Uniform Normal Structure Is Equivalent to the Jaggi* Uniform Fixed Point Property

Jaggi and Kassay proved that for reflexive Banach spaces X, normal structure is equivalent to the Jaggi fixed point property (i.e. all Jagginonexpansive 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 γ0 ∈ (1,∞) such that for all γ ∈ [1, γ0), every Jaggi* γ-uniformly Lipschitzian map T on a closed, bounded, convex subset K of X has a fixed point. Here, T is Jaggi* γ-uniformly Lipschitzian if for all T -invariant subsets G of K, for all x ∈ co(G), for all n ∈ N sup z∈G ‖Tx− Tz‖ ≤ γ sup z∈G ‖x− z‖ .