Fixed points for total asymptotically nonexpansive mappings in a new version of bead ‎space‎

Abstract

The notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $CAT(0)$ space, where the curvature is bounded from above by zero. In fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. In this paper, we define a new version of bead space and called it $CN$-bead space. Then the existence of fixed point for asymptotically nonexpansive mapping and total asymptotically nonexpansive mapping in $CN$-bead space are proved. In other word, Let $K$ be a bounded subset of complete $CN$-bead space $X$. Then the fixed point set $F(T)$, where $T$ is a total asymptotically nonexpansive selfmap on $K$, is nonempty and closed. Moreover, the fixed point set $F(T)$, where $T$ is an asymptotically nonexpansive selfmap on $K$, is ‎nonempty.