Math. Sci. Lett. 3, No. 3, (2014) 165. statistical convergence, where A is a sequence of four dimensional matrices A(uv) = a m 1 m r n 1 n s

Transcription

1 Math Sci Lett 3, No 3, ) 165 Mathematical Sciences Letters An International Journal Lacunary χ 2 A uv Convergence of p Metric Defined by mn Sequence of Moduli Musielak Nagarajan Subramanian Department of Mathematics, SASTRA University, Thanjavur , India Received: 9 Mar 2014, Revised: 18 Apr 2014, Accepted: 20 Apr 2014 Published online: 1 Sep 2014 Abstract: We study some connections between lacunary strong χa 2 uv convergence with respect to a mn sequence of moduli Musielak and lacunary χa 2 uv statistical convergence, where A is a sequence of four dimensional matrices Auv) = uv) of complex numbers Keywords: analytic sequence,χ 2 space, difference sequence space,musielak - modulus function, p metric space, mn sequences 1 Introduction Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively We write w 2 for the set of all complex sequences x mn ), where m,n N, the set of positive integers Then, w 2 is a linear space under the coordinate wise addition and scalar multiplication Some initial works on double sequence spaces is found in Bromwich 4 Later on, they were investigated by Hardy 15], Moricz 20], Moricz and Rhoades 21], Basarir and Solankan 2], Tripathy 30], Turkmenoglu 40], and many others We procure the following sets of double sequences: M u t) := x mn ) w 2 : sup m,n N x mn t mn <, C p t) := xmn ) w 2 : p lim m,n x mn l t mn = 1 f or some l C, C 0p t) := x mn ) w 2 : p lim m,n x mn t mn = 1, L u t) := x mn ) w 2 : m=1 n=1 x mn t mn <, C bp t) := C p t) M u t) and C 0bp t)=c 0p t) M u t); where t = t mn ) is the sequence of strictly positive reals t mn for all m,n N and p lim m,n denotes the limit in the Pringsheim s sense In the case t mn = 1 for all Corresponding author nsmaths@yahoocom m,n N;M u t),c p t),c 0p t),l u t),c bp t) and C 0bp t) reduce to the sets M u,c p,c 0p,L u,c bp and C 0bp, respectively Now, we may summarize the knowledge given in some document related to the double sequence spaces Gökhan and Colak 9,10] have proved that M u t) and C p t),c bp t) are complete paranormed spaces of double sequences and gave the α,β,γ duals of the spaces M u t) and C bp t) Quite recently, in her PhD thesis, Zelter 43] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences Mursaleen and Edely 22] and Tripathy 30] have independently 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 1] have defined the spaces BS,BS t),c S p,c S bp,cs r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u,m u t),c p,c bp,c r and L u, respectively, and also examined some properties of those sequence spaces and determined the α duals of the spaces BS,BV,C S bp and the βϑ) duals of the spaces CS bp and CS r of double series Basar and Sever 3] have introduced the Banach space L q of double sequences corresponding to the well-known space l q of single sequences and examined some properties of the space L q Quite recently Subramanian and Misra 29] have studied the space χm 2 p,q,u) of double sequences

2 166 N Subramanian: Lacunary χ 2 A uv Convergence of p and gave some inclusion relations The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox 19] as an extension of the definition of strongly Cesàro summable sequences Connor 5] further extended this definition to a definition of strong A summability with respect 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 respect to a modulus, and A statistical convergence In 26] the notion of convergence of double sequences was presented by A Pringsheim Also, in 12, 13], and 14] 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 paper For a,b, 0 and 0< p< 1, we have a+b) p a p + b p 1) 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 i jm,n N) A sequence x = x mn )is said to be double analytic if sup mn x mn 1/m+n < The vector space 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 0 as m,n The double gai sequences will be denoted by χ 2 Let φ =all finitesequences Consider a double sequence x = x i j ) The m,n) th section x m,n] of the sequence is defined by x m,n] = m,n i, j=0 x i ji i j for all m,n N ; where I i j denotes the double sequence whose only non zero term is a 1 i+ j)! in thei, j) th place for each i, j N An FK-spaceor a metric space)x is said to have AK property if I mn ) is a Schauder basis for X Or equivalently x m,n] x An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x=x k ) x mn )m,n N) are also continuous Let M and Φ are mutually complementary modulus functions Then, we have: i) For all u,y 0, uy Mu)+ Φy),Young s inequality)see16]] 2) ii) For all u 0, uηu)=mu)+φηu)) 3) iii) For all u 0, and 0<λ < 1, Mλ u) λ Mu) 4) Lindenstrauss and Tzafriri 18] used the idea of Orlicz function to construct Orlicz sequence space l M = x w: k=1 M xk ρ <, f or some ρ > 0, The spacel M with the norm x =in f ρ > 0 : k=1 M xk ρ ) 1, becomes a Banach space which is called an Orlicz sequence space For Mt) = t p 1 p< ), the spaces l M coincide with the classical sequence space l p A sequence f = f mn ) of modulus function is called a Musielak-modulus function A sequence g = g mn ) defined by g mn v)=sup v u f mn )u) : u 0,m,n=1,2, is called the complementary function of a Musielak-modulus function f For a given Musielak modulus function f, the Musielak-modulus sequence space t f and its subspace 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 equipped with the Luxemburg metric dx,y)= ) sup mn in f m=1 n=1 f x mn 1/m+n mn mn 1 If X is a sequence space, we give the following definitions: i)x = the continuous dual of X; ii)x α = a=a mn ) : m,n=1 a mnx mn <, f or each x X ; iii)x β = a=a mn ) : m,n=1 a mnx mn is convegent, f oreach x X ; iv)x γ = a=a mn ) : sup mn 1 M,N <, m,n=1 a mnx mn f oreachx X ; v)let X beanfk space φ; then X f = fi mn ) : f X ; vi)x δ =

3 Math Sci Lett 3, No 3, ) / wwwnaturalspublishingcom/journalsasp 167 a=a mn ) : sup mn a mn x mn 1/m+n <, f oreach x X ; X α X β,x γ are called α orköthe Toeplitz)dual of X,β or generalized Köthe Toeplitz) dual o f X,γ dual o f X, δ dual o f X respectivelyx α is defined by Gupta and Kamptan 16 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 partial sums of a double convergent series need not to be bounded The notion of difference sequence spaces for single sequences) was introduced by Kizmaz as follows Z )=x=x k ) w: x k ) Z for Z = c,c 0 andl, where x k = x k x k+1 for all k N Here c,c 0 andl denote the classes of convergent,null and bounded sclar valued single sequences respectively The difference sequence space bv p of the classical space l p is introduced and studied in the case 1 p by Başar and Altay and in the case 0 < p < 1 by Altay and Başar in 1 The spaces c ),c 0 ),l ) and bv p are Banach spaces normed by x = x 1 + sup k 1 x k and x bvp = k=1 x k p ) 1/p,1 p< ) Later on the notion was further investigated by many others We now introduce the following difference double sequence spaces defined by Z )= x=x mn ) w 2 : x mn ) Z where Z = Λ 2, χ 2 and x 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 2 Definition and Preliminaries Let mn 2) be an integer A function x :M N) M N) M N) M N)m n f actors) RC) is called a real complex mn sequence, where N,R and C denote the sets of natural numbers and complex numbers respectively Let m 1,m 2, m r,n 1,n 2,,n s N and X be a real vector space of dimension w, where m 1,m 2, m r,n 1,n 2,,n s w A real valued function d p x 11,,x m1,m 2, m r,n 1,n 2,,n s ) = d 1 x 11 ),,d n x m1,m 2, m r,n 1,n 2,,n s )) p on X satisfying the following four conditions: i) d 1 x 11 ),,d n x m1,m 2, m r,n 1,n 2,,n s )) p = 0 if and and only if d 1 x 11 ),,d m1,m 2, m r,n 1,n 2,,n s x m1,m 2, m r,n 1,n 2,,n s ) are linearly dependent, ii) d 1 x 11 ),,d m1,m 2, m r,n 1,n 2,,n s x m1,m 2, m r,n 1,n 2,,n s )) p is invariant under permutation, iii) αd1x11),,dm1,m2, mp,n1,n2,,nqxm1,m2, mr,n1,n2,,ns)) p = α d1x11),,dnxm1,m2, mp,n1,n2,,nq)) p,α R iv) d p x 11,y 11 ),x 12,y 12 ) x m1,m 2, m r,n 1,n 2,,n s,y m1,m 2, m r,n 1,n 2,,n s ))= dx x 11,x 12, x m1,m 2, m r,n 1,n 2,,n q ) p + d Y y 11,y 12, y m1,m 2, m p,n 1,n 2,,n s ) p) 1/p f or1 p< ; or) v) dx 11,y 11 ),x 12,y 12 ), x m1,m 2, m r,n 1,n 2,,n s,y m1,m 2, m r,n 1,n 2,,n s )) := supd X x 11,x 12, x m1,m 2, m r,n 1,n 2,,n s ),d Y y 11,y 12, y m1,m 2, m r,n 1,n 2,,n s ), for x 11,x 12, x m1,m 2, m r,n 1,n 2,,n s X,y 11,y 12, y m1,m 2, m r,n 1,n 2,,n s Y is called the p product metric of the Cartesian product of m 1,m 2, m r,n 1,n 2,,n s metric spaces is the p norm of the m n-vector of the norms of the m 1,m 2, m r,n 1,n 2,,n s subspaces A trivial example of p product metric of m 1,m 2, m r,n 1,n 2,,n s metric space is the p norm space is X = R equipped with the following Euclidean metric in the product space is the p norm: d 1 x 11 ),,d n x m1,m 2, mr,n 1,n 2,,ns )) E = sup ) detd m1,m 2, mr,n 1,n 2,,ns xm1,m 2, mr,n 1,n 2,,ns ) = d 11 x 11 ) d 12 x 12 ) d 1n x1,n1,n 2,,ns) d 21 x 21 ) d 22 x 22 ) d 2n x2,n1,n 2,,ns sup d m1n1 x m1n1 ) d m2n2 x m2n2 ) d m1,m 2, mr,n 1,n 2,,ns xm1,m 2, mr,n 1,n 2,,ns) where x i = x i1, x i,n1,n 2,,n s ) R n for each i=1,2, m 1,m 2 m r If every Cauchy sequence in X converges to some L X, then X is said to be complete with respect to the p metric Any complete p metric space is said to be p Banach metric space By a lacunary sequence θ =m r n s ), where m 0 n 0 = 0, we shall mean an increasing sequence of non-negative integers with h rs = m r n s m r 1 n s 1 as r,s The intervals determined by θ will be denoted by I rs =m r 1 n s 1,m r n s Let F = f mn ) be a mn sequence of moduli musielak such that lim u 0 +sup mn f mn u) = 0 Throughout this paper χa 2 uv convergence of p metric of mn sequence of musielak modulus function determinated by F will be denoted by f mn F for every m,n N The purpose of this paper is to introduce and study a concept of lacunary strong χa 2 uv convergence of p metric with respect to a mn sequence of moduli musielak indent We now introduce the generalizations of lacunary strongly χa 2 uv convergence of p metric with respect a mn sequence of musielak modulus function and investigate some inclusion relations Let A denote a sequence of the matrices A uv = ) uv) of complex numbers We write for any sequence x=x mn ),y i j uv)=a uv i j x)= m 1 m r n 1 n s uv) m 1 m r + n 1 n s )! x m1 m r n 1 n s ) 1/m 1 m r +n 1 n s if it exits for each i and uv We A uv x)= A uv i j )i x) j,ax=auv x)) uv

4 168 N Subramanian: Lacunary χ 2 A uv Convergence of p 21 Definition Let F = f i j be a mn sequence of moduli musielak, A denote the sequence of four dimensional infinte matrices of complex numbers and X be locally convex Hausdorff topological linear space whose topology is determined by a set of continuous semi norms η and X ) m 1 m r n 1 n s ) be a p metric space, q = q i j ) be double analytic sequence of strictly positive real numbers By w 2 p X) we denote the space of all sequences defined over X )) p ) In the present paper we define the following sequence spaces: )) ] p A f N, θ dx 11 ),dx 12 ),,d x m1,m2, mr 1 n1,n2, ns 1 = lim rs Nθ )) )] qi j p f i j x),, dx 11 ),dx 12 ),,d x m1,m2, mr 1 n1,n2, ns 1 = 0, where N θ x)= 1 hrs i Irs j Irs η uniformly in uv A uv i j m1 m r + n 1 n s )! x m1 mrn1 ns ) 1/m1 mr+n 1 ns ))), Λ 2qη ]= sup rs f uv N θ x), dx 11 ),dx 12 ),,d ) )] qi j x p m1,m2, m r 1n 1,n 2, n s 1 < where e= Main Results 31 proposition and Λ 2qη are linear spaces Proof: It is routine verification Therefore the proof is omitted The inclusion relation between and Λ 2qη 32 Theorem Let A be a mn sequence the four ) dimensional infinite matrices A uv = uv) of complex numbers and F = fmn i j be a mn sequence of moduli musielak If x=x mn ) lacunary strong A uv convergent to zero then x = x mn ) lacunary strong A uv convergent to zero with respect to mn sequence of moduli musielak, ie) x p m1,m 2, m r 1 n 1,n 2, n s 1 Proof: Let F = fmn i j be a mn sequence of moduli musielak and put sup fmn1) i j = T Let x = x mn ) and ε > 0 We choose 0<δ < 1 such that f i mnu)<ε j for every u with 0 u δi, j N) We can write ]= ]+ where the first part is over δ and second part is over > δ By definition of Musielak modulus fmn i j for every i j, we have )) p ] ε H 2+ 2Tδ 1 ) H 2 )) p Therefore x = x mn ) 33 Theorem Let A be a mn sequence of the four ) dimensional infinite matrices A uv = uv) of complex numbers, q = q i j ) be a mn sequence of positive real numbers with 0<in f q i j = H 1 supq i j = H 2 > and F = fmn i j be a mn sequence of moduli Musielak If f lim u,v in f i j uv) i j uv > 0, then ]= f Proof: If lim u,v in f i j uv) i j > 0, then there exists a uv number β > 0 such that f i j uv) β u for all u 0 and i, j N Let x=x m1 m r n 1 n s ) Clearly β Therefore

5 Math Sci Lett 3, No 3, ) / wwwnaturalspublishingcom/journalsasp 169 x = x m1 m r n 1 n s ) By using Theorem 32, the proof is complete We now give an example to show that in the case when β = 0 Consider A=I, unit matrix, ηx)= m 1 m r + n 1 n s )! x m1 m r n 1 n s ) 1/m 1 m r +n 1 n s,q i j = 1 for every i, j N and fmnx) i j = 1/m1 mr+n1 ns)i+1) j+1)) x m1 mrn 1 ns 1/m i, m 1 m r +n 1 n s )!) 1 mr+n1 ns j 1,x>0) in the case β > 0 Now we define x i j = h rs if i, j = m r n s for some r,s 1 and x i j = 0 other wise Then we have, ] 1 as r,s and so x = x m1 m r n 1 n s ) / The inclusion Relation between and In this section we introduce natural relationship between lacunary A uv statistical convergence and lacunary strong A uv convergence with respect to mn sequence of moduli Musielak 34 Definition Let θ be a lacunary mn sequence Then a mn sequence x = x m1 m r n 1 n s ) is said to be lacunary statistically convergent to a number zero if for every ε > 0,lim rs h 1 rs K θ ε) = 0, where K θ ε) denotes the number of elements in K θ ε)= i, j I rs :m 1 m r + n 1 n s )! x m1 m rn 1 n s 0 ) 1/m 1 m r+n 1 n s ε The set of all lacunary statistical convergent mn sequences is denoted by S θ ) Let A uv = uv) be an four dimensional infinite matrix of complex numbers Then a mn sequence x = x m1 m r n 1 n s ) is said to be lacunary A statistically convergent to a number zero if for every ε > 0,lim rs h 1 rs KA θε) =0, where KA θ ε) denotes the number of elements in KA θ ε)= i, j I rs :m 1 m r+ n 1 n s)! x m1 m rn 1 n s 0 ) 1/m1 mr+n1 ns ε The set of all lacunary A statistical convergent mn sequences is denoted by S θ A) 35 Definition Let A be a mn sequence of the four ) dimensional infinite matrices A uv = uv) of complex numbers and let q = q i j ) be a mn sequence of positive real numbers with 0<in f q i j = H 1 supq i j = H 2 < Then a mn sequence x =x m1 m r n 1 n s ) is said to be lacunary A uv statistically convergent to a number zero if for every ε > 0,lim rs h 1 rs KAθη ε) = 0, where KAθη ε) denotes the number of elements in KA θη ε)= i, j I rs :m 1 m r+ n 1 n s)! x m1 m rn 1 n s 0 ) 1/m1 mr+n1 ns ε The set of all lacunary A η statistical convergent mn sequences is denoted by S θ A,η) The following theorems give the relations between lacunary A uv statistical convergence and lacunary strong A uv convergence with respect to a mn sequence of moduli Musielak 36 Theorem Let F = f i j ) be a mn sequence of moduli Musielak Then ] if and only if lim i j f i j u)>0,u>0) Proof: Let ε > 0 and x=x m1 m r n 1 n s ) If lim i j f i j u) > 0,u>0), then there exists a number d > 0 such that f i j ε)>d for u>ε and i, j N Let ] h 1 rs d H 1KA θη ε) It follows that A f S θ Conversely, suppose that lim i j f i j u) > 0 does not hold, then there is a number t > 0 such that lim i j f i j t) = 0 We can select a lacunary mn sequence θ = m 1 m r n 1 n s ) such that f i j t) < 2 rs for any i > m 1 m r, j > n 1 n s Let A = I, unit matrix, define the mn sequence x by putting x i j = t if m 1,m 2, m r 1 n 1,n 2, n s 1 < i, j < m 1,m 2, m r n 1,n 2, n s +m 1,m 2, m r 1 n 1,n 2, n s 1 2 and x i j = 0 if m 1,m 2, m r n 1,n 2, n s +m 1,m 2, m r 1 n 1,n 2, n s 1 2 i, j m 1,m 2, m r n 1,n 2, n s We have x = x m1 m r n 1 n s ) but x / 37 Theorem Let F = f i j ) be a mn sequence of moduli Musielak Then ] if and only if sup u sup i j f i j u)< Proof: Let x

6 170 N Subramanian: Lacunary χ 2 A uv Convergence of p Suppose that hu)=sup i j f i j u) and h=sup u hu) Since f i j u) h for all i, j and u>0, we have for all u,v, h H 2h 1 rs KAθη ε) + hε) H 2 It follows from ε 0 that x Conversely, suppose that sup u sup i j f i j u) = Then we have 0 < u 11 < < u r 1s 1 < u rs <, such that f mr n s u rs ) h rs for r,s 1 Let A=I, unit matrix, define the mn sequence x by putting x i j = u rs if i, j = m 1 m 2 m r n 1 n 2 n s for some r,s = 1,2, and x i j = 0 otherwise Then we have x but x / Acknowledgement: I wish to thank the referee s for their several remarks and valuable suggestions that improved the presentation of the paper References 1] BAltay and FBaşar, Some new spaces of double sequences, J Math Anal Appl, 3091), 2005), ] MBasarir and OSolancan, On some double sequence spaces, J Indian Acad Math, 212) 1999), ] FBaşar and YSever, The space L p of double sequences, Math J Okayama Univ, 51, 2009), ] TJI ABromwich, An introduction to the theory of infinite series Macmillan and CoLtd,New York, 1965) 5] JCannor, On strong matrix summability with respect to a modulus and statistical convergence, Canad Math Bull, 322), 1989), ] PChandra and BCTripathy, On generalized Kothe-Toeplitz duals of some sequence spaces, Indian Journal of Pure and Applied Mathematics, 338) 2002), ] GGoes and SGoes Sequences of bounded variation and sequences of Fourier coefficients, Math Z, 118, 1970), ] AEsi, Lacunary strong A q convergence sequence spaces defined by a sequence of moduli, Kuwait J Sci, Vol 401) 2013), ] AGökhan and RÇolak, The double sequence spaces c P 2 p) and c PB 2 p), Appl Math Comput, 1572), 2004), ] AGökhan and RÇolak, Double sequence spaces l 2, ibid, 1601), 2005), ] MGupta and SPradhan, On Certain Type of Modular Sequence space, Turk J Math, 32, 2008), ] HJHamilton, Transformations of multiple sequences, Duke Math J, 2, 1936), ], A Generalization of multiple sequences transformation, Duke Math J, 4, 1938), ], Preservation of partial Limits in Multiple sequence transformations, Duke Math J, 4, 1939), ] GHHardy, On the convergence of certain multiple series, Proc Camb Phil Soc, ), ] PKKamthan and MGupta, Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c, New York, ] MAKrasnoselskii and YBRutickii, Convex functions and Orlicz spaces, Gorningen, Netherlands, ] JLindenstrauss and LTzafriri, On Orlicz sequence spaces, Israel J Math, ), ] IJMaddox, Sequence spaces defined by a modulus, Math Proc Cambridge Philos Soc, 1001) 1986), ] FMoricz, Extentions of the spaces 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), ] MMursaleen and OHH Edely, Statistical convergence of double sequences, J Math Anal Appl, 2881), 2003), ] MMursaleen, Almost strongly regular matrices and a core theorem for double sequences, J Math Anal Appl, 2932), 2004), ] MMursaleen and OHH Edely,Almost convergence and a core theorem for double sequences, J Math Anal Appl, 2932), 2004), ] HNakano, Concave modulars, J Math Soc Japan, 51953), ] APringsheim, Zurtheorie derzweifach unendlichen zahlenfolgen, Math Ann, 53, 1900), ] GMRobison, Divergent double sequences and series, Amer Math Soc Trans, 28, 1926), ] WHRuckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad J Math, ), ] NSubramanian and UKMisra, The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math, 46, 2010) 30] BCTripathy, On statistically convergent double sequences, Tamkang J Math, 343), 2003), ] BCTripathy and S Mahanta, On a class of vector valued sequences associated with multiplier sequences,acta Math Applicata Sinica Eng Ser), 203) 2004), ] BCTripathy and MSen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang Journal of Mathematics, 372) 2006), ] BCTripathy and AJDutta, On fuzzy real-valued double sequence spaces 2 l p F, Mathematical and Computer Modelling, ) 2007), ] BCTripathy and BSarma, Statistically convergent difference double sequence spaces, Acta Mathematica Sinica, 245) 2008), ] BCTripathy and BSarma, Vector valued double sequence spaces defined by Orlicz function, Mathematica Slovaca, 596) 2009), ] BCTripathy and AJDutta, Bounded variation double sequence space of fuzzy real numbers, Computers and Mathematics with Applications, 592) 2010), ] BCTripathy and BSarma, Double sequence spaces of fuzzy numbers defined by Orlicz function, Acta Mathematica Scientia, 31 B1) 2011),

7 Math Sci Lett 3, No 3, ) / wwwnaturalspublishingcom/journalsasp ] BCTripathy and PChandra, On some generalized difference paranormed sequence spaces associated with multiplier sequences defined by modulus function, Anal Theory Appl, 271) 2011), ] BCTripathy and AJDutta, Lacunary bounded variation sequence of fuzzy real numbers, Journal in Intelligent and Fuzzy Systems, 241) 2013), ] ATurkmenoglu, Matrix transformation between some classes of double sequences, J Inst Math Comp Sci Math Ser, 121), 1999), ] AWilansky, Summability through Functional Analysis, North-Holland Mathematical Studies, North-Holland Publishing, Amsterdam, Vol851984) 42] JYT Woo, On Modular Sequence spaces, Studia Math, 48, 1973), ] MZeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001 Nagarajan Subramanian received the PhD degree in Mathematics for Alagappa University at Karaikudi,Tamil Nadu,India and also getting Doctor of Science DSc) degree in Mathematics for Berhampur University, Berhampur,Odissa,India His research interests are in the areas of summability through functional analysis of applied mathematics and pure mathematics He has published research articles 145 in reputed international journals of mathematical and engineering sciences