The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

We study a nonlinear semigroup associated to a nonexpansive mapping on a Hadamard space and establish its weak convergence to a fixed point. A discrete-time counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literature -- both classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete, as well as continuous times.Comment: Accepted in JFPT

Two modified proximal point algorithms in geodesic spaces with curvature bounded above

We obtain existence and convergence theorems on two variants of the proximal point algorithm for proper lower semicontinuous convex functions in complete geodesic spaces with curvature bounded above.Comment: 22 pages; some minor corrections were mad