On Uniformly Gâteaux Smooth Norms and Normal Structure

It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure. A Banach space is said to have the fixed point property (FPP) if for every nonempty bounded closed convex C ⊂ X and every nonexpansive self-mapping T : C → C there is a fixed point of T in C. A Banach space is said to have the weak fixed point property (w-FPP) if for every nonempty weakly compact convex C ⊂ X there is a fixed point for every nonexpansive T : C → C. Clearly, a Banach space has w-FPP if it has FPP. The space c0 has w-FPP but does not have FPP; see [M]. These two notions obviously coincide in reflexive spaces. The classical results in metric fixed point theory state that a Banach space has w-FPP if its norm is uniformly Fréchet differentiable ([K]) or uniformly rotund ([B]). In fact, instead of uniformly rotund, it is sufficient to assume that the norm is only uniformly rotund in every direction (URED), [Z]. It is a natural question whether the uniform Fréchet differentiability can be weakened to uniform Gâteaux differentiability (UG), since the notion of UG is dual (in a sense) to URED. (In fact, UG is dual to weak∗ uniform rotundity, which is a stronger notion than URED.) We note that in a non-separable case, a theorem of [DLT] states that for any uncountable set Γ, the non-separable space c0(Γ) does not have FPP under any equivalent renorming. But it is well known that for any set Γ, c0(Γ) has an equivalent renorming that is simultaneously locally uniformly rotund, Fréchet differentiable and UG; see e.g. [DGZ, II.7.8]. Thus even norms with rather good geometrical properties do not assure FPP. In our note we show that the usual proofs of “UF, UR or URED implies w-FPP” cannot be adapted, since they prove the w-FPP by showing that UF, UR or URED implies that the norm has a normal structure. We show that, in contrast, if the norm of a Banach space is UG, it does not necessarily have a normal structure. Even more, every separable Banach space can be equivalently renormed to have a uniformly Gâteaux smooth norm that lacks asymptotic normal structure. This Received by the editors January 9, 2006. 2000 Mathematics Subject Classification. Primary 46B20.