Introduction and preliminary results

Let C be a nonempty subset of a Banach space X, with norm . A mapping is called nonexpansive if for all . We say that X has the weak fixed point property (WFPP) if every nonexpansive mapping defined on a nonempty convex and weakly compact subset K of X has a fixed point.

Perhaps, the most classical result in this area is that of Kirk [K] which asserts that every Banach space with normal structure has the WFPP. Since then, many efforts has been directed into finding geometrical conditions on a Banach space X which guarantee the WFPP for X.

Although Alspach [A] showed that lacks the WFPP, an earlier result of Day-James-Swaminathan ([DJS]) assures that there exists a renorming of with the WFPP. Hence the WFPP is not invariant under topological isomorphisms. Nevertheless, many stability results has beenestablished for the WFPP in terms of the Banach-Mazur distance and othercoefficients of Banach spaces.

Recently, J. García-Falset [Ga] showed that a Banach space Y has the WFPP if there exists a Banach space X such that , where d(X,Y) is the Banach-Mazur distance from X to Y and R(X) is the supremum of the numbers , the supremum being taken over all weakly null sequences in the unit ball of X and all points x in the unit ball of X. In the proof of this result, García-Falset uses a nice trick combined with "nonstandard" techniques, a method introduced in Fixed Point Theory by Maurey [Ma] and afterwards developped by Khamsi, Lin and Borwein and Sims, among others. (See [AK]).

Inspired on the proof of [Ga], Dominguez-Benavides [Do] has improved the stability constant . In order to do this, he defined, for a Banach space X and a nonnegative real number a, the coefficient

where the supremum is taken over all with and all weakly null sequences in the unit ball of X such that

It is shown in [Do] that a Banach space Y has the WFPP if there exists a Banach space X such that

It turns out that for , and that for a Hilbert space X, . Hence, the result of [Do] improves all previously known stability results in Hilbert spaces and , (see [BeS, By, P2, Kh, JL1]).

In this paper we improve the stability constant M(X) for a certain class of Banach spaces. Since we follow the proof of [Do], we need to state some basic results.

Suppose that C is a nonempty, convex and weakly compact subset of the Banach space X and that is nonexpansive. Standard arguments show that C contains a subset K which is minimal for the properties of being nonempty, convex, weakly compact and T-invariant, and that K contains an approximate fixed point sequence (afps in short) for T (i.e., a sequence in K such that ). The well known Goebel-Karlovitz lemma (see [GK]) ensures that if K is minimal for T and is an afps for T in K, then is diametral, i.e., for every .

We denote by the quotient space endowed with the norm , where denotes the class of . Any will be considered as an element of by identifying x with the class which contains the constant sequence . If C is a bounded subset of X and is a mapping, we denote by the set

and by the mapping defined by . With these notations we can state the following version of the Goebel-Karlovitz lemma (see [Ga, Do]):

It is shown in [Ga] that if and then and so we can define . The next lemma is also a result from [Ga]: