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1 Acta cietiau Techology IN: Uiveidade Etadual de Maigá Bail Dutta, Hee; uede Reddy, Boa; Hazah Jebil, Iqbal O two ew type of tatitical covegece ad a uability ethod Acta cietiau Techology, vol 36, ú, eeo-azo, 20, pp Uiveidade Etadual de Maigá Maigá, Bail Available i: How to cite Coplete iue Moe ifoatio about thi aticle Joual' hoepage i edalycog cietific Ifoatio yte Netwo of cietific Joual fo Lati Aeica, the Caibbea, pai ad Potugal No-pofit acadeic poject, developed ude the ope acce iitiative
2 Acta cietiau IN pited: IN o-lie: Doi: 0025/actacitecholv36i6206 O two ew type of tatitical covegece ad a uability ethod Hee Dutta *, Boa uede Reddy 2 ad Iqbal Hazah Jebil 3 Depatet of Matheatic, Gauhati Uiveity, Guwahati-780, Aa, Idia 2 Depatet of Matheatic, PGC, aifabad, Oaia Uiveity, Hydeabad-50000, Idia 3 Depatet of Matheatic, Taibah Uiveity, Aladiah Aluawwaah, Kigdo of audi Aabia *Autho fo coepodece E-ail: hee_dutta08@ediffailco ABTRACT I thi pape, we itoduce ad ivetigate elatiohip aog I -tatitically coveget, I -tatitically coveget ad I [ V,, ] uable equece epectively ove oed liea pace Keywod: diffeece opeato, ideal, filte, tatitical covegece, uability Doi ovo tipo de covegêcia etatítica e o étodo de uabilidade REUMO Itoduze-e e ivetiga-e a elação ete I -etatiticaete covegete, -etatiticaete covegete e I [ V,, ] equêcia uávei epectivaete obe epaço lieae I oatizado Palava-chave: opeado difeecial, ideal, filto, covegêcia etatítica, uabilidade Itoductio The idea of covegece of a eal equece had bee eteded to tatitical covegece by Fat (95) ad ca alo be foud i choebeg (959) If N deote the et of atual ube ad K N the K (, ) deote the cadiality of K [, ] The uppe ad the lowe atual deity of the ubet ae K defied by: K(, ) d( li up ad K(, ) d( li if If d( d( the we ay that the atual deity of K eit ad it i deoted by d( Clealy K(, ) d( li A equece ( ) of eal ube i aid to be tatitically coveget to L if fo abitay 0, the et K( ) N : L ha atual deity zeo tatitical covegece tued out to be oe of the ot active aea of eeach i uability theoy aily due to Fidy (985) ad Šalàt (980) A a geealizatio of tatitical covegece, the otio of ideal covegece wa itoduced fit by Kotyo et al (2000/200) Thi wa futhe tudied i topological pace by Lahii ad Da (2005), Da et al (2008) ad ay othe Mualee (2000) itoduced ad tudied the idea of covegece a a eteio of the [ V, ] uability itoduced by Leidle (965) tatitical covegece i a pecial cae of oe geeal I tatitical covegece tudied by Kol (99) The otio of diffeece equece pace wa itoduced by Kizaz (98), who tudied the diffeece equece pace, c ad c0 The otio wa futhe geealized by Et ad Çola c ad, (995) by itoducig the pace c0 Aothe type of geealizatio of the diffeece equece pace i due to Tipathy ad, c Ei (2006), who tudied the pace ad c0 Tipathy et al (2005) geealized the above otio ad uified thee a follow: Let, be o- egative itege, the fo Z a give equece pace we have : Z w Z, whee: ad Acta cietiau Techology Maigá, v 36,, p 35-39, Ja-Ma, 20
3 36 Dutta et al fo all N, which i equivalet to 0 the followig bioial epeetatio: 0 Taig =, we get the pace, c ad c0 tudied by Et ad Çola (995) Taig =, we get the pace, c ad c0 tudied by Tipathy ad Ei (2006) Taig = =, we get the pace, c c ad itoduced ad tudied by Kizaz (98) oe othe wo o diffeece equece ay be foud i Kaaaya ad Dutta (20), Tipathy ad Dutta (200), Tipathy ad Dutta (202), ad ay othe Recetly, ava ad Da (20) ade a ew appoach to the otio of [ V, ] -uability ad tatitical covegece by uig ideal ad itoduce ew otio, aely I [ V, ] uability ad I -tatitical covegece I thi pape, ou iteio i to geealize the eult of ava ad Da (20) by coideig diffeece equece Thoughout ( X, ) will tad fo a eal oed liea pace ad by a equece = ( ) we hall ea a equece of eleet of X N will tad fo the et of atual ube Mai eult Y A faily I 2 of ubet a o epty et Y i aid to be a ideal i Y if (i) I (ii) A, B I iply A B I (iii) A I, B A iply B I, while a adiible ideal I of Y futhe atifie { } I fo each Y If I i a ideal i Y the the collectio F( I) { M Y : M c I} fo a filte i Y which i called the filte aociated with I N Let I 2 be a otivial ideal i N The equece ( ) i X i aid to be N I coveget to X, if fo each 0 the et A( ) { N : } belog to I Fo detail, we efe to Kotyo et al (2000/200) Defiitio 2: A equece ( ) i aid to be I -tatitically coveget to L X, if fo evey 0, ad evey 0, 0 N : : L I Fo I I, = 0, =, -tatitical fi covegece coicide with tatitical covegece Let ( ) be a o-deceaig equece of poitive ube tedig to uch that The collectio of uch a, I equece will be deoted by The geealized de la Valée-Poui ea i defied by, whee I [, ] t ( ) I Defiitio 2: A equece ( ) i aid to be I-[ V,, ] uable to L X, if I li t (, ) L, whee t (, ) I ie, fo ay 0, N : t (, ) L I If I I, = 0, =, I [ V, ] uability fi becoe [ V, ] uability (LEINDLER, 965) Defiitio 3: A equece ) i aid to be I -tatitically coveget o I L, if fo evey 0 ad 0, N : ( -coveget to I : L I I thi cae we wite I li L o LI ( ) We alo wite I li L Fo I I, = 0, =, fi I -covegece agai coicide with tatitical covegece We hall deote by I (, ), (, I ) ad [ V,, ]( I) the collectio of all I -tatitically coveget, I -coveget ad I [ V,, ] uable equece epectively Acta cietiau Techology Maigá, v 36,, p 35-39, Ja-Ma, 20
4 O two ew type of tatitical covegece 37 Theoe 2: Let ( ) (i) LV [,, ]( I) L( ( I, )) (ii) If (X ), the pace of all bouded equece of X ad L( ( I, )) the LV [,, ]( I) (iii) ( I, ) X ( ) [ V,, ]( I) X ( ) Poof (i) Let 0 ad LV [,, ]( I) We have I L I & L o fo a give 0, L I I : L ie, N : I I : L N : I : L L L ice LV [,, ]( I), o the et o the ight had ide belog to I ad o it follow that L( ( I, )) (ii) uppoe that L( ( I, )) ad X We ca chooe L M, Let 0 be give Now L L L I I& L I& L M I : L Note that N : I : L A( ) I M (ay) c If ( A( )), the L 2 I Hece N : L 2 A( ) ad o I belog to I Thi how that LV [,, ]( I) (iii) The poof follow fo (i) ad (ii) Theoe 22: If liif 0, the the followig hold I (, ) (, I ) Poof Fo give 0, : L I : L I : L If li if a the fo defiitio a N : 2 i fiite Fo 0, N : I : L a N : I : L 2 a N : 2 ice I i adiible, the et o the ight had ide belog to I ad the poof follow Theoe 23: If be uch that li, the (, I ) (, I ) Poof Let 0 be give ice li, we ca chooe N uch that 2, fo all Now obeve that, fo 0 : L : L I : L 2 I : I : L L Acta cietiau Techology Maigá, v 36,, p 35-39, Ja-Ma, 20
5 38 Dutta et al 2 I : L fo all L ad L Hece N : : L N : I : L,2,, 2 If I li L the the et o the ight had ide belog to I ad o the et o the left had ide alo belog to I Thi how that i I -tatitically coveget to L ( ) Theoe 2: If X i a Baach pace, the (, I ) ( X) i a cloed ubet of (X ) Poof uppoe that ( ) i a coveget equece i (, I ) ( X) ad covege to (X ) The poof follow if we ca how that (, I ) ( X) uig the fact that evey bouded equece i alo -bouded Aue that L( ( I, )) N Tae a equece { } N of tictly deceaig poitive ube covegig to zeo We ca fid N a uch that j fo all j Chooe 0 5 Now A N : I : L F( I) ad B N : I : L F( I) ice A B F( I) ad F( I), we ca chooe The A B I : L L 2 ice ad A B F(I ) i ifiite, we ca chooe the above o that 5 (ay) Hece thee ut eit a I fo which we have iultaeouly, The it follow that L L L L L L Thi iplie that { L} i a Cauchy equece N i X, which i coplete Let L L X a We hall pove that L( ( I, )) Chooe 0 ad chooe N uch that, Now I I, : : L L L L : I L, 2 It follow that, fo ay give 0 L L N : I : L N : I : L 2 Thi how that L( ( I, )) ad coplete the poof of the theoe Cocluio The pape defie ad tudie two type of tatitical covegece ad a uability ethod Acta cietiau Techology Maigá, v 36,, p 35-39, Ja-Ma, 20
6 O two ew type of tatitical covegece 39 fo diffeece equece ove a oed pace Although we ae able to eted oe eult of ava ad Da (20), the followig futhe uggetio eai ope: I thee othe coditio uch that Theoe 22 hold? Whethe the coditio i Theoe 23 i eceay? Refeece DA, P; KOTYRKO, P; WILCZYNKI, W; MALIK, P I ad I * -covegece of double equece Matheatica lovaca, v 58, 5, p , 2008 ET, M; ÇOLAK, R O geealized diffeece equece pace oochow Joual of Matheatic, v 2,, p , 995 FAT, H u la covegece tatitique Colloquiu Matheaticu, v 2, p 2-2, 95 FRIDY, J A O tatitical covegece Aalyi, v 5,, p 30-33, 985 KARAKAYA, V; DUTTA, H O oe vecto valued geealized diffeece odula equece pace Filoat, v 25, 3, p 5-27, 20 KIZMAZ, H O cetai equece pace Caadia Matheatical Bulleti, v 2, 2, p 69-76, 98 KOLK, E The tatitical covegece i Baach pace Acta et Coetatioe Uiveitati Tatuei de Matheatica, v 928, p -52, 99 KOTYRKO, P; ŠALÀT, T; WILCZYNKI, W I-Covegece Real Aalyi Echage, v 26, 2, p , 2000/200 LAHIRI, B K; DA, P I ad I * -covegece i topological pace Matheatica Boheica, v 30, 2, p 53-60, 2005 LEINDLER, L Ube die de la Vallee-Pouche uiebaeit allge eie othogoaleihe Acta Matheatica Acadey of ciece Hugaica, v 6, p , 965 MURALEEN, M tatitical covegece Matheatica lovaca, v 50,, p -5, 2000 ŠALÀT, T O tatitical coveget equece of eal ube Matheatica lovaca, v 30, 2, p 39-50, 980 AVA, E; DA, P A geealized tatitical covegece via ideal Applied Matheatic Lette, v 2, 6, p , 20 CHOENBERG, I J The itegability of cetai fuctio ad elated uability ethod The Aeica Matheatical Mothly, v 66, 5, p , 959 TRIPATHY, B C; DUTTA, H O oe ew paaoed diffeece equece pace defied by Olicz fuctio Kyugpoo Matheatical Joual, v 50,, p 59-69, 200 TRIPATHY, B C; DUTTA, H O oe lacuay diffeece equece pace defied by a equece of olicz fuctio ad q-lacuay -tatitical covegece Aalele tiitifice ale Uiveitatii Ovidiu Cotata, eia Mateatica, v 20,, p 7-30, 202 TRIPATHY, B C; EI, A A ew type of diffeece equece pace Iteatioal Joual of ciece ad Techology, v,, p -, 2006 TRIPATHY, B C; EI, A; TRIPATHY, B K O a ew type of geealized diffeece Ceào equece pace oochow Joual of Matheatic, v 3, 3, p , 2005 Received o Mach, 202 Accepted o Jue, 202 Licee ifoatio: Thi i a ope-acce aticle ditibuted ude the te of the Ceative Coo Attibutio Licee, which peit ueticted ue, ditibutio, ad epoductio i ay ediu, povided the oigial wo i popely cited Acta cietiau Techology Maigá, v 36,, p 35-39, Ja-Ma, 20
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