Spaces of Generalized difference Lacunary I-convergent sequences

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1 NTSCI 3, No. 3, ) New Teds i athematical Scieces Spaces of Geealized diffeece Lacuay I-coveget sequeces Adem Kiliçma ad Stuti Bogohai 2 Depatmet of athematics ad Istitute fo athematical Reseach Uivesity of Puta alaysia, Sedag, Selago 43400, alaysia 2 Depatmet of athematics, Idia Istitute of Techology, Bombay, Powai:400076, umbai, ahaashta, Idia. Received: 20 Jauay 205, Revised: 8 Apil 205, Accepted: 9 ay 205 Published olie: 7 Jue 205 Abstact: I this eseach aticle, we studied some ew geealized diffeece stogly summable lacuay I-coveget -omed sequece spaces elated to l p spaces which ae defied by Olicz fuctios. Some esults ivolved with these spaces ae also ivestigated ad studied. We also give some elatios elated to these sequece spaces. Keywods: Lacuay sequece, -om, Olicz fuctio; Diffeece opeato; Ideal Covegece; de la Vallèe Poussi mea. Itoductio The cocept of the cisp set sequece space mϕ) was iitiated by W.L.C. Saget which was late o studied ad ivestigated fom the sequece space poit of view by may othe mathematicias. Recetly, some eseaches woed o some matix classes chaacteized by taig mϕ) as oe membe. Kostyo [3] itoduced the cocept of Ideal covegece as a geealizatio fom of statistical covegece. A lacuay sequece is defied as a iceasig itege sequece θ = ) such that 0 = 0 ad h = as. Fo ay lacuay sequece θ = ), the space N θ is defied as, N θ = x ) : lim h J x L = 0, fo some L. The space N θ is a BK space with the om, x ) θ = h J x. Note: Thoughout this pape, the itevals detemied by θ will be deoted by J =, ] ad the atio defied by ϕ. will be Coespodig autho ailicma@puta.upm.edu.my; stutibogohai@yahoo.com

2 2 A. Kiliçma ad S. Bogohai: Spaces of Geealized diffeece Lacuay I-coveget sequeces By a Olicz fuctio, we mea a fuctio : [0, ) [0, ), which is cotiuous, o-deceasig ad covex with 0) = 0,x) > 0, fo x > 0 ad x), as x. A sequece x l is said to be almost coveget if all of its Baach limits coicide. Let ĉ deote the space of all almost coveget sequeces. Loetz[9] itoduced the followig sequece space as, ĉ = x l : lim m t m, x) exists uifomly i whee t m, x) = x + x x m+. m + The otio of diffeece sequece spaces of cisp sets ae defied as Z ) = x = x ) : x ) Z, fo Z = l,c ad c 0, whee x = x ) = x x + ), fo all N ad late o which was geealized by may othe ecet mathematicias. These spaces ae Baach spaces, omed by,see Kizmaz [0]) x = x + x. 2 Defiitios ad Pelimiaies Let N ad X be a eal vecto space. A eal valued fuctio o X satisfyig the followig fou popeties:. z,z 2,...z ) = 0 if ad oly if z,z 2,...z ae liealy depedet; 2. z,z 2,...z ) is ivaiat ude pemutatio; 3. z,z 2,..,.z,αz ) = α z,z 2,...z, fo all α R; 4. z,z 2,...z,x + y) z,z 2,...z,x) + z,z 2,...z,y) ; is called a -om o X ad the pai X,,.,. ) is called a -omed space. The space mϕ) is defied as, mϕ) = x ) w : x mϕ) = x <. The idea of Olicz fuctio is used to costuct the sequece space, see Lidestauss ad Tzafii []) l = x ) w : = ) x <, fo some > 0 which becomes a Baach space, called as Olicz sequece space, with the followig om, x = if > 0 : = ) x Let X be a oempty set. The a family of sets I 2 X powe sets of X) is said to be a ideal if I is additive i.e. A,B I A B I ad heeditay i.e. A I,B A B I.

3 NTSCI 3, No. 3, ) / 3 Fo a lacuay sequece θ = ), a sequece x ) is said to be lacuay I-coveget if fo evey ε > 0 such that, We wite I θ limx = x. N : h J x x ε I. I this aticle, we defie some ew geealized diffeece lacuay I-coveget sequece spaces i -omed spaces elated to l p -space by usig Olicz fuctio. We will also itoduce ad examie cetai ew sequece spacies usig the above tools. 3 ai Results Let u = u ) be a sequece of eal umbes such that u > 0 fo all, ad u <. Also, let I be a admissible ideal of N ad be a olicz fuctio. I this aticle, we have itoduced the followig sequece space as, m,ϕ, q p,u,θ) I,,... ) ) ) = x : ε > 0 N : h p t m x) L,z,z 2,...z ε I. fo some > 0, z,z 2,...z X. Paticula cases: If we tae u =, fo all, we have, m,ϕ, q p,θ) I,,... ) q ) = x : ε > 0 ) N : h p t m x) L,z,z 2,...z ε) I. fo some > 0, z,z 2,...z X. Now, if we coside x) = x, the we ca easily obtai: mϕ, q p,u,θ) I,,... ) ) ) = x : ε > 0 N : h pt q m x) L,z,z 2,...z ) u ε, uifomly i m I. If x m,ϕ, q p,u,θ) I,,... ) with h ) p t m x) L,z,z 2,...z ε I as, uifomly i m, the we wite x L m,ϕ, q p,u,θ) I,,.. ).

4 4 A. Kiliçma ad S. Bogohai: Spaces of Geealized diffeece Lacuay I-coveget sequeces The followig well ow iequality will be used late. If 0 u u = H ad C = max,2 H ), the fo all ad a,b C. a + b u C a u + b u, ) Theoem. Let lim ifu > 0. The, x L implies x L m,ϕ, p,u,θ) q I,,.. ). Let lim u = u > 0. If x L m,ϕ, p,u,θ) q I,,.. ), the L is uique. Poof: Let x L. By the defiitio of Olicz fuctio, we have, fo all ε > 0, q ) ) p t m x) L,z,z 2,..z h ε I. Sice lim ifu > 0, it follows that, ) p t m x) L,z,..z h ε I Ad cosequetly, x L m,ϕ, q p,u,θ) I,,... ). Let lim s = s > 0. Suppose that, x L m,ϕ, q p,u,θ) I,,.. ) ad L L 2,z,z 2,...z ) u = a > 0. x L 2 m,ϕ, q p,u,θ) I,,.. ) Now, usig the defiitios of iequality ad Olicz fuctio, we have, )) L L 2,z,...z u h C p t m x) L,z,...z h

5 NTSCI 3, No. 3, ) / C p t m x) L 2,z,...z h Sice, N : h ) p t m x) L,z,..z ε I, ad N : h ) p t m x) L 2,z,..z ε I, Hece, Futhe, as, ad theefoe, )) ) N : h L L 2,z,..z u ε I 2) ) L L 2,z,z 2,..z u ) a u ) L L 2,z,..z u ) a u h =. 3) ) Fom the above equatios 3) ad 4)), it follows that a = 0 ad by the defiitio of a Olicz fuctio, we have a = 0. Hece L = L 2 ad this completes the poof. Theoem 2.. Let 0 < ifu u. The, m,ϕ, q p,u,θ) I m,ϕ, q p,θ) I 2. Let 0 < u u <. The, m,ϕ, q p,θ) I m,ϕ, q p,u,θ) I. Theoem 3. The iclusio m,ϕ, p q,u,θ) I m,ϕ, p,u,θ) q I is stict. I geeal, m,ϕ, i p,u,θ) I m,ϕ, q P,u,θ)I fo all i =,2,3,...p ad the iclusio is stict. Theoem 4. m,ϕ, q p,u,θ) I is a complete liea topological space,with paaom g, whee g is defied by, gx) = pq m= t m x) L,z,z 2,..z + if u H : p t m x) L,z,z 2,..z h

6 6 A. Kiliçma ad S. Bogohai: Spaces of Geealized diffeece Lacuay I-coveget sequeces whee H = max,u )). Popositio 5. m,ϕ, p,u,θ) q I m,ψ, p,u,θ) q I if ad oly if ) s ϕs Poof : Fist, pose that = K <, the we have, Kψ s. s ψ s Now, if x ) m,ϕ, q p,u,θ) I, the ϕs p t m x) L,z,z 2,..z h ε I h p t m x) L,z,z 2,..z Kψ s ε I i.e x ) m,ψ, q p,u,θ) I. ψ s ) <, fo 0 < p <. Hece, m,ϕ, q p,u,θ) I m,ψ, q p,u,θ) I. ) Covesely, pose that m,ϕ, p,u,θ) q I m,ψ, p,u,θ) q I ϕs. We should pove that = η s ) <. s ψ s s Suppose that η s ) =. The thee exists a subsequece η si ) of η s ) such that limη si ) =. The fo s i x ) m,ϕ, p,u,θ) q I, we have, h ψ s h ηsi p t m x) L,z,z 2,..z ) p t m x) L,z,z 2,..z = i which implies that x ) / m,ϕ, q p,u,θ) I, a cotadictio. This completes the poof. Coollay 6. m,ϕ, p,u,θ) q I = m,ψ, p,u,θ) q I, if ad oly if η s ) < ad 0 < p <. s ηs s ) <, whee η s = ψ s, fo Refeeces [] A. Kiliçma ad S. Bogohai, Stogly almost lacuay I-coveget sequeces. Abst. Appl. Aal. 203, At. ID , 5 pp. 40A05 [2] A. isia, -ie poduct spaces, athematische Nachichte, 40)989), pp [3] A. Şahie,. Güdal ad T. Yigit, Ideal Covegece Chaacteizatio of Completio of Liea -Nomed Spaces, Computes ad athematics with Applicatios, 63)20), [4] B.C. Tipathy, B. Hazaia, ad B. Choudhay, Lacuay I-coveget sequeces. Kyugpoo ath. J ), o. 4, A05 40A35) [5] B. Hazaia, O paaomed ideal coveget geealized diffeece stogly summable sequece spaes defied ove -omed spaces, Iteatioal Scholaly Reseach Netwo, ISRN athematical Aalysis, 20, Aticle ID 37423, 7 pages. [6] E.Savaş ad A. Kiliçma, A ote o some stogly sequece spaces, Abstact ad Applied Aalysis, 20, Aticle ID , 8 pages. [7] E.Savaş, λ m -stogly summable sequeces spaces i 2-omed spaces defied by ideal covegece ad a Olicz fuctio, Applied athematics ad Computatio, 27200), pp

7 NTSCI 3, No. 3, ) / 7 [8] E. Savas, Some double lacuay I-coveget sequece spaces of fuzzy umbes defied by Olicz fuctio. J. Itell. Fuzzy Systems ), o. 5, A45 03E72 46S40) [9] G.G.Loetz, A cotibutio to the theoy of diveget sequeces, Acta athematica, 80948), pp [0] H. Kizmaz, O cetai sequece spaces, Caadia athematical Bulleti, 242)98), pp [] J. Lidestauss ad L. Tzafii, O Olicz sequece spaces, Isael Joual of athematics, 097), pp [2]. Et, O some diffeece sequece spaces, Doğa-T. J. of athematics, 7993), [3] P. Kostyo, T. Šalǎt ad W. Wilczyńsi, O I-covegece, Real Aalysis Exchage, 262) ), pp [4] W.L.C. Saget, Some sequece spaces elated to l p spaces, J. Lod. ath. Soc, 35960), 6-7.