PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 105, Number I. January 1989 FIXED POINTS OF NONEXPANSIVE MAPPINGS IN BANACH LATTICES M. A. KHAMSI AND PH. TURPIN (Communicated by William J. Davis) Abstract. We prove the existence of a fixed point for a nonexpansive mapping operating in a convex subset of a Banach lattice E compact for some natural topology t on E . In particular, if £ is a Banach space with a 1-unconditional basis we can take for z the topology of coordinatewise convergence. 1. INTRODUCTION If B is a subset of a Banach space, a map T : B -» B is said to be nonex- pansive when the inequality ||7Xx) - T(y)\\ < ||x-y|| holds for every pair x, y in B . The main result of this paper (Corollary 1) is the following one. Let £ be a Banach space endowed with a 1-unconditional Schauder basis, i.e. a Schauder basis («>„)„>„ such that ||x0é>0 + • • • + xneH + ■ ■ ■ || < \\yoe0 + • • • + ynen + • • • || if \xn\ < \yn\ for every n (in fact, this condition will be slightly weakened). Let B be a convex nonvoid subset of E, compact for the topology of coordinatewise convergence. Then every nonexpansive map T: B —► B has a fixed point. This was proved by Lin [Ln] in the special case where B is a weakly compact convex set. The method of [Ln] is a refinement of techniques of Maurey [M, E-L-O-S]. It turns out that this method still works with the topology of coordi- natewise convergence once a key lemma of Goebel and Karlovitz [G, Ka] has been generalized (Lemmas 4 and 5). Let us notice that our proof avoids any use of ultraproducts. In fact, in the theorem below, we give a more general result, considering ar- bitrary Banach lattices E and proving the above fixed point property in convex subsets compact for some natural topology r on E. In usual spaces of mea- surable functions with order continuous norm (Corollary 2), x is the topology of convergence in measure on every set with finite measure. Received by the editors September 3, 1987 and, in revised form, January 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 47H10; Secondary 46B30, 46B05. ©1989 American Mathematical Society 0002-9939/89 $1.00+$.25 per page 102 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use