Asymptotic normal structure and fixed points of nonexpansive mappings
Jean-Bernard Baillon, Rainald Schöneberg · Proceedings of the American Mathematical Society · 1981
A mapping f f defined on a subset X X of a Banach space E E and taking values in E E is said to be nonexpansive if | f ( x ) − f ( y ) | ⩽ | x − y | \left | {f(x) - f(y)} \right | \leqslant \left | {x - y} \right | for all x , y ∈ X x,y \in X . In this paper we introduce a promising new geometric property of Banach spaces and show that it yields via a minor modification of known arguments a new fixed point theorem for nonexpansive mappings which includes Kirk’s famous result as well as a recent result of Karlovitz. We also discuss in detail a situation not covered by this result.