A geometrical coefficient implying the fixed point property and stability results

HOUSTON JOURNAL OF MATHEMATICS Volume 22, No. 4, 1996 A GEOMETRICAL COEFFICIENT IMPLYING THE FIXED POINT PROPERTY AND STABILITY RESULTS T. DOMINGUEZ BENAVIDES Communicated by Gilles Pisier. ABSTRACT. In this paper we define a new geometric constant M(X) in Banach spaces such that X has the fixed point property for nonexpansive mappings if M(X) > 1. We prove that M(X) •_ WCS(X), the inequality being strict in many important classes of Banach spaces and we obtain lower bounds for M(X) based upon either the modulus of near uniform smoothness or the modulus of the Opia] property of the conjugated space. We show that this new constant gives us stability results for the fixed point property with respect to œp-spaces which improve all previous results. Let (M, d) be a metric space. A mapping T: M -• M is said to be nonexpansive if •(r•, r•) _< •(•,•) for every x,y • M. A Banach space X is said to have the fixed point property (f.p.p.) for nonexpansive mappings if for every convex and weakly compact subset C of X, every nonexpansive mapping T: C -• C has a fixed point. In 1965 Browder [B] and Kirk [K], respectively, proved that every uniformly convex Banach space and any Banach space with normal structure has the f.p.p. In 1981 Alspach [A] showed that L• fails to have the f.p.p. Over the last 30 years many papers have appeared studying geometric properties of the Banach spaces (uniform 1991 Mathematics Subject Classification. 47H09,47H10. Key words and phrases. nonexpansive mapping, fixed point, normal structure, uniform smoothness, near uniform smoothness, Opial property. This research is partially supported by the DGICYT (research project PB 93- 1177-C01) and the Junta de Andalucia (project 1241) 835 836 T. DOMINGUEZ BENAVIDES convexity, uniform smoothness, near uniform convexity, unconditional basis, etc) which assure either normal structure or the f.p.p. (see, for instance, [Ma, GK]). A method to assure the f.p.p. for a Banach space X is to use the "proximity" of X to another Banach space Y which "strongly" satisfies this property. To use this method we need a quantification of the f.p.p. The first results in this direction were obtained by Bynum [By1] defining certain normal structure coefficients. In this paper and in latter papers [Pr3, DL] several lower bounds for the normal structure coefficients were obtained based upon the value of certain geometric coefficients (Clarkson modulus of uniform convexity, modulus of uniform smoothness, modulus of near uniform convexity, etc). These bounds can be understood as stability results for the f.p.p. Recently Garcla-Falset [Gal] defined a new geometric coefficient R(X) which assures the f.p.p. (in particular he proved that near uniformly smooth spaces have the f.p.p. in spite this spaces can fail to have normal structure) and he obtained stability results using this coefficient. In this paper, following the idea in [Gal], we define a new coefficient M(X) and we prove that X has the f.p.p. if M(X) > 1. This coefficient is, in general, equal or greater than Bynum's weakly convergent sequence coefficient WCS(X), and strictly bigger than WCS(X) in many special spaces (see Theorem 4.1 and remark after Theorem 2.5.). So we can improve a classic result in metric fixed point theory: Every Banach space with weak uniform normal structure has the f.p.p. Obviously, all lower bounds for WCS(X) also hold for M(X) and, in addition, we show lower bounds for M(X) using either the modulus of near uniform smoothness, defined in [Do], (recall that WCS(X) can be equal to I in near uniformly smooth spaces) or the Opial modulus (see [LTX]) of the dual space. In the case of/•p-Spaces we can directly obtain the value of M(X). This value gives us stability results, which are strictely better than all previous stability results in these spaces [JL, Kh, Pr2]. 1. Notations and preliminaries. In the following, X will be a Banach space, Bx the closed unit ball, of X, and $x the unit sphere. We shall often use Bynum's weakly convergent sequence coefficient WCS(X). Before introducing it, we recall some definitions. The asymptotic diameter and radius of a sequence {x•} in a Banach FIXED POINT PROPERTY 837 space X will be defined by: diama({x,,}) = limsupsup{llx,, - xmll ß n,r• • k}, k ra{Xn} = inf{lim sup IIx• - YlI'Y E {•} }, n The weakly convergent sequence coefficient of a Banach space X is defined by WCS(X) =inf{ diama({Xn}) ß {Xn} is a weakly convergent sequence which is not norm convergent}. It is known [Byl] that X has weak normal structure, that is, every weakly compact convex subset of X with more than one member is not diametral, when WCS(X) > 1. The following result shows how the coefficient WCS(X) can be useful to prove the stability of the fixed point property. Theorem 1.1 [Byl]. Let X and Y be isornorphic Banach spaces, then wcs(x) _< d(x, v)wcs(v). Several improvements of this result can be found in [Pr2]. The following form of WCS(X) [DL, DLX1] will be very important in this paper Theorem 1.2. Let X be a Banach space without the $chur property. Then: WeS(X) = inf{ limn'm;•7•m II• - Xml[ . {•} converges weakly lim sup II•ll to zero and lim I1• - xmllexists} n,m;n•m We recall that the mapping Px (t) defined by {1 } px(t) = sup •(ll• •- Yll •-IIx - YI[) - I ß II•ll • 1, Ilyll • t 838 T. DOMINGUEZ BENAVIDES is called modulus of uniform smoothness of X. A Banach space X is said to be uniformly smooth if limt-•0 px(t) = O. A more general concept is the near uniform smoothness, the dual notion of the near uniform convexity (see [Pr 1]). A Banach space X is said to be near uniformly smooth if for all s > 0 there exists q > 0 such that for each t, 0 < t < q, and for each basic sequence {xn} in Bx there exists k ) 1 such that IIx• + tx•11 < 1 + st. In [Do] a modulus of near uniform smoothness is defined in reflexive Banach spaces by F(t)=sup{inf{ Xl + tXn + Xl - txn , -1 'n>l 2 It is easy to check that 0 < F(t) < t for every t > 0. A Banach space X is said to satisfy the Opial condition [Op] if lim inf IIx• - xll < lim inf IIx• - yll for every sequence {x•} in X weakly convergent to x and every point y • x. We say that X satisfies the uniform Opial condition [Pt4] if for every c > 0, there exists an r = r(c) > 0 such that I + r < liminf IIx + x•ll for all x e X with IIxl] _> c and all weakly null sequences {xn} in X such that liminf•_• IIx•ll _> 1. In [LTX] the following modulus associated to the Opial condition has been defined: Definition 1.3. Let X be a Banach space. The modulus of Opial of X is defined as rx(c) := inf{liminfllx + xnll- 1}, c >_ o, where the infimum is taken over all x • X with x I >- c and all weakly null sequences {x•} in X with liminf IIx•ll ) 1. It is easily seen that the uniform Opial condition implies the Opial condition and that X satisfies the uniform Opial condition if and only if rx(c) > 0 for all c > O. ' {xn}weakly null in Bx} FIXED POINT PROPERTY 839 The following constant for a Banach space X is defined in [Ga2]: R(X) = sup{liminf IIx + where the supremum is taken over all weakly null sequences in Bx and over all vectors x in Bx. In [Gall the following result of existence of fixed points and stability of the f.p.p. is proved Theorem 1.4. Let X and Y isomorphic Banach spaces. If d(X, Y)R(X) < 2 then Y has the f.p.p. Finally, for a Banach space X, [X] will denote, as usual, the quotient space eoo(X)/co(X) endowed with the norm II[z]lllimsup IIz•ll, where [z•] denotes the equivalent class of {z•} • eo•(X). By identifying x • X with the class [x,x,...] we can consider X as a subset of [X]. If K is a subset of X we can consider the set [K] = {[z•] • [X]: z• • K for every n • iN}. If T is a mapping from K into K we define [r]: -. [•:] by = Iraqi. The following lemma is a basic tool in this paper: Lin's lemma 1.5 [L]. Let X be a Banach space and K be a minimal weakly compact convex subset of X which is invariant under a nonexpansive mapping T. If [W] is a nonempty closed convex subset of [K] which is invariant under [T] then sup{ll[w]- xll: [w]}- glare(K) for every x • K. 2. The coefficient M(X) and the f.p.p. In this section we are going to introduce a new coefficient in Banach spaces which yields a new fixed point theorem. As we shall see, this theorem enables us to prove the existence of a fixed point in Banach spaces without normal structure. Previously we need to define a uniparameter family of coefficients. Definition 2.1. Let X be a Banach space. For any nonnegative number a we define the coefficient R(a,X) - sup{liminf IIx• + xll} where the supremum is taken over all x E X with Ilxll _< a and all weakly null sequences in Bx such that lim•,,•;•,• IIx - x•11 _< 1. 840 T. DOMINGUEZ BENAVIDES Theorem 2.2. Let X be a Banach space and assume that for some a _• 0 we have R(a, X) • 1 + a. Then X has the fixed point property. Proof. We follow an argument similar to that in [Gal]. Assume that X fails to have the f.p.p. Then we can find a weakly compact and convex subset K of X such that diam (K) -- I and K is minimal invariant for a nonexpansive mapping T which has no fixed point and we can also find a weakly null approximated fixed point sequence {x•} of T in K. We consider the set [W]- {[z•] e [K]' I[[zd-[xdl[ <_ 1-t and limsuplimsup Ilz•-z,•l[ < t} • m where t = 1/(1 + a). It is easy to check that [W] is a closed,convex and [T]-invariant set. Furthermore [W] is non-empty because it contains [tx•]. Therefore, from Lemma 1.5 we know that sup{[[[wn]- x][ ' [w•] e [W]} -- 1 for every x e K. We take [z•] e [W] and choose a weakly convergent subsequence {y•} of {z•} such that limsup[z•[[ - lim[[y• [and lim•,,•;,•m []y• -y,•[] exists. In this way we have lim ly•-y-•l[ = limsuplimsup ][y•-y,,][ <_ limsuplimsup [[z•-z,•[] _< t. n,rn ;n•rn n rn n rn We denote the weak limit of {y• } as y. For every n e N we have Ily•- yll < lim inf,• [[y,• - •1[. Hence lim sup I[• - •11 = lim suplira sup [ly• - •ml[ < t. r[ r[ m A positive r] can be chosen such that rlR(a,X) < 1 - R(a,X)/(1 + a). For a large enough n we have 11• -•ll < t + .. Furthermore 11•1 < lim inf ][y• - x• [[ _< 1 - t. Hence t•-•v= t+. t-•-Sv<R -W,x =aa, X). Thus limsup [Iz•l[- limlly•[[ < R(a,X)(t + V) < 1 which is contradiction with Lemma 1.5. [] The following stability result, similar to those in Theorems 1.1 and 1.4, can be proved by a straightforward argument: FIXED POINT PROPERTY 841 Theorem 2.3. Let X and Y be isomorphic Banach spaces. Then R(a,Y) < d(x,Y)R(a,x) for every nonnegative number a. Definition 2.4. Let X be a Banach space. We define the coefficient M(X) as { l+a } sup R(a,X) ' a _> 0 . The following result is a direct consequence of Theorems 2.2 and 2.3: Theorem 2.5. Let X be a Banach space. If M(X) > i then X has the f.p.p. If Y is another Banach space which is isomorphic to X and d(X, Y) < M(X) then Y has the f.p.p. Remarks. (a) From Theorem 1.2 it is clear that R(0, X) -- 1/WCS(X). Thus M(X) _> WC$(X). This inequality can be strict. For instance, we consider Bynum's space X = g2,oo, that is, X is g2 with the norm max{[[x+[[, [[x-[[} where x+(n)- max{x(n), 0} and x-(n)- max{-x(n), 0} are respectively the positive and the negative part of x, and [[. [[ is the euclidean norm. Since g•,• fails to have normal structure [By2] we know that WC$(g•,•) = 1. However we shall prove in Section 4 that M(g2,•) = (b) Theorem 2.4 is also a strict improvement of the result in [Gall. Indeed, consider X = g•,l, that is, g• with the norm This space has normal structure [By2, DLX], so M(X) >_ WC'S(X) = x/-•. However, considering the vector x: el and the sequence xn = -en+l it is clear that R(•2,1) = 2. 3. Lower bounds for M(X). Since M(X) _> WCS(X), all lower bounds for WCS(X) based upon the Clarkson modulus of convexity, the modulus of near uniform convexity and the modulus of uniform smoothness (see [By1, Pr3, DL]) also hold for M(X). We shall give in this section several new bounds which do not longer hold for WC$(X). 842 T. DOMINGUEZ BENAVIDES Theorem 3.1. Let X be a refiezive Banach space and denote F=inf l+F(s)-•'sß[0,1] . Then R(a,X) _< I + aF if a _< 2 and R(a,X) _< a + 2F - I if a _> 2. In particular, M(X) _> 3/(1 + 2F) and M(X) > I if F'(0) < 1/2. Proof. The statement is obvious if a = 0. Assume 2 _> a > 0. Let {xn} be a weakly null sequence in Bx and x ß X be a vector such that x I = r _< a. Taking subsequences we can assume that lim IIxn + xll exists. For an arbitrary positive number •7, a number t ß [0, r/2] can be chosen such that 1--+F <F+o. With these assumptions we have [[x + x,l[ = r - + -- _< r - + -x, +(l-t). If {x•} --> 0 it is clear that liminf[lx+x•ll_<r(l+F(2•)) +(i-t)_<r•I+rF+l_<l+aF+a•7. (Note that r(t) _> 0 implies F _> 1/2 > 0 and thus rF < aF). If {x•} does not converge to zero, we can assume that the sequence {yn} defined by Yl = x, Yn '- Xn-1 for n > 1 is a basic sequence with arbitrary basic constant c > I (see [LT, page 5]). Hence, we have 1112•+2t•11 < 1 1 IIx+t11- _ (1111+11+2tx11)< (cll-2txll+ll+2tnll). Taking again subsequences we can assume -+ x• + - x• -I<F -- +7. FIXED POINT PROPERTY 843 Thus ----x,• + -+--x,• +(l-t)_< IIx+x•ll_• r r r r rc t 1] [l+F(2•)+r/]+(l-t) <-c[r(l+F(2•)+r/-•) + +(c-1)t _< c(1 + rF + 2rv) + (c- 1)a _< c(1 + aF + 2av) + (c- 1)a. Hence R(a,X) _< c(1 + aF + 2av) + (c- 1)a. Since c > 1 and r/> 0 are arbitrary we obtain R(a,X) _< 1 + aF. If a _> 2 we have Applying the above result for the sequence 2x/r + x,• we have R(a, X) _< (a- 2) + 1 + 2r = a- 1 + 2r. Taking a = 2 we obtain M(X) _> 3/(1 + 2r). Finally, if F'(0) < 1/2 it is clear that F < 1. [] We have not used in the proof of Theorem 3.1 the condition lim I1• - x,• _< 1. This condition lets us improve the result: Theorem 3.2. Let X be a reflexive Banach space and denote swcs(x) } F•=inf l+F(s)- 2 'sE [0,1] . Then M(X) _> 3/(1+2F'). In particular, M(X) > 1 ifF'(O) < WC$(X)/2. Proof. We use the same arguments as those in the proof of Theorem 3.1, noting that the condition lim•,• ;•#,• x•-x• ] < 1 lets us assume lira I•l <_ 1/WC$(X). [] It is an open question for us if 1/F is a lower bound for M(X) in a similar way as the lower bound for WC$(X) obtained in [Pt3], using the modulus of uniform smoothness.