Structure and Some Geometric Properties of Nakano Difference Sequence Space

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1 Stuctue ad Soe Geoetic Poeties of Naao Diffeece Sequece Sace N Faied ad AA Baey Deatet of Matheatics, Faculty of Sciece, Ai Shas Uivesity, Caio, Egyt awad_baey@yahooco Abstact: I this ae, we exted the class of sequece saces of -bouded vaiatio ad Başa [Uaiia MathJ55((3,36-47]; whee itoduced by Altay to the sace of all sequeces x that ( x x belogs to the sequece sace itoduced by Naao ( whee ( is a sequece of ositive ubes with,we defie a odula fuctioal o this sace ad show that whe equied with the Luxebug o is a Baach sace ad locally uifoly otud whe ( such, theefoe ossessig - oety ad otud Fially we fid Guaii s odulus of covexity fo the sace [Joual of Aeica Sciece;6(:3-8] (ISSN: Keywods: Guaii s odulus of covexity, -oety, R-oety, covex odula, Luxebug o, locally uifoly otud Itoductio Let (, be a eal Baach sace ad let B( (ese S( be the closed uit ball (es uit shee of A oit x S( is a -oit of B( if fo ay sequece x i such that Li x, ( the wea covegece of W x to x ( wite x x Li x x ilies that If evey oit of S( is a -oit of B(; the is said to have oety (Kadec-Klee A oit x S( is called a extee oit of B(, if fo ay y, z S(, the equality x= y z ilies y=z A Baach sace is said to be Rotud (R if evey oit of S( is a extee oit of B( A oit x S( is called a locally uifoly otud (LUR-oit, if fo ay sequece ( x i B( such that Li x x, thee holds that Li x x If evey oit of S( is a LUR-oit of B(, the the sace is called locally uifoly otud (LUR It is ow that if is LUR, the it is otud (R ad ossesses oety ( oweve the covese of this last stateet is ot tue i geeal Fo these geoetic otios ad thei ole i atheatics we efe to the oogahs [], [], [3], [4] Soe of these geoetic oeties wee studied fo olicz saces i [5], [6], [7], ad [8] By, we shall deote the sace of all eal o colex sequeces ad the set of atual ubes will deote by = {,,, } Fo a eal vecto sace ove the eal Nubes a fuctio : [, ] is called a odula, if it satisfies the followig coditios: (i ( x x x,, fo all with (ii ( x ( x, htt://wwwaeicascieceog 3 edito@aeicascieceog

2 (iii ( x y ( x ( y, ;, x, y, Futhe, the odula is called covex if (iv ( x y ( x ( y, ; x, y Let (, be a oed liea sace, coside Claso s odulus of covexity (Claso[] ad Day [] defied by x y ( if ; x, y S(, x y, whee (,] The iequality ( fo all chaacteizes the uifoly covex saces I [], Guaii s odulus of covexity is defied by ( if if [,] x ( y ; x, y S(, x y, whee It is easily show that ( ( ( fo ay Also if (, The is uifoly covex, ad if (, the is stictly covex by The Naao sequece sace ( is defied ( x ( x : ( tx fo soe t, whee ( x x ad ( of ositive eal ubes with o is a sequece, The sace ( is a Baach sace with the x if t : ( t x If is bouded, we ca sily wite ( P ( x : x Also, soe geoetic oeties of ( wee studied i [], [3] The idea of diffeece sequece was fist itoduced by Kizaz [3] Wite x x x fo all with diffeece oeato defied by ( x x x ad : be the x Defiitio: Let ( be a sequece of, we defie the ositive eal ubes with followig sequece sace ( {x : ( x, fo soe } ( x, whee o x P with the x x if t : ( t If ( is bouded, we ca sily wite P ( { x : x } The sace is a aaoed sace by the aao g ( ( x x, see [6], whee su ( Thoughout this ae, the sequece is cosideed to be bouded, ad let su with Fo ay bouded sequece of ositive ubes, we have a b fo all Taig Lea : ( a b P P (, whee, the (, see [9] The fuctioal is covex odula o ( Poof: It ca be oved with stadad techiques i a siila way as i [4, 5] Lea : htt://wwwaeicascieceog 4 edito@aeicascieceog

3 Fo x ( the followig oeties:, the odula o ( x satisfies (i If <<, the (x ad ( x ( x (ii If >, the x x ( (iii If, the ( x ( x ( x Poof: It ca be oved with stadad techiques i a siila way as i [4, 5] Lea 3: Fo ay, x ( the followig assetios ae satisfied: (i If x, the ( x x (ii If (iii x, the ( x x x If ad oly if ( x (iv If ad x, the ( x (v If ad x, the ( x Poof : It ca be oved with stadad techiques i a siila way as i [4,5] Lea 4: Let ( x be a sequece i ( (i If x Li, the Li ( x, (ii if ( x Li, the Li x Poof :(i Suose that Li x The fo ay (, thee exists o such that By lea (3, x ( ( x ( ilies that ( x Li (ii If x Li, the thee is a (, subsequece x such that ( ad a x This ilies that Li ( x ad hece ( x Li Mai esults Theoe: ( is a Baach sace with esect to Luxebug o defied by x if : x Poof: Let x ( x (,,,, be a Cauchy sequece i ( is coveget accodig to the Luxebug o Thus (, o such that x x, we obtai By Lea 3(i, ( x x x x ( That is P x ( x (, fixed we get that x ( x ( ( x ( Fo,, ad the sequece is a Cauchy sequece of eal ubes Let x( Li x (, the Li x ( x( Theefoe, P x ( x( is ( x x Li x x By the followig calculatios, we obtai that x( P That P x( x ( x ( P P x ( x ( x ( <, we see that the sequece x ( x( ( theoe x coveges to This coletes the oof of htt://wwwaeicascieceog 5 edito@aeicascieceog

4 Theoe: Let ( B( ( x ad ( B( ( x y ( y ( y If Li, the Li x oof: See [5 Poositio6], fo all Theoe3: Let ( B( ( x x x Li x ( x( If Li, the Poof: Fo each,,, let S ( sg( x ( x( if x ( x(, ad S ( if x ( x(, hece we have x x x ( x( Li = x ( S( S( ( Let P x( ( = S( x( ( = S ( x(,, (,( ( ad The ad fo ( we ( ( have Li Fo Theoe (, we have Li (3, fo all Now we shall ove that Li x ( x(, Fo (3 we have at that Li s ( x ( s ( x( ilies that Li x ( x( This Assue that Li x ( x( The we get Li sice s ( ( x ( x( s ( ( x (4 ( x( ( (,, It follows that fo (3 ad (4 that s( ( x ( x( as This ilies Li x ( x( we get x ( x( Li Theoe4: The sace ( is LUR Poof: Let ( B( ( x x ad ( x( S( ( be such that x x Li x x Li ( Li x ( i x( i, by lea (4-i we have s ( ( x, by theoe (3 we have i, fo, P ( x ( x(, ( P x( 3( (5 Sice sice, also such that ( x( Li ( x ( x( ( x ( x(, ad Li x ( x(,, thee exists such that P ( x ( (6 P P ( x( P 3( htt://wwwaeicascieceog 6 edito@aeicascieceog

5 , sice Li x ( x( cotiuous oeato we get ( x( x, As a esult, we have ( x ( x( P 3 (7 The, fo (5, (6 ad (7 it follows that, we have ( x x x ( x( = P ( x ( x( P ( x P x( x( 3 x( P M 3(, this shows that Sice is ( x( Li ( x x ece by lea 4 (ii, we have Li x x Theoe5: Guaii s odulus of covexity fo the oed sace, Poof: Fo is x ( (, we get that ( x x ( x, ad, Coside the two Sequeces x ( ( (, (,,,, P whee ad P y ( ( (, (,,,,, we get that x (( (,(,,,,, ad (( (,(,,,, y Sice x x ( ( the x S ad (, y y ( ( y S( ad y x y, sice x if x ( y if x ( y if, the [ ( ( ( ( ( ( ( ( ] if [( ( ( ] [ ( ], Cosequetly fo, we get ( Coollay ( ( (i If, the is stictly covex,, hece (ii if, the ( (iii ad hece if, is uifoly covex,, the ( ( htt://wwwaeicascieceog 7 edito@aeicascieceog

6 Coesodig autho N Faied Deatet of Matheatics, Faculty of Sciece, Ai Shas Uivesity, Caio, Egyt _faied@hotailco Refeeces ST Che, Geoety of Olicz saces, Dissetatios Math, 996, 356 YA Cui, udziu ad C Meg, o soe local geoety of Olicz sequece saces equied the Luxebug os, Acta Sci Math ugaica, 8(-(998, YA Cui ad udzi, o the Baach-sas ad wea Baach-sas oeties of soe Baach sequece saces, Act Sci Math (Szeged, 65(999, J Diestel, geoety of Baach sacesselected toics, sige- Velag, (984 5 R Gzaslewicz, udzi ad W Kuc, Extee oits i Olicz saces, Caad J Math Bull, 44(99, udzi, Olicz saces without stogly extee oits ad without -oit, Caad Math Bull, 35(99, -5 7 udzi ad D Pallasche, o soe covexity oeties of Olicz sequece saces, Math Nach, 86(997, J Musiela, Olicz saces ad odula saces, lectue otes i Math 34, Sige-velag, (983 9 A Ahedov ad F Basa, the fie secta of the Cesáo oeato C ove the sequece sace, ( Uiv 5(8, 35-47, Math J Oayaa JA Claso, Uifoly covex saces, tasactios of the Aeica Matheatical Society, vol 4, o 3, , 936 MM Day, Uifoly covexity i facto ad cojugate saces, Aals of Matheatics, vol 45, , 944 VI Guaii, Diffeetial oeties of the covexity oduli of Baach saces, Mathatichesie Issledovaiya, vol 4-48, Kizaz, o cetai sequece saces, Caad Math Bull, 4((98, W Saha ad S Suatai, Soe geoetic oeties of Cesao sequece sace, Kyug-oo Math J, 43(3, SSuatai, o soe covexity oeties of geealized Cesao sequece saces, Geogia Matheatical Joual, (3, Nube, 93-6 YAlti, MIşi, Ad Rçola,A ew sequece sace defied by a odulus, Studia Uiv Babeş-Bolyai, Matheaitca, Volue LIII Nube,Jue 8 /5/ htt://wwwaeicascieceog 8 edito@aeicascieceog