# UNIFORM NORMAL STRUCTURE IS EQUIVALENT TO THE JAGGI* UNIFORM FIXED POINT PROPERTY

Santos, Eduardo Castillo and Lennard, Chris and Sims, Brailey UNIFORM NORMAL STRUCTURE IS EQUIVALENT TO THE JAGGI* UNIFORM FIXED POINT PROPERTY. Journal of Mathematical Analysis and Applications . ISSN 0022-247X (Unpublished)

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Jaggi and Kassay proved that for re exive Banach spaces X, normal structure is equivalent to the Jaggi fixed point property (i.e. all Jaggi-nonexpansive maps on closed, bounded, convex sets in X have a fixed point); which we note is equivalent to a natural variation: the Jaggi* fixed point property. In the spirit of this result, we prove that for all Banach spaces X, uniform normal structure is equivalent to the Jaggi* uniform fixed point property: i.e. there exists a constant $\gamma_0 \in (1\infty)$ such that for all $\gamma \in [1,\gamma_0)$, every Jaggi* $\gamma$-uniformly Lipschitzian map $T$ on a closed, bounded, convex subset $K$ of $X$ has a fixed point. Here, $T$ is Jaggi* $\gamma$-uniformly Lipschitzian if for all $T$-invariant subsets $G$ of $K$, for all $x \in \={co}(G), for all$n \in N\$ \[ sup_{z\in G} \|T^nx-T^nz\|\leq sup_{z\in G}\|x-z\|.