Research Article Generalized Vector-Valued Sequence Spaces Defined by Modulus Functions

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1 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 00, Article ID 45789, 7 pages doi:0.55/00/45789 Research Article Geeralized Vector-Valued Sequece Spaces Defied by Modulus Fuctios Mahmut Işik Departmet of Statistics, Firat Uiversity, 39 Elazığ, Turkey Correspodece should be addressed to Mahmut Işik, misik63@yahoo.com Received 7 Jue 00; Accepted 6 December 00 Academic Editor: Alberto Cabada Copyright q 00 Mahmut Işik. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We itroduce the vector-valued sequece spaces w Δ m,f,q,p,u, w Δ m,f,q,p,u, ad w 0 Δ m,f,q,p,u, S q u ad S q 0u, usig a sequece of modulus fuctios ad the multiplier sequece u u k of ozero complex umbers. We give some relatios related to these sequece spaces. It is also show that if a sequece is strogly Δ m u q -Cesàro summable with respect to the modulus fuctio f the it is Δ m u q -statistically coverget.. Itroductio Let w be the set of all sequeces real or complex umbers ad l, c, adc 0 be, respectively, the Baach spaces of bouded, coverget, ad ull sequeces x x k with the usual orm x sup x k,wherek N {,,...}, the set of positive itegers. The studies o vector-valued sequece spaces are doe by Das ad Choudhury, Et, Etetal. 3, Leoard 4, Rath ad Srivastava 5, J.K.SrivastavaadB.K.Srivastava 6, Tripathyetal. 7, 8, ad may others. Let E k,q k be a sequece of semiormed spaces such that E E k for each k N. We defie w E {x x k : x k E k for each k N}.. It is easy to verify that w E is a liear space uder usual coordiatewise operatios defied by x y x k y k ad αx αx k,whereα C. Let u u k be a sequeces of ozero scalar. The for a sequece space E, the multiplier sequece space E u, associated with the multiplier sequece u, isdefiedas E u { x k w : u k x k E}.

2 Joural of Iequalities ad Applicatios The otio of a modulus was itroduced by Nakao 9.Werecallthatamodulusf is afuctiofrom 0, to 0, such that i f x 0 if ad oly if x 0, ii f x y f x f y for x, y 0, iii f is icreasig, iv f is cotiuous from the right at 0. It follows that f must be cotiuous everywhere o 0,. Maddox 0 ad Ruckle used a modulus fuctio to costruct some sequece spaces. After the some sequece spaces defied by a modulus fuctio were itroduced ad studied by Bilgi, Pehliva ad Fisher 3, Waszak 4, Bhardwaj 5, Altı 6, ad may others. The otio of differece sequece spaces was itroduced by Kızmaz 7 ad it was geeralized by Et ad Çola8.Letm be a fixed positive iteger. The we write X Δ m {x x k : Δ m x k X}. for X l, c ad c 0,wherem N, Δ m x Δ m x k Δ m x, Δ 0 x x k ad so we have m Δ m x k ν m ν x k ν. ν 0.3. Mai Results I this sectio, we prove some results ivolvig the sequece spaces w 0 Δ m,f,q,p,u, w Δ m,f,q,p,u,adw Δ m,f,q,p,u. Defiitio.. Let E k,q k be a sequece of semiormed spaces such that E E k for each k N, p a sequece of strictly positive real umbers, Q q k asequeceof semiorms, F f k a sequece of modulus fuctios, ad u u k ay fixed sequece of ozero complex umbers u k. We defie the followig sequece spaces: w 0 Δ m,f,q,p,u ) { x x k : x k E k : [ fk qk u k Δ m x k )] } 0, as, w Δ m,f,q,p,u ) { x x k : x k E k : [ fk qk u k Δ m x k l )] 0, as,l E k }, w Δ m,f,q,p,u ) { [ x x k : x k E k :sup fk qk u k Δ m x k )] } <..

3 Joural of Iequalities ad Applicatios 3 Throughout the paper Z will deote ay oe of the otatio 0, or. If f k f ad q k q for all k N, we will write w Z Δ m,f,q,p,u istead of w Z Δ m,f,q,p,u. If f x x ad for all k N, we will write w Z Δ m,q,u istead of w Z Δ m,f,q,p,u. If x w Δ m,f,q,p,u, we say that x is strogly Δ m u q -Cesàro summable with respect to the modulus fuctio f ad we will write x k l w Δ m,f,q,p,u ad l will be called Δ m u q -limit of x with respect to the modulus f. The proofs of the followig theorems are obtaied by usig the kow stadard techiques; therefore, we give them without proofs. Theorem.. Let the sequece be bouded. The the spaces w Z Δ m,f,q,p,u are liear spaces. Theorem.3. Let f be a modulus fuctio ad the sequece be bouded; the w 0 Δ m,f,q,p,u ) w Δ m,f,q,p,u ) w Δ m,f,q,p,u ). ad the iclusios are strict. Theorem.4. w 0 Δ m,f,q,p,u is a paraormed eed ot total paraorm) space with [ g Δ x sup fk qk u k Δ m x k )] ) /M,.3 where M max, sup. Theorem.5. Let F f k ad G g k be ay two sequeces of modulus fuctios. For ay bouded sequeces p ad t t k of strictly positive real umbers ad for ay two sequeces of semiorms q q k ad r r k,wehave i w Z Δ m,f,q,u w Z Δ m,f g,q,u ; ii w Z Δ m,f,q,p,u w Z Δ m,f,r,p,u w Z Δ m,f,q R, p, u ; iii w Z Δ m,f,q,p,u w Z Δ m,g,q,p,u w Z Δ m,f G, Q, p, u ; iv If q k is stroger tha r k for each k N, thew Z Δ m,f,q,p,u w Z Δ m,f,r,p,u ; v If q k equivalet to r k for each k N, thew Z Δ m,f,q,p,u w Z Δ m,f,r,p,u ; vi w Z Δ m,f,q,p,u w Z Δ m,f,r,p,u /. Proof. i We will oly prove i for Z 0 ad the other cases ca be proved by usig similar argumets. Let ε>0 ad choose δ with 0 <δ<suchthatf t <εfor 0 t δ ad for all k N. Writey k g q k u k Δ m x k ad cosider [ )] [ )] [ )] f yk f yk f yk,.4

4 4 Joural of Iequalities ad Applicatios where the first summatio is over y k δ ad secod summatio is over y k >δ.sicef is cotiuous, we have [ )] f yk <ε..5 By the defiitio of f,wehavefory k >δ, f y k ) < f y k δ..6 Hece [ )] f yk δ f y k..7 From.5 ad.7,weobtaiw 0 Δ m,f,q,u w 0 Δ m,f g,q,u. The followig result is a cosequece of Theorem.5 i. Corollary.6. Let f be a modulus fuctio. The w Z Δ m,q,u w Z Δ m,f,q,u. Theorem.7. Let 0 < t k ad t k / be bouded; the w Z Δ m,f,q,t,u w Z Δ m,f,q,p,u. Proof. If we take w k f k q k u k Δ m x k t k for all k ad usig the same techique of Theorem 5 of Maddox 9, itiseasytoprovethetheorem. Theorem.8. Let f be a modulus fuctio; if lim t f t /t β>0, thew Δ m,q,p,u w Δ m,f,q,p,u. Proof. Omitted. 3. Δ m u q -Statistical Covergece The otio of statistical covergece were itroduced by Fast 0 ad Schoeberg, idepedetly. Over the years ad uder differet ames, statistical covergece has bee discussed i the theory of Fourier aalysis, ergodic theory, ad umber theory. Later o it was further ivestigated from the sequece space poit of view ad liked with summability theory by Šalát, Fridy 3, Coor 4, Mursalee 5, Işik 6, Malkowsky ad Savas 7, ad may others. I recet years, geeralizatios of statistical covergece have appeared i the study of strog itegral summability ad the structure of ideals of bouded cotiuous fuctios o locally compact spaces. Statistical covergece ad its geeralizatios are also coected with subsets of the Stoe-Čech compactificatio of the atural umbers. Moreover, statistical covergece is closely related to the cocept of covergece i probability. The otio depeds o the desity of subsets of the set N of atural umbers.

5 Joural of Iequalities ad Applicatios 5 AsubsetE of N is said to have desity positive itegers which is defied by δ E if δ E lim χ E k exists, 3. where χ E is the characteristic fuctio of E. It is clear that ay fiite subset of N have zero atural desity ad δ E c δ E. I this sectio, we itroduce Δ m u q -statistically coverget sequeces ad give some iclusio relatios betwee Δ m u q -statistically coverget sequeces ad w f, q, p, u summable sequeces. Defiitio 3.. Asequecex x k is said to be Δ m u q -statistically coverget to l if for every ε>0, δ { k N : q u k Δ m x k l ε }) I this case, we write x k l S q u Δ m.thesetofallδ m u q -statistically coverget sequeces is deoted by S q u Δ m.ithecasel 0, we will write S q 0u Δm istead of S q u Δ m. Theorem 3.. Let f be a modulus fuctio; the i If x k l w Δ m,q,u, thex k l S q u Δ m ; ii If x l Δ m u q ad x k l S q u Δ m, thex k l w Δ m,q,u ; iii S q u Δ m l Δ m u q w Δ m,q,u l Δ m u q, where l Δ m u q {x w X :su q u k Δ m x k < }. Proof. Omitted. I the followig theorems, we will assume that the sequece p is bouded ad 0 <h if k su H<. Theorem 3.3. Let f be a modulus fuctio. The w Δ m,f,q,p,u S q u Δ m. Proof. Let x w Δ m,f,q,p,u ad let ε>0begive. Let ad deote the sums over k with q u k Δ m x k l ε ad q u k Δ m x k l <ε, respectively. The [ f q uk Δ m x k l )] [ f q uk Δ m x k l )] [ ] pk f ε [f ε ] h, [ ] H ) mi f ε { k : q u k Δ m x k l ε } [f ε ] h, [ ] H mi f ε ). 3.3 Hece x S q u Δ m.

6 6 Joural of Iequalities ad Applicatios Theorem 3.4. Let f be bouded; the S q u Δ m w Δ m,f,q,p,u. Proof. Suppose that f is bouded. Let ε>0ad ad be deoted i previous theorem. Sice f is bouded, there exists a iteger K such that f x <K,forallx 0. The [ f q uk Δ m x k l )] [ f q uk Δ m x k l )] [ f q uk Δ m x k l )] ) max K h,k H) [ ] pk f ε max K h,k H) { k : q u k Δ m x k l ε } max f ε h,f ε H). 3.4 Hece x w Δ m,f,q,p,u. Theorem 3.5. S q u Δ m w Δ m,f,q,p,u ifadolyiff is bouded. Proof. Let f be bouded. By Theorems 3.3 ad 3.4,wehaveS q u Δ m w Δ m,f,q,p,u. Coversely suppose that f is ubouded. The there exists a sequece t k of positive umbers with f t k k,fork,,... If we choose t k, i k,i,,..., u i Δ m x i 0, otherwise, 3.5 the we have {k : u kδ m x k ε} 3.6 for all ad so x S q u Δ m,butx / w Δ m,f,q,p,u for X C, q x x ad forall k N. This cotradicts to S q u Δ m w Δ m,f,q,p,u. Refereces N. R. Das ad A. Choudhury, Matrix trasformatio of vector valued sequece spaces, Bulleti of the Calcutta Mathematical Society, vol. 84, o., pp , 99. M. Et, Spaces of Cesàro differece sequeces of order r defied by a modulus fuctio i a locally covex space, Taiwaese Joural of Mathematics, vol. 0, o. 4, pp , M. Et, A. Gökha, ad H. Altiok, O statistical covergece of vector-valued sequeces associated with multiplier sequeces, Ukraïs kiĭ Matematichiĭ Zhural, vol. 58, o., pp. 5 3, 006, traslatio i Ukraiia Mathematical Joural, vol. 58, o., pp , I. E. Leoard, Baach sequece spaces, Joural of Mathematical Aalysis ad Applicatios, vol. 54, o., pp , 976.

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