Some Geometric properties in Orlicz- Cesaro Spaces

@ESEA'[gE"t:RTICLE Some Geometric SdenceAsia31 (2005): 173-177 properties in Orlicz- Cesaro Spaces Narin Petrot and Suthep Suantai* Department of Mathematics, Chiang Mai University,Chiang Mal. 50200, Thailand. * Corresponding author, E-mail: [email protected] Received 12 Nov 2004 Accepted 28 Jon 2005 ABSTRACT: On the Orlicz- Cesaro sequence spaces( ces<f> ) which are defined by using Orlicz function <I>, we show that the space ces<f>equipped with both Amemiya property and uniform Kadec-Klee property if <I> and Luxemburg norms possesses uniform Opial satisfy the 52 -condition. KEYWORDS: Orlicz-Cesaro sequence spaces,uniform Kadec-Klee property, uniform Opial property, Amemiya norm, Luxemburg norm. A BanachspaceX is said to have the apial property (see [3]) if for any weakly null sequence (xn) and INTRODUCTION In the whole paper Nand IR. stand for the sets of natural numbers and of real numbers, respectively.The spaceof all real sequencesis denoted by 1°. Let (X,II'ID be a real normed space and B(X)(S(X)) be the closed unit ball (the unit sphere) of X. A Banach space(X,II'ID which is a subspace of 1° is said to be a Kothe sequencespace,if: CDfor anyx E 1°andy E X suchthat I x(o 1::;1y(o 1 for all i EN, we have x E X and Ilxll::; Ilyll ' (ii) there is x E X with x( i) "* o for all i E N An element x from a Kothe sequence spaceX is called order continuous if for any sequence (xn) in X+ (the positive cone of X) such that xn ::;Ixl for all n E Nand Xn -? o coordinatewise, we have IlxnII-? 0. A Kothe sequence space X is said to be order continuousif any x E X is ordercontinuous.ltiseasyto see that x E X is order continuous if and only ifll(O,O,...,O,x(n+l),x(n+2),...)II-?O asn-?oo. A Banach space X is said to have the Kadec-Klee property (or H-property) if every weakly convergent sequence on the unit sphere is convergent in norm. Recall that a sequence { Xn} c X is said to be [;separatedsequencefor someIi> 0 if sep(xJ = inf{llxn -xm":Wi'm} >6". A Banach space is said to have the uniform KadecKleeproperty (write (UKK) for short) if for every 6"> 0 there exists 15> 0 such that for every sequence(xn) in SeX) with sep(xn»1i and Xn--~-H, we have IIxII < 1- 15.Every (UKK) Banachspacehas H-property (see [1]) The Opial property is important because Banach spaceswith this property have the weak fixed point property (see [2]). Opial has proved in [3] that the sequencespaces £p (l < P < 00) have this condition but Lp[O,2n"j(p"* 2, 1 < P < 00)do not. every x*-O in X, we have lim infllxn II < lim infllxn + xii. n-->oo n-->oo A Banachspace X is said to have the uniform Opial property (see [4]) if for each & > 0 there exists T > 0 such that for any weakly null sequence (xn) in S(X) and x E X with Ilxll;:::& the following inequality holds: l+TS liminfllxn +xll. n->", Fora real vector spaceX, a function 9'J1:X ~ [0,00] is called a modular if it satisfiesthe following conditions: CO 9'J1(x) (ii)9'J1(ax) = a if and only if x = 0, = 9'J1(x) for all scalar a with lal=l, (iii)9'J1(ax + j1y)::; 9'J1(x)+ 9'J1(y), for all x,y E X andalla,j1~O with a+j1=l. The modular 9'J1is called convexif (iii)9'J1(ax+j1y)::;a9'J1(x)+j19'J1(y), for all X,YEX andalla,j1~O witha+j1=l. For any modular 9'J1on X, the space X!m =(x E X:9'J1(AX) ~ a as A ~ OJ, is called the modular space. A sequence(xn) of elementsof X!m is called modular convergent to x E X!m if there exists a A> a such that 9'J1(A(Xn- x)) ~ 0, as n ~ 00. If 9'J1is a convex modular, the function Ilxll = inf{A > 0:9'J1(/'i)::; I}, and 1 IIxIL = inf-(l + 9'J1(kx)), bok are two norms on X!)J!'which are called the Luxembur~norm and the Amemiyanorm, respectively In addition, IIxlls IlxiiAs 211xll for all x E X!)J!(see [5]). Theorem 1.1 Let(xn)cX!m thenllxnII-"'~O (or equivalently IlxnIIA~ 0) if and onlyifM(A,(xn)) ~ 0, as n ~ 00 , for every A > 0 . 174 SdmaAsia 31 (2005) Proof. See[6, Theorem1.3(a)]. This space was first introduced by Shiue [9]. It is A modular 9Jt is said to satisfy the /).2-condition useful in the theory of Matrix operators and others (see ( 9Jt e /).2) if for any [; > 0 there exist constants K ;;::2 [ 10] and [ll]). Somegeometric properties ofthe Cesaro and a > 0 such that sequence spaces cesp were studied by many authors. 9Jt(2x)::;; K9Jt(x) + E For an Orlicz function <Dthe Orlicz- Cesaro sequence forallxeX!)JI with 9Jt(x)::;;a. space,cestf> , is defined by If 9Jt satisfies the /).2-condition for all a > 0 ces<l> ={XE£o: p<I>(AX) <00, 3A>O}, with K ;;::2 dependenton a, we say that 9Jt satisfies where 1n thestrong /).2-condition ( 9Jt e /).; ). (-Llx(i)1 ) n Pq,(X)=L~=IcI> Theorem 1.2 Convergencesin norm and in modular areequivalentin X!D!if !m E .£\2' Proof. See[7, Lemma2.3]. isaconvexmodularon Theorem 1... If rot E 6.~, then for any 8 > 0 there exists li = li( 8) > 0 such that Ilxll ~ 1 + li whenever rot(x)~ 1+8. Proof. See [7, Lemma 2.4]. A map<1>:R ~ [0,00] is said to beanOrlicz.function if it is even, convex, continuous and vanishing at 0 and <1>( u) ~ 00 as u ~ 00. Furthermore, Orlicz function we say that an . <1>(u) if hm -= <1> is an N' -function .-->"" 00. U The Orlicz sequence space, £ 11>'where <1>is an Orlicz function £11> ={xe£O :I<I>SAx)<oo 3A>O}, where III>(x) = L<1>(x(i)) is a convex modular i=l II> is a Banachspacewith norm II'II,~> andAmemiyanorm by K(x) the IIxlL 1 =k"(l + (kx)), Iq, set of both Luxemburg II'II,~ (see [5]). Denoted all k > 0 k>0 such that it is well known that K(xh= 0 for all x E fell whenever <1>is an N' -function (see [8]). An Orlicz function <1>is said to satisfy the 82condition (we will write <1> E 02 for short) if there exist constants K ~ 2 and Uo> 0 such that the inequality <1>(2u)$ K<1>( u) holds for every u E IR satisfying lu\ $uo' For 1< P < co, the Cesaro sequence space (write, cesp' for short) is defined by cesp={XEfO cesq,.Thesubspace E", of Pq,(AX) < 00, VA>O}. It is worth noting that if cI>e 82, then pq, e A~ and ces",= E",. To simplify notations, we put ces~= (ces",,Ill) and ces~= (cesq"llt). In thecasewhen cI>(t)=ItlP,(p > 1) the Orlicz- Cesaro sequence spaceces<J) becomes the Cesaro sequencespacecespand the Luxemburg norm is that one defined by (1.1). From now on, for x e [° and i e N we let xI = (x(1),x(2),...,x(i),O,O,...), xlN-i= (O,O",.,x(i + l),x(i + 2),x(i + 3)",.), and suppx ={ieN: x(i):;t:O}. RESULTS is defined as on f 11>' Thenf ' ces", is defined by E", ={xe[o: Theorem 1.3 IfVJlE~~ then for any L>O and 8 > 0, there exists8 > 0 such that IVJl(u+ v)-VJl(u)1 <8 wheneveru,veX9J!with!.m(u):S:Land !.m(v):S:8. Proof. See[7. Lemma2.1]. 1=1 :L:={~~IX(i)IJ ( )) Ilxll= ~l ~~lx(i)1 p Lemma 2.1 If <I> is an Nt -function, then for each x E ce~~there exists k E JR such that 1- IIxIIA ="k(l+ p<I>(kx)). Proof. Foroo each x=(~~IX(i)ll=l x = (X(U):l E ces<I>we have ER<I>'Observe that IlxIIm~=llxllf~, and <I>isan Nt-function, by[8~Corollary2.3] there exists k E IR such that 114<4=llxlli~ =i(l+I<I>OvC)) = i( 1 + L~=l <1>(; L~=lIX(i)I)) =i(1 + p<I>(kx)). This completesthe proof of our Lemma. <co}, equipped W (i~ht~e.norm We first give an important fact forllxllA on ces~. ~ (1.1) Proposition 2.2 Supposethat <1> is an N' -function and let {xn} be a boundedsequence in ces~such thatxn~x for somexEces~. IfknEK(xn) SdenceAsia31 (2005) 175 and kn ~ 00, then x = 0. Proof. For each n E N,1] > 0, put 1] > O,G(n.~) G(n,ry) ={iEl~ First, we claim that for each :+L:;=t!Xn(})I2':lJ}. = 0 for alllarge n E N. Otherwise, without loss of generality, we may assume that G(n,ry) *- 0 for all n E N and for some 1] > O. Then IlxnllA=f(l+P<l)(knxn))~ <D(:nXn) (iEG(n,~))' By applying the assumption that <D is an Nr -function, we obtai~llxn IIA-;. 00, which contradicts to the fact that {xn} is bounded, hence we have the claim. By the claim, we have ~I;=llx /i)I-;. 0 as n -;. 00 for all i E N. ! have xn 0) -;. xCi) for all i E N, so it This implies that xn ci) -;. 0 as n -;. 00 for all i E N. Since xn ~ x, we follows that xCi) =0 for all i E N. Lemma 2.3 For any Orlicz function <D,we have E<I) ~ {x E ces<I) : Ilx- xliIIA-;. O}. Proof. Write A = { X E ces<I) : Ilx that p", xli IIA -;. O}. Let x E E", and e > 0 . Since x E E<I),there exists io E N such (( x - xI,)/ e) < e for alli > io' Therefor e, by the definition ofll'ijA we have , , le-l(x-xl,l :::;l+p",((x-xl,)/e)<l+e for all i > io' This yields II( x - xli )1 A -;. 0 as i -;. 00 since e is arbitrary Hence x E A, proving the Lemma. Theorem 2.4 The space ces~ is (UKK) if <D is an N' -function which satisfies the 62 - condition. Proof. For a given e > 0, by Theorem 1.2 there exists 6 E (0,1) such that IlyliA~ implies p", (y) ~ 26. Given xn E B( ces~) ,xn -;. x weakly and Ilxn - xmIIA~ e( n ::f= m), we shall complete the proof by showing ~ =0, that IIxIIA :::;1- 6. Indeed, if x then it is clear. So, we assumex::f=O. In this case,by Proposition 2.2 we have that {kn} is bounded, where kn E K( xn)' Passingto a subsequenceif necessarywe may assume that kn -;. k for some k > O. Since <DE 62, Lemma 2.3 assures that there exists j E N such that convergence {xn} of implies that Xn -;. x coordinatewise, we deduce Ih IL ~ IlxiiA - that x/i) 6. Since the weak -;. xCi) uniformly on {1,2,...,j}. Consequently, there exists no EN such that II(xn-xm\L :::;~ foralln,m~no' which implies II(Xn -xmt1 This gives orlIXmIN_,IIA ~~ hence pq, ( XnIN-I ~~ for all n,m~no,m*n. foralln,m~n(),m*n, whiChyieldsIlXn,N-,11~~ )~ 2b. Without lossof generalitywe mayassumethat IIXn"-1 t ~ ~, forinfinitelymanynEN, for all n E N. By using the convexity of <t>and the inequality <t>(a + b) ~ <t>(a) + <t>(b), a,b E JR+together with the fact that kn ~ 1, we have 1- 20 ( ) 211xn IIA - P<1> XnlN-i 211xnliA-i-P<1>(knXnIN-i) n ( ( =k:+k:~<1> 1-~ xn(r) +k: ( ) ( 2k:+k: -:- ~ 1 1 00 k ; 1 I) I ( (1-~ xn(r) -;=tl<1>(-:-~ xn(J+r) ;=tl<1> ( ) ( (-:- ~ k 00 i-j =k:+k:~<1> 1-~ xn(r) -k:i=tl<1> 1-~ xn(J+r) I) 1 1 1 j 1 j =k:+k:~<1> 1 1 j k n I k P<1> (knXnli) n 1[ I) 1 k 00 ; I) 1 00 k ;-J ; 1 xn(r) I) 100 n 00 +k:;=tl ~i-+in P<1> I)] I k ; k j k ;-j 1[ -:-~ xn(r)1 +k: ;=tl<1>-:-~IXn(r)I+-:-~IXn(J+r)1 ~<1> =i-+i- ; l k <1> (knx,J j l xn(r) I) 21h L 2/IxliA -0, k ;-j -;=tl<1> -:-~IXn(J+r)1 00 )] 176 ScimceAsia 31 (2005) hence IlxiiA~1-8. function <Dwe have that P<t> (knXljo) ~ 2knP<t> (XI!,). Thus inequalities (2.1) and (2.3) imply that Theorem2.5 If <I>is an N' -function which satisfies 0 - condition, then ces~ has the uniform opial )?roperty IIX+Xnt Proof. Take any G> 0 and x E ces~ with IlxlL :?:G. Let (xn) be weakly null sequencein S( ces~). By <I>E 02' ( IIII and Theorem 1 2x)here is ~E (0,1) independentof x that P 2" > c;. Also, such <1>E 52, we by have ces~=E<I>.By Lemma 2.3, x is an order continuous element, this allows us to find that ~ >1--+--- 4 +2P<D[ X~" (1j .j" nIN-1"A )-f IX(i)l +22:<1>-2:)=1 j ;=1 2 14~ 4 j -- ~ 2 ~ 8 =1+ ~ Ih-i"IL<.f and > x N such )0 E 21IXnIN-j" L 2 for all n > no' which deduces n-->oo lim infllx + xn IIA 21 +~. tB2l ] ~ L: <I> ( ~ j i=1 2. <"8' 00 It follows that Theorem 2.6 If <1> is an Orlicz function which satisfies 62 -condition, then ces~ has the uniform opiai )=.1,+] ~~ t<I> [ ~ )-1 t IX(i)I ] + 00 <I> ) 1=1 2.-L t IX(i)l [~ ) i=l 2. property Proof. Take any G > o and x E ces<D with IlxIIL2 G. Let (xn) be weakly null sequencein S( ces~). By <1> E 62, we have P<DE,1;. Thus by Theorem 1.2, there is 7]E (0,1) independent of x such that 7] < P<D (x) < 00. Also, by P<DE,1;, Theorem 1.3 assertsthatthere exists 7]1E (0,7]) such that j )-),,+1 (l j" .j IX(i)l :5:I<l> -Ij=l j 1=1 2 which implies j, .. 71; -:<:;I<D 8 }=1 j +-,~ 8 [1j IX(i)I j 2 -I-. 1=1 Ipq,(y+z)- pq,(y)1<%' (2.4) whenever pq,(y) S 1 and pq,(z) S 1]]. Since pq,(x) < 00, we choose jo EN such that (2.1) J Fromxn~O, '1eh~vexn i)~O foralliEN, . plies that P$ Xnlj, } ~ O. By Theorem 1.2 we which im \ 00 j=t+l<D ( Ji=t+l 1 j 1 x(i) I haveIh, IIA~ 0, so the~ exists no EN such that IIXnlj,t <4 ) <i=t+l<D( J~ 1 jl x(i) I) <~. 1] (2.5) 00 This gives for all n>no' Therefore, 1]< ~<D(] ~IX(i)I)+ jJ+l <D(] ~Hi)l) Ilx+xnllA =11(x+xn\" +(x+xnt",L ~Ih" +xnIN-;" IIA -llxIN-h' IIA -llxnl;" IL S t<D ( ~tlx(i)I (2.2) > II -- ~ - IIxI" +X nIN-;" A 2. Sincecp is an N'-function, This .11 N-r" A =~ kn together ( (kn(xl +Xnl- ))) . with (2.2) r" and n n ( knXI )+~Pq, ( k knXnl I" n ::::llxn'N-I"L +fPq,(knX'j,,)-f We may assumen without the . )_I2 4 1] 31] 4 4 (1j J~Xn(i)+X(i). I) (2.6) I for all n > no' since the weak convergence implies the coordinatewise convergence. Again by Xn~O, there existsnl >no such thatpq, I. <1]1 for all x nl" n > np so from (2.4) we obtain ( ) Ipq,(XnIN-j"+Xnlj,,)-Pq,(XnIN-j,,)I<%, of generality that kn ::::~. Since 2kn::::1, by convexity of Orlicz j" 4s~<D (2.3) loss I ) ;=1 31] fact suppz= 0, N-I" . 1]1 L<D -:L X(I) I) >1]-->1]--=-. This together with the assumption that Xn~O, there exists no EN such that N )0 that P<!> (Y + z) 2 p<!>(y) + p<!>(z) ifsuppyl1 we have Ilx+xnll A ::::~+~Pq, k k (1 j j" by Lemma 2.1 there j=1 l+P<!> ) +~,4 which implies exists kn > 0 such that Ihr" +Xnl ) ;=1 )=1 since Pq, (xn) =1. Hence, 1-!Z. 4 = pq,(xn) _!Z. < pq,(XnIN-i") = 4 .f J=J" +1 <D t ( ~) I=J,,+1 Ixn(i )1) , SdmceAsia 31 (2005) 177 for all n > nj. This together with (2.4), (2.5) and (2.6) imply that for any n > nj, (j~ l j, I) lj pq,(xn +x)=~<I> xn(i)+x(i) > .~<I>(~ ~IXn (i)+x(i)I)+ 31] ;:::-+ 4 '? 31] + 4 ): 00 . . . ( Ii <1> ~_L .. /-1,,+1 I (l-!l 4)-!l4 (j~ l lj xn(i)+x(i) jJ+l <I>(~iJ+1IXn(i)+X(i)l) xn(i)I) ) l-Jc,+1 00 +i=t+l<I> -- 1] 4 =l+~ 4" By P<1> E L1;, and by Theorem 1.4, there is T depending on 17only such that Ilx. + xilL ~ 1 + T. Corollary 2. 7 ([ 12, Theorem 2]) For any 1 < P < 00, the spacecesphasthe uniform Opial property ACKNOWLEDGMENTS The author would like to thank the Thailand Research Fund(RGJ Project) for the financial support during the preparation of this paper. The first author was supported by The Royal Golden Jubilee Grant PHD/ 0018/2546 and Graduate School, Chiang Mai University. R REFERENCES EFERENCES 1. Huff R (1985) Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math. 10 10, 743-9. 2. Gossez JP and Lami Dozo E (1972) Some geometric properties related to fixed point theory for nonexpensive mappings. Pacific J. Math. 40 40, 565-73. 3. Opial Z (1967) Weak convergence of the sequence of successive approximations for non expensive mappings. Bull. 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