We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum X ⊕ Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.2010 Mathematics Subject Classification. 47H10, 46B20, 47H09.
We show a Wolff-Denjoy type theorem in complete geodesic spaces in the spirit of Beardon's framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in R n or C n with respect to a large class of metrics including Hilbert's and Kobayashi's metrics. The results are generalized to 1-Lipschitz compact mappings in infinitedimensional Banach spaces.
It is shown that if S is a commuting family of weak * continuous nonexpansive mappings acting on a weak * compact convex subset C of the dual Banach space E, then the set of common fixed points of S is a nonempty nonexpansive retract of C. This partially solves a long-standing open problem in metric fixed point theory in the case of commutative semigroups.
A Banach space X is said to have property (S m ) if every metrically convex set A C X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.
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