On Some New Entire Sequence Spaces

Transcription

1 J. Aa. Num. Theor. 2, No. 2, (2014) 69 Joural of Aalysis & Number Theory A Iteratioal Joural O Some New Etire Sequece Spaces Kuldip Raj 1, ad Ayha Esi 2, 1 School of athematics, Shri ata Vaisho Devi Uiversity, Katra J & K, Idia 2 Departmet of athematics, Sciece ad Art Faculty, Adiyama Uiversity, Adiyama, Turey Received: 24 Feb. 2014, Revised: 26 Apr. 2014, Accepted: 29 Apr Published olie: 1 Jul Abstract: I this paper we itroduce etire sequece spaces ad aalytic sequece spaces o semiormed spaces defied by a usiela-orlicz fuctio ad study some toplological properties ad iclusio relatios betwee these spaces. We also mae a effort to study these sequece spaces over -ormed spaces. Keywords: paraorm space, Orlicz fuctio, usiela-orlicz fuctio, solid, mootoe, etire sequece space, aalytic sequece space. subjclass[2000] 40A05, 40C05, 40D05. 1 Itroductio A Orlicz fuctio : [0, ) [0, ) is a cotiuous, o-decreasig ad covex fuctio such that (0) = 0, (x) > 0 for x > 0 ad (x) as x. Lidestrauss ad Tzafriri [17] used the idea of Orlicz fuctio to defie the followig sequece space. Let w be the space of all real or complex sequeces x=(x ), the l = x w: ( x ) < which is called a Orlicz sequece space. Also l is a Baach space with the orm x =if > 0 : ( x ) 1. Also, it was show i [17] that every Orlicz sequece space l cotais a subspace isomorphic to l p (p 1). The 2 coditio is equivalet to (Lx) L(x), for all L with 0<L<1. A Orlicz fuctio ca always be represeted i the followig itegral form x (x)= η(t)dt 0 where η is ow as the erel of, is right differetiable for t 0, η(0) = 0, η(t) > 0, η is o-decreasig ad η(t) as t. A sequece = ( ) of Orlicz fuctios is called a usiela-orlicz fuctio see ([18],[20]). A sequece N =(N ) of Orlicz fuctios defied by N (v)=sup v u (u) : u 0,,2,... is called the complemetary fuctio of the usiela-orlicz fuctio. For a give usiela-orlicz fuctio, the usiela-orlicz sequece space t ad its subspace h are defied as follows t = x w:i (cx)<, for some c>0, h = x w:i (cx)<, for all c>0, where I is a covex modular defied by I (x)= (x ),x=(x ) t. We cosider t equipped with the Luxemburg orm ( x ) x =if >0:I 1 or equipped with the Orlicz orm 1 ) x 0 = if 1+I (x) : >0. ( Correspodig author uldipraj68@gmail.com, aesi23@hotmail.com

2 70 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces Let X be a liear metric space. A fuctio p : X R is called paraorm, if 1.p(x) 0, for all x X, 2.p( x)= p(x), for all x X, 3.p(x+y) p(x)+ p(y), for all x,y X, 4.if (λ ) is a sequece of scalars with λ λ as ad (x ) is a sequece of vectors with p(x x) 0 as, the p(λ x λ x) 0 as. A paraorm p for which p(x) = 0 implies x = 0 is called total paraorm ad the pair (X, p) is called a total paraormed space. It is well ow that the metric of ay liear metric space is give by some total paraorm (see [28], Theorem , P-183). For more details about sequece spaces see( [1], [3], [5], [15], [16], [21], [22], [23], [24], [25], [26], [27]). A complex sequece, whose th term is x is deoted by (x ). Let ϕ be the set of all fiite sequeces. A sequece x=(x ) is said to be aalytic if sup x 1 <. The vector space of all aalytic sequeces will be deoted by Λ. A sequece x is called etire sequece if lim x 1 = 0. The vector space of all etire sequeces will be deoted by Γ. Let σ be a oe-oe mappig of the set of positive itegers ito itself such that σ m ()=σ(σ m 1 ()),m=1,2,3,. A cotiuous liear fuctioal φ o Λ is said to be a ivariat mea or a σ mea if ad oly if 1.φ(x) 0whe the sequece x=(x ) has x 0 for all, 2.φ(e)=1 where e=(1,1,1, ) ad 3.φ(x σ() )=φ(x ) for all x Λ. For certai ids of mappigs σ, every ivariat mea φ exteds the limit fuctioal o the space C of all coverget sequeces i the sese that φ(x)= limx for all x C. Cosequetly C V σ, where V σ is the set of aalytic sequeces all of those σ meas are equal. If x=(x ), set T x=(tx) 1 =(x σ() ). It ca be show that V σ = x=(x ) : lim m t m (x ) 1 = L uiformly i, L=σ lim (x ) 1, where t m (x)= (x + Tx + + T m x ) 1 m+1 Give a sequece x =x its th sectio is the sequece x () =x 1,x 2, x,0,0,, δ () =(0,0,,1,0,0, ), i the th place ad zeros elsewhere. The space ( cosistig ) of all those sequeces x i w such that x 1/ 0 as for some arbitrary fixed > 0 is deoted by Γ ad is ow as usiela-orlicz space of etire sequeces. The space Γ is a metric space ( x y 1/ ) with the metric d(x, y) = sup for all x=x ad y=y i Γ. The space cosistig of all those sequeces x i w such. ( ( x that sup ( 1/ ))) < for some arbitrarily fixed > 0 is deoted by Λ ad is ow as usiela-orlicz space of aalytic sequeces. A sequece space E is said to be solid or ormal if (α x ) E wheever (x ) E ad for all sequeces of scalars (α ) with α 1 (see [20]). The followig iequality will be used throughout the paper. Let p=(p ) be a sequece of positive real umbers with 0 p sup p = G, K = max(1,2 G 1 ) the a + b p K a p + b p for all ad a,b C. (1.1) Also a p max(1, a G ) for all a C. Let =( ) be a usiela-orlicz fuctio, X be locally covex Hausdorff topological liear space whose topology is determied by a set of cotiuous semiorms q. The symbol Λ(X), Γ(X) deotes the space of all aalytic ad etire sequeces recpectively defied over X. I this paper we defie the followig classes of sequeces: Λ (p,σ,q,s)= x Λ(x) : sup s[ ( xσ () 1 ))] p < uiformly i, 0, s 0 ad for some > 0, Γ (p,σ,q,s)= x Γ(x) : s[ ( xσ () 1 ))] p 0 as uiformly i 0, s 0 ad for some > 0. If we tae p=(p )=1, we get Λ (σ,q,s)= x Λ(x) : sup s[ ( xσ () 1, Γ (σ,q,s)= x Γ(x) : 0, s 0 ad for some > 0, ))] < uiformly i s[ ( xσ () 1 ))] 0 as. uiformly i 0, s 0 ad for some > 0 The mai purpose of this paper is to study some etire ad aalytic sequece spaces o semiormed spaces defied by a usiela-orlicz fuctio = ( ). We study some topological properties ad iclusio relatios betwee the spaces Λ (p,σ,q,s) ad Γ (p,σ,q,s) i the secod sectio of this paper. I the third sectio we mae a effort to study some properties of these sequece spaces over -ormed spaces. 2 Some topological properties of spaces Λ (p,σ,q,s) ad Γ (p,σ,q,s) Theorem 2.1Let = ( ) be a usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive

3 J. Aa. Num. Theor. 2, No. 2, (2014) / 71 real umbers. The the spaces Γ (p,σ,q,s) ad Λ (p,σ,q,s) are liear spaces over the field of complex umbers C. Proof. Let x=(x ), y=(y ) Γ (p,σ,q,s). The there exist positive umbers 1 ad 2 such that ad s[ ( xσ () 1 ))] p 0 as 1 s[ ( yσ () 1 ))] p 0 as. 2 Let 3 = max(2 α 1,2 β 2 ). Sice = ( ) is o decreasig, covex ad q is a semiorm so by usig iequality(1.1), we have s[ ( αx σ () + β y σ () 1 3 )) K + K ] p s[ ( αxσ () + β y σ () ) 1 )] p ( 2 p s[ ( xσ q () 1 )) 1 ( yσ + () 1 ))] p 2 s[ ( xσ () 1 )) ( yσ + () 1 ))] p as. s[ ( xσ () 1 ))] p 1 s[ ( yσ () 1 ))] p 2 Thus αx+β y Γ (p,σ,q,s). Hece Γ (p,σ,q,s) is a liear space. Similarly, we ca show that Λ (p,σ,q,s) is a liear space. Theorem 2.2Suppose = ( ) is usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive real umbers. The the space Γ (p,σ,q,s) is a paraormed space with the paraorm defied by g(x) = if pm : sup s[ ( x σ () 1 ] p )) 1 1, uiformly i > 0, > 0, where = max(1,sup p ). Proof. Clearly g(x) 0,g(x) = g( x) ad g(θ) = 0, where θ is the zero sequece of X. Let (x ), (y ) Γ (p,σ,q,s). Let 1, 2 > 0 be such that sup s[ ( xσ () 1 ))] p 1 1 ad sup s[ ( yσ () 1 ))] p 1. 1 Let = ad by usig iowsi s iequality, we have sup s[ ( x σ () + y σ () 1 ))] p 1 1 Hece g(x+y) sup 1 if ( ) pm : sup s[ ( x σ () + y σ () if ( 1 ) pm : sup s[ ( x σ () if ( 2 ) pm : sup s[ ( y σ () s[ ( x σ () 1 ))] p 1 2 sup s[ ( y σ () 1 ))] p ))] p 1, 1, 2 > 0,m N ))] p 1, 1 > 0, m N ))] p 1, 2 > 0,m N. Thus we have g(x+y)g(x)+g(y). Hece g satisfies the triagle iequality. Now g(λx) = if () pm = if (r λ ) pm : sup s[ ( x σ () 1 ))] p 1, > 0, m N 1 : sup s[ ( x σ () 1 ))] p 1, r> 0, m N, 1 where r= λ. Hece Γ (p,σ,q,s) is a paraormed space. Theorem 2.3Let = ( ) be a usiela-orlicz fuctio. The Γ (p,σ,q,s) Λ (p,σ,q,s) Γ (p,σ,q,s). Proof. The proof is trivial so we omit. Theorem 2.4Γ (p,σ,q,s) Λ (p,σ,q,s). Proof. The proof is trivial so we omit. Theorem 2.5Let 0 p r ad let r p be bouded. The Γ (r,σ,q,s) Γ (p,σ,q,s). Proof. Let x Γ (r,σ,q,s). The s[ ( xσ () 1 ))] r 0 as. (2.1) Let t = s[ ( xσ () 1 ))] q ad λ = p r. Sice p r, we have 0λ 1. Tae 0<λ < λ. Defie t, if t 1 u = 0, if t < 1

4 72 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces ad t = u +v, v λ 0, if t 1 v = t, if t < 1 t λ v λ. Sice tλ = u λ +vλ. It follows that uλ = u λ + vλ, the tλ u t, t + v λ. Now s[[ ( x σ () 1 ))] r ] λ s[ ( ( x σ () 1 ))] r q = s[ ( x σ () 1 )) r ] p /r s[ ( ( x σ () 1 ))] r q But = s[ ( x σ () 1 ))] p s[ ( ( x σ () 1 ))] r. q s[ ( xσ () 1 ))] r 0 as (by(2.1)). Therefore s[ ( xσ () 1 ))] p 0 as. Hece x Γ (p,σ,q,s). From (2.1), we get Γ (r,σ,q,s) Γ (p,σ,q,s). Theorem 2.6(i) Let 0 < if p p 1. The Γ (p,σ,q,s) Γ (σ,q,s), (ii) let 1 p sup p <. The Γ (σ,q,s) Γ (p,σ,q,s). Proof.(i) Let x Γ (p,σ,q,s). The s[ ( xσ () 1 ))] p 0 as. (2.2) Sice 0<if p p 1, s[ ( xσ () 1 ))] s[ ( xσ () 1 ))] p 0 as. (2.3) From (2.2) ad (2.3) it follows that, x Γ (σ,q,s). Thus Γ (p,σ,q,s) Γ (σ,q,s). (ii) Let p 1 for each ad sup p < ad let x Γ (σ,q,s). The s[ ( xσ () 1 ))] 0 as. (2.4) Sice 1 p sup p <, we have s[ ( xσ () 1 ))] p s[ ( ( xσ q () 1 ))] s[ ( xσ () 1 ))] p 0 as. This implies that x Γ (p,σ,q,s). Therefore Γ (σ,q,s) Γ (p,σ,q,s). Theorem 2.7Suppose s[ ( xσ () 1 ))] p x 1/, the Γ Γ (p,σ,q,s). Proof. Let x Γ. The we have, x 1/ 0 as. (2.5) But s[ ( xσ () 1 ))] p x 1/, by our assumptio, implies that s[ ( xσ () 1 ))] p 0 as. by(2.5) The x Γ (p,σ,q,s) ad Γ Γ (p,σ,q,s). Theorem 2.8Γ (p,σ,q,s) is solid. Proof. Let x y ad let y = (y ) Γ (p,σ,q,s), because =( ) is o-decreasig s[ ( xσ () 1 ))] p s[ ( ( yσ q () 1 ))] p. Sice y Γ (p,σ,q,s). Therefore, ad hece s[ ( ( yσ () 1 ))] p 0 as s[ ( xσ () 1 ))] p 0 as. Therefore x=x Γ (p,σ,q,s). Theorem 2.9Γ (p,σ,q,s) is mootoe. Proof. The proof is trivial. 3 Sequece spaces over - ormed spaces The cocept of 2-ormed spaces was iitially developed by Gähler[11] i the mid of 1960 s, while that of -ormed spaces oe ca see i isia[19]. Sice the, may others have studied this cocept ad obtaied various results, see Guawa ([12,[13]) ad Guawa ad ashadi [14]. Let N ad X be a liear space over the field R, where R is field of reals of dimesio d, where d 2. A real valued fuctio,, o X satisfyig the followig four coditios: 1. x 1,x 2,,x = 0 if ad oly if x 1,x 2,,x are liearly depedet i X;

5 J. Aa. Num. Theor. 2, No. 2, (2014) / x 1,x 2,,x is ivariat uder permutatio; 3. αx 1,x 2,,x = α x 1,x 2,,x for ay α R, ad 4. x+x,x 2,,x x,x 2,,x + x,x 2,,x is called a -orm o X, ad the pair (X,,, ) is called a -ormed space over the field R. For example, we may tae X = R beig equipped with the -orm x 1,x 2,,x E = the volume of the -dimesioal parallelopiped spaed by the vectors x 1,x 2,,x which may be give explicitly by the formula x 1,x 2,,x E = det(x i j ), where x i = (x i1,x i2,,x i ) R for each i = 1,2,,. Let(X,,, ) be a -ormed space of dimesio d 2 ada 1,a 2,,a be liearly idepedet set i X. The the fuctio,, o X 1 defied by If we tae p=(p )=1, we get x 1,x 2,,x 1 = max x 1,x 2,,x 1,a i : i=1,2,, Λ (σ,q,s,.,,. ) = is ow as a ( 1)-orm o X with respect to a 1,a 2,,a. Let N ad X be a real vector space of dimesio d, where 2 d. Let β 1 be the collectio of liearly idepedet sets B with 1 elemets. For B β 1, let us defie q B (x 1 )= x 1,x 2, x, x 1 X. The q B is a semiorm o X ad the family q=q B : B β 1 of semiorms geerates a locally covex topology o X. The semiorms q B have the followig properties: 1.er B )=the liear spa of B. 2.For B β 1, y Bad x X\the liear spa of B we have q B x\y (y)=q B (x). See ([10]) A sequece(x ) i a -ormed space(x,,, ) is said to coverge to some L X if lim x L,z 1,,z 1 =0 for every z 1,,z 1 X. A sequece(x ) i a -ormed space(x,,, ) is said to be Cauchy if lim x x p,z 1,,z 1 =0 for every z 1,,z 1 X.,p If every Cauchy sequece i X coverges to some L X, the X is said to be complete with respect to the -orm. Ay complete -ormed space is said to be -Baach space. For more details about ormed spaces oe ca see ([2], [4], [6], [7], [8], [9]) ad refereces therei. Let = ( ) be a usiela-orlicz fuctio, X be locally covex Hausdorff topological real liear ormed space whose topology is determied by a set of cotiuous semiorms q. The symbol Λ(X), Γ(X) deotes the space of all aalytic ad etire sequeces respectively defied over X. I this sectio, for each z 1,,z 1 X we defie the followig classes of sequeces: Λ (p,σ,q,s,.,,. )= x Λ(x) : sup s[ ( (x σ () ) 1, Γ (p,σ,q,s,.,,. ) = s[ ( (x σ () ) 1 ))] p,z 1,,z 1 < uiformly i 0, s 0 for some > 0, x Γ(x) : ))] p,z 1,,z 1 0 as uiformly i 0, s 0 for some > 0. sup s[ ( (x σ () ) 1 ))],z 1,,z 1, 0 for some > 0, Γ (σ,q,s,.,,. ) = s[ ( (x σ () ) 1 x Λ(x) : < uiformly i 0, s x Γ(x) : ))],z 1,,z 1 0 as uiformly i 0, s 0 for some > 0. I the preset sectio we study some topological properties of the spaces Λ (p,σ,q,s,.,,. ) ad Γ (p,σ,q,s,.,,. ) ad also examie some iclusio relatio betwee these spaces. Theorem 3.1Let = ( ) be a usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive real umbers. The the spaces Γ (p,σ,q,s,.,,. ) ad Λ (p,σ,q,s,.,,. ) are liear space over the field of real umbersr. Proof. Let x = (x ), y = (y ) Γ (p,σ,q,s,.,,. ). The there exist positive umbers 1 ad 2 such that ad s[ ( (x σ () ) 1 ))] p,z 1,,z 1 0 as 1 s[ ( (y σ () ) 1 ))] p,z 1,,z 1 0 as. 2 Let 3 = max(2 α 1,2 β 2 ). Sice = ( ) is o decreasig, covex ad q is a semiorm ad by usig

6 74 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces iequality(1.1), we have s[ ( (αxσ () + β y σ () ) 1 ) 3 s[ ( α(xσ () ) + (y σ () ) 3 3 ) 1 1 ( 2 p s[ q ( (x σ () ) 1 )),z 1,,z ( (y σ () ) 1 ))] p,z 1,,z 1 2 s[ ( (x σ () ) 1 )),z 1,,z ( (y σ () ) 1 ))] p,z 1,,z 1 2 ] p,z 1,,z 1 K s[ ( (x σ () ) 1 ))] p,z 1,,z K s[ ( (y σ () ) 1 ))] p,z 1,,z as. )] p,z 1,,z 1 Thus αx + β y Γ (p,σ,q,s,.,,. ). Hece Γ (p,σ,q,s,.,,. ) is a liear space. Similarly, we ca prove Λ (p,σ,q,s,.,,. ) is a liear space. Theorem 3.2Suppose = ( ) is usiela-orlicz fuctio ad p = (p ) be a sequece of strictly positive real umbers. The the space Γ (p,σ,q,s,.,,. ) is a paraormed space with the paraorm defied by g(x) = if pm : sup s[ ( (xσ ()) 1 )] p,z 1,,z 1 ) 1, 1 uiformly i > 0, > 0, where = max(1,sup p ). Proof. Clearly g(x) 0,g(x) = g( x) ad g(θ) = 0, where θ is the zero sequece of X. Let (x ), (y ) Γ (p,σ,q,s,.,,. ). Let 1, 2 > 0 be such that ad sup s[ ( (x σ () ) 1 ))] p,z 1,,z sup s[ ( (y σ () ) 1 ))] p,z 1,,z The sup s[ ( (x σ () + y σ () ) ))] p,z 1,,z 1 1 sup s[ ( (x σ () ) 1 ))] p,z 1,,z sup s[ ( (y σ () ) 1 ))] p,z 1,,z Hece g(x+y) if 1 ( ) pm : sup 1 1, 2 > 0, m N if ( 1 ) pm : sup 1 1 > 0, m N + if ( 2 ) pm : sup 1. 2 > 0, m N s[ ( (x σ () + y σ () ) 1 ))] p,z 1,,z 1 1, s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1, s[ ( (y σ () ) 1 ))] p,z 1,,z 1 1, Thus we have g(x+y)g(x)+g(y). Hece g satisfies the triagle iequality. Now g(λ x) = if () pm = if (r λ ) pm : sup s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1, > 0, m N 1 : sup s[ ( (x σ () ) 1 ))] p,z 1,,z 1 1,r > 0,m N, 1 where r = λ. Hece Γ (p,σ,q,s,.,,. ) is paraormed space. Theorem 3.3Let = ( ) be a usiela-orlicz fuctio. The Γ (p,σ,q,s,.,,. ) Λ (p,σ,q,s,.,,. ) Γ (p,σ,q,s,.,,. ). Proof. It is easy to prove so we omit the proof. Theorem 3.4 Γ (p,σ,q,s,.,,. ) Λ (p,σ,q,s,.,,. ). Proof. It is easy to prove so we omit the proof. Theorem 3.5Γ (p,σ,q,s,.,,. ) is solid. Proof. Let x y ad let y = (y ) Γ (p,σ,q,s,.,,. ), sice = ( ) is o-decreasig, so s[ ( (x σ ()) 1 ))] p,z 1,,z 1 s[ ( (y σ ()) 1 p.,z 1,,z 1 ))] a

7 J. Aa. Num. Theor. 2, No. 2, (2014) / 75 Sice y Γ (p,σ,q,s,.,,. ). Therefore, s[ ( (y σ () ) 1 p,z 1,,z 1 ))] 0 as. So that s[ ( (x σ () ) 1 p,z 1,,z 1 ))] 0 as. Therefore x = (x ) Γ (p,σ,q,s,.,,. ). Hece Γ (p,σ,q,s,.,,. ) is solid. Theorem 3.6Γ (p,σ,q,s,.,,. ) is mootoe. Proof. The proof is trivial so we omit it. Refereces [1] H. Dutta ad B.S. Reddy, O some sequece spaces, Tamsui Oxford Joural of Iformatio ad athematical Scieces, 28 (1) (2012), pp [2] H. Dutta, A Orlicz extesio of differece sequeces o real liear -ormed spaces, Joural of Iequalities ad Applicatios, 2013 (2013), art. o [3] H. Dutta ad F. Başar, A geeralizatio of Orlicz sequece spaces by Ces`sro mea of order oe, Acta athematica Uiversitatis Comeiaae, 80(2) (2011), pp [4] H. Dutta ad B.S. Reddy, O o-stadard -orm o some sequece spaces, It. J. Pure Appl. ath., 68(1) (2011), pp [5] H. Dutta ad T. Bilgi, Strogly (V λ,a, vm, p)-summable sequece spaces defied by a Orlicz fuctio, Applied athematics Letters, 24(7) (2011), pp [6] H. Dutta, B.S. Reddy ad S.S. Cheg, Strogly summable sequeces defied over real ormed spaces, Applied athematics E - Notes, 10(2010), pp [7] H. Dutta, O -ormed liear space valued strogly (C, 1)- summable differece sequeces, Asia-Europea Joural of athematics, 3(4) (2010), pp [8] H. Dutta, O sequece spaces with elemets i a sequece of real liear -ormed spaces, Applied athematics Letters, 23(9) (2010), pp [9] H. Dutta, A applicatio of lacuary summability method to -orm, Iteratioal Joural of Applied athematics ad Statistics, 15(09) (2009), pp [10] H. Dutta, O sequece spaces with elemets i a sequece of real liear -ormed spaces, Applied athematics Letters, 23(9) (2010), pp [11] S. Gahler, Liear 2-ormietre Rume, ath. Nachr., 28 (1965), pp [12] H. Guawa, O -Ier Product, -Norms, ad the Cauchy-Schwartz Iequality, Sci. ath. Jap., 5 (2001), pp [13] H. Guawa, The space of p-summable sequece ad its atural -orm, Bull. Aust. ath. Soc., 64 (2001), pp [14] H. Guawa ad., ashadi,o -ormed spaces, It. J. ath. ath. Sci., 27 (2001), pp [15] P. K. Kamtha ad. Gupta, Sequece spaces ad series, Lecture Notes i Pure ad Applied athematics, 65 arcel Deer, Ic., New Yor,(1981). [16] V. Karaaya ad H. Dutta, O some vector valued geeralized differece modular sequece spaces, Filomat, 25(3) (2011), pp [17] J. Lidestrauss ad L. Tzafriri, O Orlicz sequece spaces, Israel J. ath; 10, (1971). [18] L. aligrada, Orlicz spaces ad iterpolatio, Semiars i athematics 5, Polish Academy of Sciece, [19] A. isia, -ier product spaces, ath. Nachr., 140 (1989), pp [20] J. usiela, Orlicz spaces ad modular spaces, Lecture Notes i athematics, 1034,(1983). [21] S. D. Prashar ad B. Choudhary, Sequece spaces defied by Orlicz fuctios, Idia J. Pure Appl. ath. 25(14) (1994), [22] K. Raj, A. K. Sharma ad S. K. Sharma, A Sequece space defied by usiela-orlicz fuctios, It. J. Pure Appl. ath., 67 (2011), [23] K. Raj, S. K. Sharma ad A. K. Sharma, Differece sequece spaces i -ormed spaces defied by usiela-orlicz fuctios, Arme. J ath., 3 (2010), pp [24] K. Raj ad S. K. Sharma, Geeralized differece sequece spaces defied by usiela-orlicz fuctio, Iteratioal J. of ath. Sci. & Egg. Appls., 5 (2011), pp [25] K. Raj ad S. K. Sharma, Some differece sequece spaces defied by sequece of modulus fuctio, It. Joural of athematical Archive, 2 (2011), pp [26] B. C. Tripathy ad H. Dutta, Some differece paraormed sequece spaces defied by Orlicz fuctios, Fasciculi athematici, Nr 42 (2009), [27] B.C. Tripathy, ad H. Dutta, O some lacuary differece sequece spaces defied by a sequece of orlicz fuctios ad q-lacuary m statistical Covergece, Aalele Stiitifice ale Uiversitatii Ovidius Costata, Seria atematica, 20(1) (2012), pp [28] A. Wilasy, Summability through Fuctioal Aalysis, North- Hollad ath. Stud. (1984). Kuldip Raj ia a assistat professor, wors atschool of athematics, Shri ata Vaisho Devi Uiversity, Katra , J&K, Idia. His qualificatio is Ph. D. (1999, athematics) i the area of Fuctioal aalysis, Operator theory, Sequece, Series ad Summability.His total teachig experiece is 15 years. His research experiece is 20 years. He has 100 published papers.

8 76 Kuldip Raj, Ayha Esi: O Some New Etire Sequece Spaces Ayha Esi was bor i Istabul, Turey, o arch 5, Ayha Esi got his B.Sc. from Iou Uiversity i 1987 ad. Sc. ad Ph.D. degree i pure mathematics from Elazig Uiversity, Turey i 1990 ad 1995, respectively. His research iterests iclude Summability Theory, Sequeces ad Series i Aalysis ad Fuctioal Aalysis. I 2000, Esi was appoited to Educatio Faculty i Gaziatep Uiversity. I 2002, Esi was appoited as the head of Departmet of athematics i Sciece ad Art Faculty i Adiyama of the Iou Uiversity. I 2006, Esi joied the Departmet of athematics of Adiyama Uiversity.