A mapping T : C → X defined
on a subset C of a Banach space X, with norm ∥⋅∥, is said to be nonexpansive if
∥Tx − Ty∥≦∥x − y∥ for all x,y ∈ C. If C is assumed to be convex and weakly
compact and if T : C → C then one of the main open questions is whether T has a
fixed point in C, i.e., whether there exists x ∈ C so that Tx = x. If X is reflexive and
uniformly convex or, more generally, if X is reflexive and has normal structure then
the answer is affirmative. Our purpose is to give an example of a classical reflexive
space which does not have normal structure and for which the answer is nevertheless
affirmative.
