Narrow Results

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space .

In this paper we investigate the geometry of when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, is not non-positively curved, our results show that have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in .

Let $X$ be a complete CAT(0) space, $K\ne \varnothing$ be a closed and convex subset of $X$ and $T:K\to C(K)$ be a multivalued nonexpansive mapping. We prove that the sequence of Ishikawa iteration converges to an endpoint of $T$. This improves, extends and unifies some recently announced results of the current literature.

In this paper, we introduce a viscosity-type proximal point algorithm comprising of a finite composition of resolvents of monotone bifunctions and a generalized asymptotically nonspreading mapping recently introduced by Phuengrattana [Appl. Gen. Topol.18 (2017) 117–129]. We establish a strong convergence result of the proposed algorithm to a common solution of a finite family of equilibrium problems and fixed point problem for a generalized asymptotically nonspreading and nonexpansive mappings, which is also a unique solution of some variational inequality problems in an Hadamard space. We apply our result to solve convex feasibility problem and to approximate a common solution of a finite family of minimization problems in an Hadamard space.

We give an exposition of a fixed-point property of random groups of the triangular model obtained in the joint work with Izeki and Nayatani.⁹