Ultradiversification of Diversities

In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.

On approximating curves associated with nonexpansive mappings

Let X be a Banach space with metric d. Let T, N : X → X be a strict d-contraction and a d-nonexpansive map, respectively. In this paper we investigate the properties of the approximating curve associated with T and N. Moreover, following [3], we consider the approximating curve associated with a holomorphic map f : B → α B and a ρ-nonexpansive map M : B → B, where B is the open unit ball of a complex Hilbert space H, ρ is the hyperbolic metric defined on B and 0 ≤ α < 1. We give conditions on f and M for this curve to be injective, and we show that this curve is continuous.

A Denjoy–Wolff theorem for Hilbert metric nonexpansive maps on polyhedral domains

For a polyhedral domain $\Sigma \subset \mathbb{R}^n$, and a Hilbert metric nonexpansive map T:Σ→Σ which does not have a fixed point in Σ, we prove that the omega limit set ω(x;T) of any point x ∈ Σ is contained in a convex subset of the boundary ∂Σ. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone $\mathbb{R}^n_+$ which demonstrate that there are Hilbert metric nonexpansive maps on an open simplex with omega limit sets that can contain any convex subset of the boundary.

Convergence theorems forI-nonexpansive mapping

We establish the weak convergence of a sequence of Mann iterates of anI-nonexpansive map in a Banach space which satisfies Opial's condition.