Spaces Defined by Musielak

Transcription

1 Alied Mathematical Sciences, Vol 7, 2013, no 82, HIKARI Ltd, wwwm-hikaricom htt://dxdoiorg/ /ams The Difference of χ 2 over Metric Saces Defined by Musielak N Kavitha Deartment of Mathematics University College of Engineering constitutnet College of Anna University) Pattukkottai, India kavitha977@yahoocomsg N Saivaraju Deartment of Mathematics Sri Angalamman College of Engineering and Technology Trichiraalli , India saivaraju@yahoocom N Subramanian Deartment of Mathematics SASTRA University, Thanjavur , India nsmaths@yahoocom Coyright c 2013 N Kavitha et al This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited

2 4082 N Kavitha, N Saivaraju and N Subramanian Abstract In this aer, we introduce the sequence saces f Δn m ) and Λ2q f Δn m ) defined by Musielak We study some toological roerties and rove some inclusion relations between these saces Mathematics Subject Classification: 40A05, 40C05, 40D05 Keywords: analytic sequence, double sequences, χ 2 sace, difference sequence sace, Musielak - modulus function, metric sace 1 Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, resectively We write w 2 for the set of all comlex sequences x mn ), where m, n N, the set of ositive integers Then, w 2 is a linear sace under the coordinate wise addition and scalar multilication Some initial works on double sequence saces is found in Bromwich 2 Later on, they were investigated by Hardy 3, Moricz 7, Moricz and Rhoades 8, Basarir and Solankan 1, Triathy 11, Turkmenoglu 12, and many others We rocure the following sets of double sequences: M u t) := x mn ) w 2 : su m,n N x mn tmn < }, C t) := x mn ) w 2 : lim m,n x mn l tmn =1for some l C }, C 0 t) := x mn ) w 2 : lim m,n x mn tmn =1 }, L u t) := x mn ) w 2 : m=1 n=1 x mn tmn < }, C b t) :=C t) M u t) and C 0b t) =C 0 t) M u t); where t = t mn ) is the sequence of strictly ositive reals t mn for all m, n N and lim m,n denotes the limit in the Pringsheim s sense In the case t mn = 1 for all m, n N; M u t), C t), C 0 t), L u t), C b t) and C 0b t) reduce to the sets M u, C, C 0, L u, C b and C 0b, resectively Now, we may summarize the knowledge given in some document related to the double sequence saces Gökhan and Colak 14,15 have roved that M u t) and C t), C b t) are comlete aranormed saces of double sequences and gave the α,β,γ duals of the saces M u t) and C b t) Quite recently, in her PhD thesis, Zelter 16 has essentially studied both the theory of toological double sequence saces and the theory of summability of double sequences Mursaleen and Edely 17 and Triathy 11 have indeendently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences Altay and Basar 20 have defined the saces BS, BS t), CS, CS b, CS r and BV of double sequences consisting of all double series whose sequence of artial sums are in the saces M u, M u t), C, C b, C r and L u, resectively, and also examined some roerties of those sequence saces and determined the α duals of the saces

3 The difference of χ 2 over metric saces defined by Musielak 4083 BS, BV, CS b and the β ϑ) duals of the saces CS b and CS r of double series Basar and Sever 21 have introduced the Banach sace L q of double sequences corresonding to the well-known sace l q of single sequences and examined some roerties of the sace L q Quite recently Subramanian and Misra 22 have studied the sace χ 2 M, q, u) of double sequences and gave some inclusion relations The class of sequences which are strongly Cesàro summable with resect to a modulus was introduced by Maddox 6 as an extension of the definition of strongly Cesàro summable sequences Connor 23 further extended this definition to a definition of strong A summability with resect to a modulus where A =a n,k ) is a nonnegative regular matrix and established some connections between strong A summability, strong A summability with resect to a modulus, and A statistical convergence In 24 the notion of convergence of double sequences was resented by A Pringsheim Also, in 25-26, and 27 the four dimensional matrix transformation Ax) k,l = m=1 n=1 amn kl x mn was studied extensively by Robison and Hamilton We need the following inequality in the sequel of the aer For a, b, 0 and 0 <<1, we have 11) a + b) a + b The double series m,n=1 x mn is called convergent if and only if the double sequence s mn )is convergent, where s mn = m,n i,j=1 x ijm, n N) A sequence x =x mn )is said to be double analytic if su mn x mn 1/m+n < The vector sace of all double analytic sequences will be denoted by Λ 2 A sequence x =x mn ) is called double gai sequence if m + n)! x mn ) 1/m+n 0asm, n The double gai sequences will be denoted by χ 2 Let φ = allfinitesequences} Consider a double sequence x =x ij ) The m, n) th section x m,n of the sequence is defined by x m,n = m,n i,j=0 x iji ij for all m, n N ; where I ij denotes the double sequence whose only non zero term is a 1 i+j)! in the i, j) th lace for each i, j N An FK-saceor a metric sace)x is said to have AK roerty if I mn ) is a Schauder basis for X Or equivalently x m,n x An FDK-sace is a double sequence sace endowed with a comlete metrizable; locally convex toology under which the coordinate maings x =x k ) x mn )m, n N) are also continuous Let M and Φ are mutually comlementary modulus functions Then, we have: i) For all u, y 0, 12) uy M u)+φy), Y oung s inequality)see13

4 4084 N Kavitha, N Saivaraju and N Subramanian ii) For all u 0, 13) uη u) =M u)+φη u)) iii) For all u 0, and 0 <λ<1, 14) M λu) λm u) Lindenstrauss and Tzafriri 5 used the idea of Orlicz function to construct Orlicz sequence sace l M = x w : ) } k=1 M xk ρ <, for some ρ > 0, The sace l M with the norm x = inf ρ>0: ) } k=1 M xk ρ 1, becomes a Banach sace which is called an Orlicz sequence sace For M t) =t 1 < ), the saces l M coincide with the classical sequence sace l A sequence f =f mn ) of modulus function is called a Musielak-modulus function A sequence g =g mn ) defined by g mn v) =su v u f mn )u) :u 0},m,n=1, 2, is called the comlementary function of a Musielak-modulus function f For a given Musielak modulus function f, the Musielak-modulus sequence sace t f and its subsace h f are defined as follows } t f = x w 2 : I f x mn ) 1/m+n 0 as m, n, } h f = x w 2 : I f x mn ) 1/m+n 0 as m, n, where I f is a convex modular defined by I f x) = m=1 n=1 f mn x mn ) 1/m+n,x=x mn ) t f We consider t f equied with the Luxemburg metric )) } d x, y) =su mn inf m=1 n=1 f xmn 1/m+n mn mn 1 If X is a sequence sace, we give the following definitions: i)x = the continuous dual of X; ii)x α = a =a mn ): m,n=1 a mn x mn <, for eachx X } ; iii)x β = a =a mn ): m,n=1 a mn x mn is convegent, foreachx X } ; iv)x γ = a =a mn ):su mn 1 } M,N m,n=1 a mnx mn <, foreachx X ; v)letx beanf K sace φ; then X f = fi mn ):f X } ;

5 vi)x δ = The difference of χ 2 over metric saces defined by Musielak 4085 } a =a mn ):su mn a mn x mn 1/m+n <, foreachx X ; X α X β,x γ are called α orköthe T oelitz)dual of X, β or generalized Köthe T oelitz) dual ofx, γ dual of X, δ dual ofx resectivelyx α is defined by Guta and Kamtan 13 It is clear that X α X β and X α X γ, but X β X γ does not hold, since the sequence of artial sums of a double convergent series need not to be bounded The notion of difference sequence saces for single sequences) was introduced by Kizmaz as follows Z Δ) = x =x k ) w :Δx k ) Z} for Z = c, c 0 and l, where Δx k = x k x k+1 for all k N Here c, c 0 and l denote the classes of convergent,null and bounded sclar valued single sequences resectively The difference sequence sace bv of the classical sace l is introduced and studied in the case 1 by Başar and Altay and in the case 0 <<1 by Altay and Başar in 20 The saces c Δ),c 0 Δ),l Δ) and bv are Banach saces normed by x = x 1 + su k 1 Δx k and x bv = k=1 x k ) 1/, 1 < ) Later on the notion was further investigated by many others We now introduce the following difference double sequence saces defined by Z Δ) = x =x mn ) w 2 :Δx mn ) Z } where Z = Λ 2,χ 2 and Δx mn = x mn x mn+1 ) x m+1n x m+1n+1 ) = x mn x mn+1 x m+1n + x m+1n+1 for all m, n N The generalized difference double notion has the following reresentation: Δ m x mn = Δ m 1 x mn Δ m 1 x mn+1 Δ m 1 x m+1n +Δ m 1 x m+1n+1, and also this generalized difference double notion has the following binomial reresentation: Δ m x mn = m m m) ) m i=0 j=0 1)i+j i j x m+i,n+j 2 Definition and Preliminaries Let n N and X be a real vector sace of dimension w, where n w A real valued function d x 1,,x n )= d 1 x 1 ),,d n x n )) on X satisfying the following four conditions: i) d 1 x 1 ),,d n x n )) = 0 if and and only if d 1 x 1 ),,d n x n ) are linearly deendent, ii) d 1 x 1 ),,d n x n )) is invariant under ermutation, iii) αd 1 x 1 ),,d n x n )) = α d 1 x 1 ),,d n x n )),α R iv) d x 1,y 1 ), x 2,y 2 ) x n,y n ))=d X x 1,x 2, x n ) + d Y y 1,y 2, y n ) ) 1/ for1 < ; or) v) d x 1,y 1 ), x 2,y 2 ), x n,y n )) := su d X x 1,x 2, x n ),d Y y 1,y 2, y n )}, for x 1,x 2, x n X, y 1,y 2, y n Y is called the roduct metric of the Cartesian roduct of n metric saces is the norm of the n-vector of the norms of the n subsaces A trivial examle of roduct metric of n metric sace is the norm sace is X = R equied with the following Euclidean metric in the roduct sace is the norm:

6 4086 N Kavitha, N Saivaraju and N Subramanian d 1 x 1 ),,d n x n )) E = su detd mn x mn )) ) = d 11 x 11 ) d 12 x 12 ) d 1n x 1n ) d 21 x 21 ) d 22 x 22 ) d 2n x 1n ) su d n1 x n1 ) d n2 x n2 ) d nn x nn ) where x i =x i1, x in ) R n for each i =1, 2, n If every Cauchy sequence in X converges to some L X, then X is said to be comlete with resect to the metric Any comlete metric sace is said to be Banach metric sace Let X be a linear metric sace A function w : X R is called aranorm, if 1) w x) 0, for all x X; 2) w x) =w x), for all x X; 3) w x + y) w x)+wy), for all x, y X; 4) If σ mn ) is a sequence of scalars with σ mn σ as m, n and x mn ) is a sequence of vectors with w x mn x) 0asm, n, then w σ mn x mn σx) 0asm, n A aranorm w for which w x) = 0 imlies x = 0 is called total aranorm and the air X, w) is called a total aranormed sace It is well known that the metric of any linear metric sace is given by some total aranorm see 32, Theorem 1042, 183) The zero sequence is denoted by θ and = mn ) is a sequence of strictly ositive real numbers Further the sequence 1 mn) will be reresented by tmn ) Let f =f mn ) be a Musielak-modulus function and = mn ) be any bounded sequence of ositive real numbers and let X, q) be a seminormed sace seminormed by q In the resent aer, we define the following sequenc saces: Let us consider μ mn x) = q m + n)!δ n m )1/m+n) mn tmn f Δn m )=x =x mn) X :f μ mn x)) 0, asm,n }, f Δn m)=x =x mn ) X : su mn f μ mn x)) < } If we take = mn )=1, we have

7 The difference of χ 2 over metric saces defined by Musielak 4087 f Δn m)=x =x mn ) X :f μ mn x)) 0, asm,n }, f Δn m)=x =x mn ) X : su mn f μ mn x)) < } The following inequality will be used throughout the aer If 0 mn su mn = K, D = max 1, 2 K 1) then 21) a mn + b mn mn D a mn mn + b mn mn } for all m, n and a mn,b mn C Also a mn max 1, a K) for all a C In this aer we study some toological roerties of the above sequence saces 3 Main Results 31 Theorem Let f =f mn ) be a Musielak-modulus function, = mn ) be a double analytic sequence of strictly ositive real numbers, the sequence saces f, d x 1),dx 2 ),,dx n 1 )) and f, d x 1),dx 2 ),,dx n 1 )) are linear saces Proof: It is routine verification Therefore the roof is omitted 32 Theorem Let f =f mn ) be a Musielak-modulus function, = mn ) be a double analytic sequence of strictly ositive real numbers, the sequence sace f, d x 1),dx 2 ),,dx n 1 )) is a aranormed sace with resect to the aranorm defined) by } g x) =inf f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 1 =0 Proof: Clearly g x) 0 for x = x mn ) f, d x 1),dx 2 ),,dx n 1 )) Since f mn 0) = 0, we get g 0) = 0 Conversely, suose that g x) = 0, then ) } inf f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 1 =0 Suose that μ mn x) 0 for each m, n N Then μ mn x), d x 1 ),dx 2 ),,dx n 1 )) )) 1/H It follows that f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) which is a contradiction Therefore μ mn x) =0 Let )) 1/H f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 1 and f mn μ mn y), d x 1 ),dx 2 ),,dx n 1 )) )) 1/H 1 Then by using Minkowski s inequality, we have f mn μ mn x + y), d x 1 ),dx 2 ),,dx n 1 )) )) 1/H

8 4088 N Kavitha, N Saivaraju and N Subramanian f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) )) 1/H + f mn μ mn y), d x 1 ),dx 2 ),,dx n 1 )) )) 1/H So we have ) g x + y) =inf f mn μ mn x + y), d x 1 ),dx 2 ),,dx n 1 )) ) } inf f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 1 + ) } inf f mn μ mn y), d x 1 ),dx 2 ),,dx n 1 )) 1 Therefore, g x + y) g x)+g y) } 1 Finally, to rove that the scalar multilication is continuous Let λ be any comlex number By definition, ) } g λx) =inf f mn μ mn λx), d x 1 ),dx 2 ),,dx n 1 )) 1 Then ) g λx)=inf λ t) qmn/h qmn } : u mn f mn μ mn λx), d x 1 ),dx 2 ),,dx n 1 )) 1 where t = 1 λ Since λ qmn max 1, λ sumn ), we have g λx) max 1, λ sumn ) inf ) } t qmn/h : u mn f mn μ mn λx), d x 1 ),dx 2 ),,dx n 1 )) 1 This comletes the roof 33 Theorem i) If the sequence f mn ) satisfies uniform Δ 2 condition, then α = g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ii) If the sequence g mn ) satisfies uniform Δ 2 condition, then α g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) = Proof: Let the sequence f mn ) satisfies uniform Δ 2 condition, we get 31) α g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) To rove the inclusion α g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )), let a Then for all x mn } with x mn )

9 The difference of χ 2 over metric saces defined by Musielak 4089 we have 32) x mn a mn < m=1 n=1 Since the sequence f mn ) satisfies uniform Δ 2 condition, then y mn ), we get m=1 n=1 by 32) Thus ϕ rs a mn ) f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) = g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) and hence a mn ) g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) This gives that ϕ rsy mna mn Δ m λ mnm+n)! 33) α g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) we are granted with 31) and 33) α = g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) α ii) Similarly, one can rove that g, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) if the sequence g mn ) satisfies uniform Δ 2 condition 34 Proosition If 0 < mn <r mn < for each m and n, then fr, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) Proof: The roof is standard, so we omit it 35 Proosition i) If 0 < inf mn mn < 1 then Λ 2f, μ 2 mn x), d x 1 ),dx 2 ),,dx n 1 )) ii) If 1 mn su mn <, then Λ 2 f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) Proof: The roof is standard, so we omit it ) ) 36 Proosition Let f = f mn and f = f mn are sequences of Musielak functions, we have, μ f mn x), d x 1 ),dx 2 ),,dx n 1 )), μ f mn x), d x 1 ),dx 2 ),,dx n 1 )), μ f +f mn x), d x 1 ),dx 2 ),,dx n 1 )) Proof: The roof is easy so we omit it <

10 4090 N Kavitha, N Saivaraju and N Subramanian 37 Proosition For any sequence of Musielak functions f =f mn ) and q =q mn ) be double analytic sequence of strictly ositive real numbers Then Proof: The roof is easy so we omit it 38 Proosition The sequence sace Proof: Let x =x mn ) is solid, ie) su mn < Let α mn ) be double sequence of scalars such that α mn 1 for all m, n N N Then we get su mn f, μ mn αx), d x 1 ),dx 2 ),,dx n 1 )) su mn This comletes the roof is mono- 39 Proosition The sequence sace tone Proof: The roof follows from Proosition Proosition If f =f mn ) be any Musielak function Then f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ ϕ if and only if su rs r,s 1 ϕ < rs Proof: Let x f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ ϕ and N = su rs r,s 1 ϕ < rs Then we get f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ rs = N f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ rs =0 Thus x f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ and x f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ Then f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ Conversely, suose that ϕ <ɛ,for every ɛ>0 Suose that su rs r,s 1 ϕ = rs ϕ, then there exists a sequence of members rs jk ) such that lim jk j,k ϕ = Hence, we have jk f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ rs = Therefore x/ f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ, which is a contradiction This comletes the roof

11 The difference of χ 2 over metric saces defined by Musielak Proosition If f =f mn ) be any Musielak function Then f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ = f, μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ϕ Proof: It is easy to rove so we omit if and only if su r,s 1 ϕ rs ϕ rs <, su r,s 1 ϕ rs ϕ rs > 312 Proosition The sequence sace is not solid Proof: The result follows from the following examle Examle: Consider x =x mn )= m+n 1 m+n 1 m+n 1 m+n 1 m+n 1 m+n α mn =, for all m, n N 1 m+n 1 m+n 1 m+n Let Hence Then α mn x mn / is not solid 313 Proosition The sequence sace is not monotone Proof: The roof follows from Proosition 312 A sequence x =x mn ) is said to be ϕ statistically convergent or s ϕ statistically convergent to 0 if for every ɛ>0, lim rs f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ) qmn ɛ } =0 where the vertical bars indicates the number of elements in the enclosed set In this case we write s ϕ limx =0orx mn 0s ϕ ) and s ϕ = x : 0 R : s ϕ limx =0} 314 Proosition For any sequence of Musielak functions f =f mn ) and = mn ) be double analytic sequence of strictly ositive real numbers Then s 2q

12 4092 N Kavitha, N Saivaraju and N Subramanian Proof:Let x and ɛ>0 Then ) f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) f μ ) } mn mn x), d x 1 ),dx 2 ),,dx n 1 )) ɛ from which it follows that x s 2q To show that s 2q strictly contain We define x =x mn )byx mn )=mn if rs ϕrs + mn rs and xmn ) = 0 otherwise Then x/ and for every ɛ 0 <ɛ 1), f μ ) } mn mn x), d x 1 ),dx 2 ),,dx n 1 )) ɛ = ϕ rs ϕ rs 0asr, s ) ie x 0 s 2q, where denotes the greatest integer function On the other hand, ) f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) as r, s ie x mn 0 This comletes the roof ) 315 Theorem Suose f = f mn and f ) are Musielak-modulus functions satisfying the Δ 2 condition then we have the following results: i) If mn ) Λ 2 then f Δn m ) χ2q f f Δn m ) ii) f Δn m) f Δn m) f +f Δn m) Proof: If x =x mn ) f Δn m) then ) f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 0asm, n Suose ) y mn = f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) for all m, n N Choose δ>0 be such that 0 <δ<1, then for y mn δ we have y mn < ymn δ < 1+ ymn δ Now f satisfies Δ 2 condition so that there exists J 1 such that f mn y mn ) < Jymn 2δ f mn 2) = jymn δ f mn 2) ) We obtain f mn f mn ) μ mn x), d x 1 ),dx 2 ),,dx n 1 )) )} = f f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) ) = f mn μ mn y mn ), d x 1 ),dx 2 ),,dx n 1 )) 0, as m, n Similarly, we can rove the other cases ii) Suose x =x mn ) f Δn m) f Δn m), then ) f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 0asm, n and

13 The difference of χ 2 over metric saces defined by Musielak 4093 ) f mn μ mn x), d x 1 ),dx 2 ),,dx n 1 )) 0asm, n The above inequality follows h f mn f i mn µ mn x), d x 1),dx 2),,dx n 1)) D nhn f mn oi h io µ mn x), d x 1),dx 2),,dx n 1)) + f mn µ mn x), d x 1),dx 2),,dx n 1)) Hence f Δn m ) f Δn m ) χ2q f +f Δn m ) Cometing Interests: Author have declared that no cometing interests exist References 1 MBasarir and OSolancan, On some double sequence saces, J Indian Acad Math, 212) 1999), TJI ABromwich, An introduction to the theory of infinite series Macmillan and CoLtd,New York, 1965) 3 GHHardy, On the convergence of certain multile series, Proc Camb Phil Soc, ), MAKrasnoselskii and YBRutickii, Convex functions and Orlicz saces, Gorningen, Netherlands, JLindenstrauss and LTzafriri, On Orlicz sequence saces, Israel J Math, ), IJMaddox, Sequence saces defined by a modulus, Math Proc Cambridge Philos Soc, 1001) 1986), FMoricz, Extentions of the saces c and c 0 from single to double sequences, Acta Math Hung, 571-2), 1991), FMoricz and BERhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math Proc Camb Phil Soc, 104, 1988), HNakano, Concave modulars, J Math Soc Jaan, 51953), WHRuckle, FK saces in which the sequence of coordinate vectors is bounded, Canad J Math, ), BCTriathy, On statistically convergent double sequences, Tamkang J Math, 343), 2003), ATurkmenoglu, Matrix transformation between some classes of double sequences, J Inst Math Com Sci Math Ser, 121), 1999), PKKamthan and MGuta, Sequence saces and series, Lecture notes, Pure and Alied Mathematics, 65 Marcel Dekker, In c, New York, AGökhan and RÇolak, The double sequence saces c P 2 ) and c PB 2 ), Al Math Comut, 1572), 2004), AGökhan and RÇolak, Double sequence saces l 2, ibid, 1601), 2005), MZeltser, Investigation of Double Sequence Saces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ of Tartu, Faculty of Mathematics and Comuter Science, Tartu, MMursaleen and OHH Edely, Statistical convergence of double sequences, J Math Anal Al, 2881), 2003),

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