Andrzej Wiśnicki

Products of uniformly noncreasy spaces

We show that finite products of uniformly noncreasy spaces with a strictly monotone norm have the fixed point property for nonexpansive mappings. It gives new and natural examples of superreflexive Banach spaces without normal structure but with the fixed point property.

On the fixed points of nonexpansive mappings in direct sums of Banach spaces

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 of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.

Towards the fixed point property for superreflexive spaces

Let C be a nonempty, bounded, closed and convex subset of a Banach space X andlet T : C —> C be a nonexpansive mapping, that is —, Ty\\ \\Tx ^ \\x-y\\ for all x,y £ C.We say that X has the fixed point property (FPP in short) if every such mapping has afixed point.Fixed point theory for nonexpansive mappings has its origins in 1965, when Browder[3] proved that a Hilbert space has FPP. In the same year Browder [4] and Gohde [11]showed that all uniformly convex spaces have FPP and Kirk [16] proved more generalresult stating that all Banach spaces with the so-called normal structure have the fixedpoint property for weakly compact, convex sets. In particular, all reflexive spaces withnormal structure have FPP. The problem whether reflexivity implies the fixed pointproperty and the converse question, in spite of many investigations in this direction, areboth still open.However, there are some partial results concerning this problem. In [18] Maureyused the Banach space ultraproduct construction to prove the fixed point property forall reflexive subspace Z<i[0,1]s of H.e also showed that isometries in superreflexive spacesalways have FPP. Note that quite recently Dowling and Lennard [6] have proved thatevery nonreflexive subspace of Li[0,1] fails FPP.The ultrapower techniques of Maurey have been extended by many authors and alot of strong and deep results in metric fixed point theory have been obtained in this way(see for instance [1, 7, 9, 17, 19, 20]).

A common fixed point theorem for a commuting family of weak* continuous nonexpansive mappings

It is shown that if

The fixed point property in direct sums and modulus \(R(a,X)\)

S

On the structure of fixed-point sets of asymptotically regular semigroups

is a commuting family of weak

The super fixed point property for asymptotically nonexpansive mappings

^{\ast }

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

continuous nonexpansive mappings acting on a weak

Banach ultrapowers and multivalued nonexpansive mappings

^{\ast }

On the super fixed point property in product spaces

compact convex subset