DUALITY MAP CHARACTERISATIONS FOR OPIAL CONDITIONS

Transcription

1 DUALITY MAP CHARACTERISATIONS FOR OPIAL CONDITIONS TIM DALBY AND BRAILEY SIMS We characterise Opial's coditio, the o-strict Opial coditio, ad the uiform Opial coditio for a Baach space X i terms of properties of the duality mappig from X ito X'. I 967, Opial [4] itroduced the folloig coditio o a Baach space X If (2,) coverges ealy to x the lim if - limif x, - 2 forallx#x. This coditio has bee used i the study of the existece of fixed poits for oexpasive maps. For example, Gossez ad Lami Dozo [2] have sho that Opial's coditio implies ea ormal structure ad hece the ea fixed poit property. A eaer coditio, o-strict Opial, is that (x) covergig ealy to x implies lim if - limif llx - 2 for all x. Agai, this coditio is associated ith the ea fixed poit property. See, for example, Sims [7]. I the opposite directio Prus [5] i 992 itroduced the uiform Opial coditio. For c > 0 defie the Opial modulus of X to be ~ ( c = ) if limif llx : x >/ C, x 0, ad limif llxll 2 { I The ~ ( c is ) a icreasig fuctio of c, ad e say X has the uiform Opial property if ~ ( c ) > 0, for c > 0, i hich case e have + ~ ( c 6 ) limif 2, + xll Received 25th July, 995 We ish to tha Michael Smyth for his isightful commets regardig the proof of Lemma 3. Copyright Clearace Cetre, Ic. Serial-fee code: $A

2 44 T. Dalby ad B. Sims [2] heever x - 0, limif llxll 3 I, ad /x 2 C. For < p < oo, the space tp satisfies the uiform Opial coditio hilst Lp[O,, p # 2, fails eve the o-strict Opial coditio. A gauge, p, is a cotiuous strictly icreasig real-valued fuctio o [0, oo) satisfyig p(0) = 0 ad lim p(t) = oo. A mappig J, : X -+ X* is called a duality t-00 mappig ith gauge fuctio p if for every x E X If p(t) = t e rite J istead of J,. X is said to have a ealy cotiuous duahty map if there exists a gauge,u such that the duality map J, is sigle-valued ad sequetially cotiuous from X ith the ea topology to X* ith the ea * topology. Gossez ad Lami Dozo [2], i 972, shoed that a Baach space ith a ealy cotiuous duality map satisfies Opial's coditio. Recetly, Li, Ta ad Xu [3] improved o this by shoig that such a space has the uiform Opial coditio. More recetly still, Beavides, Acedo ad Xu [I.] have produced a example, tp,,, that satisfies the uiform Opial coditio but fails to have a ealy cotiuous duality map. This aturally raises the questio of a duality map characterisatio of the uiform Opial coditio. Sims [6] i 985 characterised Opial's coditio i terms of the asymptotic ature of J(x) here (2,) is a o-ull ealy coverget sequece. More precisely e have the folloig. THEOREM. A Baach space satisfies Opial's coditio if ad oly if heever (2,) coverges ealy to a o-zero Limit x,, for xc E J(x) e have limif xc(x,) > 0. A examiatio of the proof shos that the folloig is also true. THEOREM 2. A Baach space satisfies the o-strict Opial coditio if ad oly if heever (2,) coverges ealy to a o-zero Limit x,, for x; E J(x) e have limif xt(x,) 2 0. Here e complete the cycle by extedig the techiques of [6] to obtai a characterisatio of the uiform Opial coditio. We begi by shoig that the uiform Opial coditio is determied i the folloig ay. Note: the subsequetial form of this characterisatio is ot eeded for our later proofs, but is icluded for its potetial utility.

3 [3 Duahty map characterisatios LEMMA 3. For a Baach space X the folloig are equivalet: (i) X has the uiform Opial coditio. (ii) There exists a strictly positive fuctio p such that heever x, 0, lim JJx, = ad JJxll 2 C, there exists a subsequece (x) ith limif x, + xi b + p(c). PROOF: Clearly (i) implies (ii) ad (ii) implies Opial's coditio. No suppose X has (ii) but fails to have the uiform Opial coditio. The there exists a c > 0 ad, for each r E N, a sequece x: - 0, as --+ oo, ith ad a xm ith Jlxmll 2 c SO that rm := limif JJx:II 2 m limif (Jx: + xm(l < + -. NOTE. By passig to a subsequece e ca, ad shall, assume that both of the above lim if 's are i fact limits. Also, sice X has Opial's coditio, NOW, let yr = X:/T, ad ym = xm/rm, the yz - 0, as + oo, lim I ~ Y ~ I I I ( y m l ( 2 c/rm 2 c/2, hile =, The sequece (yz) for m > lip(c/2) cotradicts (ii), so (ii) implies (i). 0 We shall say that a Baach space X has Property (D) if there exists a i creasig strictly positive fuctio a o (0, oo) such that heever x - x, # 0, lim (Jx, - x,/( =, ad xz E J(x), e have limif xi(x,) 2 a(l(x, (). OBSERVATION. From Theorem it is clear that (D) implies Opial's coditio. We o sho that (D) is ecessary for the uiform Opial coditio.

4 46 T. Dalby ad B. Sims [4 LEMMA 4. If X has uiform Opial coditio the X has property (D) ith ~(t) = tt(t) PROOF: Let x - x, xt E J(xj such that f 0 ith lim llx - xmll = ad suppose there exists limifx;(x,) < (xw/ T(/Ix,II). The there exists a subsequece (x,~) ith m x ( x ) < x T ( x ) uiform Opial coditio, By the limif llx( = lim:f /I(x - x,) +xl\ 2 + T(~Wl) = lip - x + ~(~) ", 2 limif -(x - xm) + ~(~~) l~ * " 2 limif llx ( - limsup ~ ( 2, ) l\xl/ +~(Ilx,ll). Thus, cotradictig the choice of 0 We o use a modificatio of a argumet suggested i [2] ad developed i [6] to establish a coverse to Lemma 4. THEOREM 5. A Baach space X has the uiform Opial coditio if ad oly if its duality map satisfies property (D). PROOF: ( j) has bee established i lemma 4. (e) We use the characterisatio of the uiform Opial coditio give i Lemma 3. Thus, let (2,) be a ea ull sequece ith llx -t. The, for x f 0 here g:(t) := lim h+t+ IIx + hzl2 - I ~ X, + tx2 h-t

5 [5 Duality map characterisatios 4 7 is the upper Gateaux derivative at t of the covex fuctio t H 2 ilx + t ~~, ad so is a icreasig fuctio of t, equal to max {x:(x) : x: E J(x + tx)). No, for ay E > 0, 2, +EX - EX # 0 ad so, sice (D) implies Opial's coditio, e see from Theorem that for sufficietly large, g :(~) > 0. Thus for sufficietly large Sice the g;(t) are uiformly bouded it follos that limif /x + x ~~' ) limif llxl2 + 2irif g:(t)dt Thus, X satisfies (ii) of lemma 3 ith [I] T. Domiguez Beavides, G. L6pez Acedo ad H-K Xu, 'Qualitative ad quatitative properties for the space Lp,q ', (preprit). [2] J.P. Gossez ad E. Lami Dozo, 'Some geometric properties related to the fixed poit theory for oexpasive mappigs', Pacific J. Math. 40 (972), [3] P-K. Li, K-K. Ta ad H-K. Xu, 'Demiclosedess priciple ad asymptotic behaviour for asymptotically oexpasive mappigs', Noliear Aal. (to appear). [4] Z. Opial, 'Wea covergece of the sequece of successive approximatios of oexpasive mappigs', Bull. Amer. Math. Soc. 73 (967), [5] S. Prus, 'Baach spaces ith the uiform Opial property', Noliear Aal. 8 (992), [6] B. Sims, 'A support map characterizatio of the Opial coditios', Proc. Cetre Math. Aal. Austral. Nat. Uiv. 9 (985), [7] B. Sims, 'A class of spaces ith the ea ormal structure', Bull. Austral. Math. Soc. 49 (994), Departmet of Mathematics The Uiversity of Necastle Ne South Wales 2308 Australia bsimsqfrey.ecaste.edu.a~