit and somenew spaces of double sequencesrdquo Turkish Journal of Mathemat-ics vol 36 no 1 pp 121ndash130 2012

[6] M Basarir ldquoOn the strong almost convergence of doublesequencesrdquo PeriodicaMathematica Hungarica vol 30 no 3 pp177ndash181 1995

[7] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[8] F Cunjalo ldquoAlmost convergence of double subsequencesrdquoFilomat vol 22 no 2 pp 87ndash93 2008

[9] K Kayaduman and C Cakan ldquoThe cesaro core of doublesequencesrdquo Abstract and Applied Analysis vol 2011 Article ID950364 9 pages 2011

[10] Mursaleen ldquoAlmost strongly regular matrices and a core the-orem for double sequencesrdquo Journal of Mathematical Analysisand Applications vol 293 no 2 pp 523ndash531 2004

[11] Mursaleen and O H H Edely ldquoAlmost convergence and acore theorem for double sequencesrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 532ndash540 2004

[12] M Zeltser M Mursaleen and S A Mohiuddine ldquoOn almostconservative matrix methods for double sequence spacesrdquoPublicationes Mathematicae Debrecen vol 75 no 3-4 pp 387ndash399 2009

[13] A Pringsheim ldquoZur theorie der zweifach unendlichen zahlen-folgenrdquo Mathematische Annalen vol 53 no 3 pp 289ndash3211900

[14] B Altay and F Basar ldquoSome new spaces of double sequencesrdquoJournal of Mathematical Analysis and Applications vol 309 no1 pp 70ndash90 2005

[15] H J Hamilton ldquoTransformations of multiple sequencesrdquo DukeMathematical Journal vol 2 no 1 pp 29ndash60 1936

[16] S A Mohiuddine and A Alotaibi ldquoSome spaces of doublesequences obtained through invariant mean and related con-ceptsrdquo Abstract and Applied Analysis vol 2013 Article ID507950 11 pages 2013

[17] Mursaleen and S A Mohiuddine ldquoDouble 120590-multiplicativematricesrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 991ndash996 2007

[18] M Mursaleen and S A Mohiuddine ldquoOn 120590-conservative andboundedly 120590-conservative four-dimensional matricesrdquo Com-puters amp Mathematics with Applications vol 59 no 2 pp 880ndash885 2010

[19] P Schaefer ldquoInfinite matrices and invariant meansrdquo Proceedingsof the AmericanMathematical Society vol 36 pp 104ndash110 1972

[20] GM Robison ldquoDivergent double sequences and seriesrdquo Trans-actions of the American Mathematical Society vol 28 no 1 pp50ndash73 1926

[21] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966

[22] P Schaefer ldquoMatrix transformations of almost convergentsequencesrdquo Mathematische Zeitschrift vol 112 pp 321ndash3251969

[23] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991

[24] F Moricz ldquoExtensions of the spaces 119888 and 1198880from single to

double sequencesrdquoActaMathematicaHungarica vol 57 no 1-2pp 129ndash136 1991

[25] MMursaleen and S AMohiuddine ldquoRegularly 120590-conservativeand 120590-coercive four dimensional matricesrdquoComputers ampMath-ematics with Applications vol 56 no 6 pp 1580ndash1586 2008

[26] C Eizen and G Laush ldquoInfinite matrices and almost conver-gencerdquoMathematica Japonica vol 14 pp 137ndash143 1969

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Lemma 7 Let 119860(119904 119905) = (119886119898119899119895119896

(119904 119905)) 119904 119905 = 0 1 2 be asequence of infinite matrices such that

(i) 119860(119904 119905) lt 119867 lt +infin for all 119904 119905 and

(ii) for each 119895 119896 lim119898119899

119886119898119899119895119896

(119904 119905) = 0 uniformly in 119904 119905

Then

lim119898119899

sum

119895

sum

119896

119886119898119899119895119896

(119904 119905) 119909119895119896

= 0 uniformly in 119904 119905 for each 119909 isin Linfin

(23)

if and only if

lim119898119899

sum

119895

sum

119896

10038161003816100381610038161003816119886119898119899119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (24)

Theorem8 Amatrix119861 = (119887119901119902119895119896

) is (Λ)almost coercive if andonly if

(AC1) 119861 = sup

119901119902sum119895119896|119887119901119902119895119896

| lt infin

(AC2) lim119898119899rarrinfin

120572(119898 119899 119904 119905 119895 119896) = 119906119895119896 for each 119895 119896 (uni-

formly in 119904 119905)

(AC3) lim119898119899rarrinfin

suminfin

119895=1suminfin

119896=1(1120582119898120583119899)| sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

minus119906119895119896| = 0 uniformly in 119904 119905

In this case the FΛ-limit of 119861119909 is suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

for every(119909119895119896) isin L

infin

Proof Sufficiency Assume that conditions (AC1)ndash(AC

3)

hold For any positive integers 119869 119870

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816119906119895119896

10038161003816100381610038161003816=

119869

sum

119895=1

119870

sum

119896=1

lim119898119899rarrinfin

1

120582119898120583119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

= lim119898119899rarrinfin

1

120582119898120583119899

119869

sum

119895=1

119870

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le lim sup119898119899

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816119887119901+119904119902+119905119895119896

10038161003816100381610038161003816

le 119861

(25)

This shows that suminfin

119895=1suminfin

119896=1|119906119895119896| converges and that

suminfin

119895=1suminfin

119896=1119906119895119896119909119895119896

is defined for every double sequence119909 = (119909

119895119896) isin L

infin

Let (119909119895119896) be any arbitrary bounded double sequence For

every positive integers119898 11989910038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

(1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896)119909119895119896

10038171003817100381710038171003817100381710038171003817100381710038171003817

=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

le sup119904119905

[

[

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

le 119909 sup119904119905

[

[

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

]

]

(26)

Letting 119901 119902 rarr infin and using condition (AC3) we get

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

997888rarr

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896 (27)

Hence 119861119909 isin FΛwithF

Λ- lim119861119909 = sum

infin

119895=1suminfin

119896=1119906119895119896119909119895119896

Necessity Let 119861 be (Λ)almost coercive matrix This impliesthat a four-dimensional matrix 119861 is (Λ)almost conservativethen we have conditions (AC

1) and (AC

2) from Theorem 6

Now we have to show that (AC3) holds

Suppose that for some 119904 119905 we have

lim sup119898119899

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901+119904119902+119905119895119896

minus 119906119895119896]

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 119873 gt 0

(28)

Since 119861 is finite therefore119873 is also finite We observe thatsince suminfin

119895=1suminfin

119896=1|119906119895119896| lt +infin and 119861 is (Λ)almost coercive

the matrix 119860 = (119886119901119902119895119896

) where 119886119901119902119895119896

= 119887119901119902119895119896

minus 119906119895119896

isalso (Λ)almost coercive matrix By an argument similar tothat ofTheorem 21 in [26] for single sequences one can find119909 isin L

infinfor which 119860119909 notin F

Λ This contradiction implies the

necessity of (AC3)

Now we use Lemma 7 to show that this convergence isuniform in 119904 119905 Let

ℎ119898119899119895119896

(119904 119905) =1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896] (29)

and let119867(119904 119905) be thematrix (ℎ119898119899119895119896

(119904 119905)) It is easy to see that119867(119904 119905) le 2119861 for every 119904 119905 and from condition (AC

2)

lim119898119899

ℎ119898119899119895119896

(119904 119905) = 0 for each 119895 119896 uniformly in 119904 119905 (30)

For any 119909 isin Linfin

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

= FΛ- lim119861119909 minus

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(31)

Abstract and Applied Analysis 5

and the limit exists uniformly in 119904 119905 since 119861119909 isin FΛ

Moreover this limit is zero since10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le119909sum

infin

119895=1suminfin

119896=1

10038161003816100381610038161003816sum119901isin119869119898

sum119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896]10038161003816100381610038161003816

120582119898120583119899

(32)

Hence

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816ℎ119898119898119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (33)

This shows that matrix 119861 = (119887119901119902119895119896

) satisfies condition (AC3)

In the following theorem we characterize the four-dimensional matrices of type (L

1FΛ) where

L1=

119909 = (119909119895119896)

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt infin

(34)

the space of all absolutely convergent double series

Theorem 9 A matrix 119861 isin (L1FΛ) if and only if it satisfies

the following conditions

(i) sup119898119899119904119905119895119896

|(1120582119898120583119899) sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

| lt infin

and the condition (CR2) of Theorem 6 holds

Proof Sufficiency Suppose that conditions (i) and (CR2)

hold For any double sequence 119909 = (119909119895119896) isin L

1 we see that

lim119898119899rarrinfin

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

=

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(35)

uniformly in 119904 119905 and it also converges absolutely Further-more

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (36)

converges absolutely for each119898 119899 119904 and 119905 Given 120598 gt 0 thereexist 119895

∘= 119895∘(120598) and 119896

∘= 119896∘(120598) such that

sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt 120598 (37)

By the condition (CR2) we can find119898

∘ 119899∘isin N such that

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin (38)

for all 119898 gt 119898∘and 119899 gt 119899

∘ uniformly in 119904 119905 Now by using

the conditions (37) (38) and (CR2) we get

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816

(39)

for all119898 gt 119898∘ 119899 gt 119899

∘and uniformly in 119904 119905 Hence (37) holds

Necessity Suppose that 119861 isin (L1FΛ) The condition (CR

2)

follows from the fact that 119864 isin L1 where 119864 = (119890

(119895119896)

) with119890(119895119896)

= 1 for all 119895 119896 To verify the condition (i) we define acontinuous linear functional 119871

119898119899119904119905(119909) onL

1by

119871119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (40)

Now

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(41)

and hence

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(42)

For any fixed 119895 119896 isin N we define a double sequence 119909 = (119909119894ℓ)

by

119909119894ℓ

=

sgn( 1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) for (119894 ℓ) = (119895 119896)

0 for (119894 ℓ) = (119895 119896)

(43)

Then 1199091= 1 and

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(44)

so that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 ge sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

6 Abstract and Applied Analysis

It follows from (42) and (45) that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 = sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(46)

Since 119861 isin (L1FΛ) we have

sup119898119899119904119905

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816

= sup119898119899119904119905

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(47)

Hence by the uniform boundedness principle we obtain

sup119898119899119904119905

1003817100381710038171003817119871119898119899119904119905 (119909)1003817100381710038171003817 = sup119898119899119904119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(48)

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no(130-100-D1434) The authors therefore acknowledge withthanks DSR technical and financial support

References

[1] S BanachTheorie des Operations Lineaires 1932[2] G G Lorentz ldquoA contribution to the theory of divergent

sequencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948[3] S A Mohiuddine ldquoAn application of almost convergence in

approximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[4] F Moricz and B E Rhoades ldquoAlmost convergence of doublesequences and strong regularity of summability matricesrdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 104 no 2 pp 283ndash294 1988

[5] M Mursaleen and S A Mohiuddine ldquoBanach limit and somenew spaces of double sequencesrdquo Turkish Journal of Mathemat-ics vol 36 no 1 pp 121ndash130 2012

[6] M Basarir ldquoOn the strong almost convergence of doublesequencesrdquo PeriodicaMathematica Hungarica vol 30 no 3 pp177ndash181 1995

[7] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[8] F Cunjalo ldquoAlmost convergence of double subsequencesrdquoFilomat vol 22 no 2 pp 87ndash93 2008

[9] K Kayaduman and C Cakan ldquoThe cesaro core of doublesequencesrdquo Abstract and Applied Analysis vol 2011 Article ID950364 9 pages 2011

[10] Mursaleen ldquoAlmost strongly regular matrices and a core the-orem for double sequencesrdquo Journal of Mathematical Analysisand Applications vol 293 no 2 pp 523ndash531 2004

[11] Mursaleen and O H H Edely ldquoAlmost convergence and acore theorem for double sequencesrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 532ndash540 2004

[12] M Zeltser M Mursaleen and S A Mohiuddine ldquoOn almostconservative matrix methods for double sequence spacesrdquoPublicationes Mathematicae Debrecen vol 75 no 3-4 pp 387ndash399 2009

[13] A Pringsheim ldquoZur theorie der zweifach unendlichen zahlen-folgenrdquo Mathematische Annalen vol 53 no 3 pp 289ndash3211900

[14] B Altay and F Basar ldquoSome new spaces of double sequencesrdquoJournal of Mathematical Analysis and Applications vol 309 no1 pp 70ndash90 2005

[15] H J Hamilton ldquoTransformations of multiple sequencesrdquo DukeMathematical Journal vol 2 no 1 pp 29ndash60 1936

[16] S A Mohiuddine and A Alotaibi ldquoSome spaces of doublesequences obtained through invariant mean and related con-ceptsrdquo Abstract and Applied Analysis vol 2013 Article ID507950 11 pages 2013

[17] Mursaleen and S A Mohiuddine ldquoDouble 120590-multiplicativematricesrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 991ndash996 2007

[18] M Mursaleen and S A Mohiuddine ldquoOn 120590-conservative andboundedly 120590-conservative four-dimensional matricesrdquo Com-puters amp Mathematics with Applications vol 59 no 2 pp 880ndash885 2010

[19] P Schaefer ldquoInfinite matrices and invariant meansrdquo Proceedingsof the AmericanMathematical Society vol 36 pp 104ndash110 1972

[20] GM Robison ldquoDivergent double sequences and seriesrdquo Trans-actions of the American Mathematical Society vol 28 no 1 pp50ndash73 1926

[21] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966

[22] P Schaefer ldquoMatrix transformations of almost convergentsequencesrdquo Mathematische Zeitschrift vol 112 pp 321ndash3251969

[23] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991

[24] F Moricz ldquoExtensions of the spaces 119888 and 1198880from single to

double sequencesrdquoActaMathematicaHungarica vol 57 no 1-2pp 129ndash136 1991

[25] MMursaleen and S AMohiuddine ldquoRegularly 120590-conservativeand 120590-coercive four dimensional matricesrdquoComputers ampMath-ematics with Applications vol 56 no 6 pp 1580ndash1586 2008

[26] C Eizen and G Laush ldquoInfinite matrices and almost conver-gencerdquoMathematica Japonica vol 14 pp 137ndash143 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

and the limit exists uniformly in 119904 119905 since 119861119909 isin FΛ

Moreover this limit is zero since10038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

ℎ119898119899119895119896

(119904 119905) 119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

le119909sum

infin

119895=1suminfin

119896=1

10038161003816100381610038161003816sum119901isin119869119898

sum119902isin119868119899

[119887119901119902119895119896

minus 119906119895119896]10038161003816100381610038161003816

120582119898120583119899

(32)

Hence

lim119898119899

infin

sum

119895=1

infin

sum

119896=1

10038161003816100381610038161003816ℎ119898119898119895119896

(119904 119905)10038161003816100381610038161003816= 0 uniformly in 119904 119905 (33)

This shows that matrix 119861 = (119887119901119902119895119896

) satisfies condition (AC3)

In the following theorem we characterize the four-dimensional matrices of type (L

1FΛ) where

L1=

119909 = (119909119895119896)

infin

sum

119895=0

infin

sum

119896=0

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt infin

(34)

the space of all absolutely convergent double series

Theorem 9 A matrix 119861 isin (L1FΛ) if and only if it satisfies

the following conditions

(i) sup119898119899119904119905119895119896

|(1120582119898120583119899) sum119901isin119869119898

sum119902isin119868119899

119887119901+119904119902+119905119895119896

| lt infin

and the condition (CR2) of Theorem 6 holds

Proof Sufficiency Suppose that conditions (i) and (CR2)

hold For any double sequence 119909 = (119909119895119896) isin L

1 we see that

lim119898119899rarrinfin

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

=

infin

sum

119895=1

infin

sum

119896=1

119906119895119896119909119895119896

(35)

uniformly in 119904 119905 and it also converges absolutely Further-more

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (36)

converges absolutely for each119898 119899 119904 and 119905 Given 120598 gt 0 thereexist 119895

∘= 119895∘(120598) and 119896

∘= 119896∘(120598) such that

sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816lt 120598 (37)

By the condition (CR2) we can find119898

∘ 119899∘isin N such that

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin (38)

for all 119898 gt 119898∘and 119899 gt 119899

∘ uniformly in 119904 119905 Now by using

the conditions (37) (38) and (CR2) we get

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

infin

sum

119895=1

infin

sum

119896=1

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895le119895∘

sum

119896le119896∘

[

[

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

]

]

119909119895119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ sum

119895gt119895∘

sum

119896gt119896∘

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

minus 119906119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119909119895119896

10038161003816100381610038161003816

(39)

for all119898 gt 119898∘ 119899 gt 119899

∘and uniformly in 119904 119905 Hence (37) holds

Necessity Suppose that 119861 isin (L1FΛ) The condition (CR

2)

follows from the fact that 119864 isin L1 where 119864 = (119890

(119895119896)

) with119890(119895119896)

= 1 for all 119895 119896 To verify the condition (i) we define acontinuous linear functional 119871

119898119899119904119905(119909) onL

1by

119871119898119899119904119905

(119909) =1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896 (40)

Now

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(41)

and hence

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 le sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(42)

For any fixed 119895 119896 isin N we define a double sequence 119909 = (119909119894ℓ)

by

119909119894ℓ

=

sgn( 1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

) for (119894 ℓ) = (119895 119896)

0 for (119894 ℓ) = (119895 119896)

(43)

Then 1199091= 1 and

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1199091

(44)

so that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 ge sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

6 Abstract and Applied Analysis

It follows from (42) and (45) that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 = sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(46)

Since 119861 isin (L1FΛ) we have

sup119898119899119904119905

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816

= sup119898119899119904119905

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(47)

Hence by the uniform boundedness principle we obtain

sup119898119899119904119905

1003817100381710038171003817119871119898119899119904119905 (119909)1003817100381710038171003817 = sup119898119899119904119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(48)

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no(130-100-D1434) The authors therefore acknowledge withthanks DSR technical and financial support

References

[1] S BanachTheorie des Operations Lineaires 1932[2] G G Lorentz ldquoA contribution to the theory of divergent

sequencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948[3] S A Mohiuddine ldquoAn application of almost convergence in

approximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[4] F Moricz and B E Rhoades ldquoAlmost convergence of doublesequences and strong regularity of summability matricesrdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 104 no 2 pp 283ndash294 1988

[5] M Mursaleen and S A Mohiuddine ldquoBanach limit and somenew spaces of double sequencesrdquo Turkish Journal of Mathemat-ics vol 36 no 1 pp 121ndash130 2012

[6] M Basarir ldquoOn the strong almost convergence of doublesequencesrdquo PeriodicaMathematica Hungarica vol 30 no 3 pp177ndash181 1995

[7] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[8] F Cunjalo ldquoAlmost convergence of double subsequencesrdquoFilomat vol 22 no 2 pp 87ndash93 2008

[9] K Kayaduman and C Cakan ldquoThe cesaro core of doublesequencesrdquo Abstract and Applied Analysis vol 2011 Article ID950364 9 pages 2011

[10] Mursaleen ldquoAlmost strongly regular matrices and a core the-orem for double sequencesrdquo Journal of Mathematical Analysisand Applications vol 293 no 2 pp 523ndash531 2004

[11] Mursaleen and O H H Edely ldquoAlmost convergence and acore theorem for double sequencesrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 532ndash540 2004

[12] M Zeltser M Mursaleen and S A Mohiuddine ldquoOn almostconservative matrix methods for double sequence spacesrdquoPublicationes Mathematicae Debrecen vol 75 no 3-4 pp 387ndash399 2009

[13] A Pringsheim ldquoZur theorie der zweifach unendlichen zahlen-folgenrdquo Mathematische Annalen vol 53 no 3 pp 289ndash3211900

[14] B Altay and F Basar ldquoSome new spaces of double sequencesrdquoJournal of Mathematical Analysis and Applications vol 309 no1 pp 70ndash90 2005

[15] H J Hamilton ldquoTransformations of multiple sequencesrdquo DukeMathematical Journal vol 2 no 1 pp 29ndash60 1936

[16] S A Mohiuddine and A Alotaibi ldquoSome spaces of doublesequences obtained through invariant mean and related con-ceptsrdquo Abstract and Applied Analysis vol 2013 Article ID507950 11 pages 2013

[17] Mursaleen and S A Mohiuddine ldquoDouble 120590-multiplicativematricesrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 991ndash996 2007

[18] M Mursaleen and S A Mohiuddine ldquoOn 120590-conservative andboundedly 120590-conservative four-dimensional matricesrdquo Com-puters amp Mathematics with Applications vol 59 no 2 pp 880ndash885 2010

[19] P Schaefer ldquoInfinite matrices and invariant meansrdquo Proceedingsof the AmericanMathematical Society vol 36 pp 104ndash110 1972

[20] GM Robison ldquoDivergent double sequences and seriesrdquo Trans-actions of the American Mathematical Society vol 28 no 1 pp50ndash73 1926

[21] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966

[22] P Schaefer ldquoMatrix transformations of almost convergentsequencesrdquo Mathematische Zeitschrift vol 112 pp 321ndash3251969

[23] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991

[24] F Moricz ldquoExtensions of the spaces 119888 and 1198880from single to

double sequencesrdquoActaMathematicaHungarica vol 57 no 1-2pp 129ndash136 1991

[25] MMursaleen and S AMohiuddine ldquoRegularly 120590-conservativeand 120590-coercive four dimensional matricesrdquoComputers ampMath-ematics with Applications vol 56 no 6 pp 1580ndash1586 2008

[26] C Eizen and G Laush ldquoInfinite matrices and almost conver-gencerdquoMathematica Japonica vol 14 pp 137ndash143 1969

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 Abstract and Applied Analysis

It follows from (42) and (45) that

10038171003817100381710038171198711198981198991199041199051003817100381710038171003817 = sup119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(46)

Since 119861 isin (L1FΛ) we have

sup119898119899119904119905

1003816100381610038161003816119871119898119899119904119905 (119909)1003816100381610038161003816

= sup119898119899119904119905

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

infin

sum

119895=1

infin

sum

119896=1

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

119909119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(47)

Hence by the uniform boundedness principle we obtain

sup119898119899119904119905

1003817100381710038171003817119871119898119899119904119905 (119909)1003817100381710038171003817 = sup119898119899119904119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

120582119898120583119899

sum

119901isin119869119898

sum

119902isin119868119899

119887119901+119904119902+119905119895119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

lt infin

(48)

Conflict of Interests

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

Acknowledgment

This work was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no(130-100-D1434) The authors therefore acknowledge withthanks DSR technical and financial support

References

[1] S BanachTheorie des Operations Lineaires 1932[2] G G Lorentz ldquoA contribution to the theory of divergent

sequencesrdquo Acta Mathematica vol 80 pp 167ndash190 1948[3] S A Mohiuddine ldquoAn application of almost convergence in

approximation theoremsrdquo Applied Mathematics Letters vol 24no 11 pp 1856ndash1860 2011

[4] F Moricz and B E Rhoades ldquoAlmost convergence of doublesequences and strong regularity of summability matricesrdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 104 no 2 pp 283ndash294 1988

[5] M Mursaleen and S A Mohiuddine ldquoBanach limit and somenew spaces of double sequencesrdquo Turkish Journal of Mathemat-ics vol 36 no 1 pp 121ndash130 2012

[6] M Basarir ldquoOn the strong almost convergence of doublesequencesrdquo PeriodicaMathematica Hungarica vol 30 no 3 pp177ndash181 1995

[7] F Basar and M Kirisci ldquoAlmost convergence and generalizeddifferencematrixrdquoComputers ampMathematics with Applicationsvol 61 no 3 pp 602ndash611 2011

[8] F Cunjalo ldquoAlmost convergence of double subsequencesrdquoFilomat vol 22 no 2 pp 87ndash93 2008

[9] K Kayaduman and C Cakan ldquoThe cesaro core of doublesequencesrdquo Abstract and Applied Analysis vol 2011 Article ID950364 9 pages 2011

[10] Mursaleen ldquoAlmost strongly regular matrices and a core the-orem for double sequencesrdquo Journal of Mathematical Analysisand Applications vol 293 no 2 pp 523ndash531 2004

[11] Mursaleen and O H H Edely ldquoAlmost convergence and acore theorem for double sequencesrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 532ndash540 2004

[12] M Zeltser M Mursaleen and S A Mohiuddine ldquoOn almostconservative matrix methods for double sequence spacesrdquoPublicationes Mathematicae Debrecen vol 75 no 3-4 pp 387ndash399 2009

[13] A Pringsheim ldquoZur theorie der zweifach unendlichen zahlen-folgenrdquo Mathematische Annalen vol 53 no 3 pp 289ndash3211900

[14] B Altay and F Basar ldquoSome new spaces of double sequencesrdquoJournal of Mathematical Analysis and Applications vol 309 no1 pp 70ndash90 2005

[15] H J Hamilton ldquoTransformations of multiple sequencesrdquo DukeMathematical Journal vol 2 no 1 pp 29ndash60 1936

[16] S A Mohiuddine and A Alotaibi ldquoSome spaces of doublesequences obtained through invariant mean and related con-ceptsrdquo Abstract and Applied Analysis vol 2013 Article ID507950 11 pages 2013

[17] Mursaleen and S A Mohiuddine ldquoDouble 120590-multiplicativematricesrdquo Journal of Mathematical Analysis and Applicationsvol 327 no 2 pp 991ndash996 2007

[18] M Mursaleen and S A Mohiuddine ldquoOn 120590-conservative andboundedly 120590-conservative four-dimensional matricesrdquo Com-puters amp Mathematics with Applications vol 59 no 2 pp 880ndash885 2010

[19] P Schaefer ldquoInfinite matrices and invariant meansrdquo Proceedingsof the AmericanMathematical Society vol 36 pp 104ndash110 1972

[20] GM Robison ldquoDivergent double sequences and seriesrdquo Trans-actions of the American Mathematical Society vol 28 no 1 pp50ndash73 1926

[21] J P King ldquoAlmost summable sequencesrdquo Proceedings of theAmerican Mathematical Society vol 17 pp 1219ndash1225 1966

[22] P Schaefer ldquoMatrix transformations of almost convergentsequencesrdquo Mathematische Zeitschrift vol 112 pp 321ndash3251969

[23] F Basar and I Solak ldquoAlmost-coercivematrix transformationsrdquoRendiconti di Matematica e delle sue Applicazioni vol 11 no 2pp 249ndash256 1991

[24] F Moricz ldquoExtensions of the spaces 119888 and 1198880from single to

double sequencesrdquoActaMathematicaHungarica vol 57 no 1-2pp 129ndash136 1991

[25] MMursaleen and S AMohiuddine ldquoRegularly 120590-conservativeand 120590-coercive four dimensional matricesrdquoComputers ampMath-ematics with Applications vol 56 no 6 pp 1580ndash1586 2008

[26] C Eizen and G Laush ldquoInfinite matrices and almost conver-gencerdquoMathematica Japonica vol 14 pp 137ndash143 1969

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