A Fixed Point Theorem for Mappings which do not Increase Distances

Abstract

In this article, we prove a fixed point theorem for cyclic relatively nonexpansive mappings in the setting of generalized semimetric spaces by using a geometric notion of seminormal structure and then we conclude a result in uniformly convex Banach spaces. We also discuss on the stability of seminormal structure in generalized semimetric spaces.2010 MSC: 47H10; 46B20.

Fixed point property and normal structure for Banach spaces associated to locally compact groups

“…It is also well-known (Browder's Theorem [6]) that uniformly convex Banach spaces have the fpp. In [14] Kirk, extending Browder's Theorem, showed that a weakly compact convex subset of a Banach space with normal structure has the fpp. In [2] Alspach exhibited a weakly compact convex subset K of the Lebesgue space L 1 [0,1] and an isometry T : K → K without a fixed point, proving, thereby, that the space L 1 [0, 1] does not have the fpp.…”

The fixed point property for some uniformly nonoctahedral Banach spaces

“…It is known that the condition eo(X) < 1 implies that X has the w-fpp. In fact, those Banach spaces X with £o(^O < 1 have normal structure and then Kirk's theorem applies [7]. Nevertheless, it remains unknown whether the w-fpp holds for every uniformly nonsquare Banach space X (that is, £o(X) < 2).…”

Uniform asymptotic normal structure, the uniform semi‐Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I

“…It was introduced by Brodskii and Milman [6] and applied in Kirk's wellknown fixed point theorem [24]. Asymptotic normal structure appeared for the first time in a paper by Baillon and Schöneberg [4] in which they generalized Kirk's theorem.…”