Convexity and metric fixed point theory (Q2711859)

This article deals with two concepts, abstract convexity structures and metric convexity, and their relationship to metric fixed point theory. The authors discuss a series of known and new results about fixed points of nonexpansive mappings and mappings with similar properties in the framework of the general theory of metric spaces involving suitable assumptions on ball intersections. Among many curious results of this article one can findNEWLINENEWLINENEWLINEthe extension of the Goebel theorem about the existence of fixed points for a mapping \(T\) with Lipschitz constant \(\Lambda(T)< 2\) and with the nonexpansive square \(T^2\) in bounded hyperconvex metric spaces,NEWLINENEWLINENEWLINEthe extension of the Kirk theorem about the existence of fixed points for a mapping \(T\) whose iterates \(T^n\) with sufficiently large \(n\) are nonexpansive also in bounded hyperconvex metric spaces.NEWLINENEWLINENEWLINEFurther, the authors prove that \(\text{Fix }T\) of a nonexpansive and rotative operator \(T\) in a metric space of hyperbolic type is a nonempty nonexpansive retract (an operator \(T\) is rotative if \(d(x, T^nx)\leq ad(x, Tx)\) for some \(a< n\) and \(n\in\mathbb{N}\)), and the theorem about the existence of fixed points for a mapping \(T\) with Lipschitz constant \(\Lambda(T)< 2\) and with the nonexpansive and rotative square \(T^2\) in metric spaces of hyperbolic type. At last, for nonexpansive (or asymptotically nonexpansive and Lipschitzian) operators \(T\) in spherically complete ultrametric spaces it is proved that either \(T\) has a fixed point, or there exists a ball of radius \(r>0\) such that \(\limsup_{n\to\infty} d(x,T^n x)= r\) for all \(x\) from this ball.NEWLINENEWLINENEWLINEThe authors also analyze properties (normality, compactness, and so on) of abstract convex structures and formulate some open problems in this field.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00012].