On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature

Convergence of nonlinear semigroups under nonpositive curvature

The present paper is devoted to semigroups of nonexpansive mappings on metric spaces of nonpositive curvature. We show that the Mosco convergence of a sequence of convex lsc functions implies convergence of the corresponding resolvents and convergence of the gradient flow semigroups. This extends the classical results of Attouch, Brezis and Pazy into spaces with no linear structure. The same method can be further used to show the convergence of semigroups on a sequence of spaces, which solves a problem of [Kuwae and Shioya, Trans. Amer. Math. Soc., 2008].

$α$-Firmly Nonexpansive Operators on Metric Spaces

We extend to $p$-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise $α$-averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying $α$-firm nonexpa…

Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(κ)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $σ$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-…

Continuous Selections of Lipschitz Extensions in Metric Spaces

This paper deals with the study of parameter dependence of extensions of Lipschitz mappings from the point of view of continuity. We show that if assuming appropriate curvature bounds for the spaces, the multivalued extension operators that assign to every nonexpansive (resp. Lipschitz) mapping all its nonexpansive extensions (resp. Lipschitz extensions with the same Lipschitz constant) are lower semi-continuous and admit continuous selections. Moreover, we prove that Lipschitz mappings can be extended continuously even when imposing the condition that the image of the extension belongs to the…

On generic convergence of successive approximations of mappings with convex and compact point images

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

等距叶上的测度收缩性质与单调纤维

Measure contraction property on isometric leaves and monotone fibres

本文证明非扩张映射的等距叶上测度收缩性质不变，并推广到单调纤维，同时通过反例否证Klartag猜想，揭示余维数一情形下的继承条件。

Monotone vector fields and generation of nonexpansive semigroups in complete CAT(0) spaces

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT(0) spaces stands in opposed to the monotonicity defined earlier in CAT(0) spaces by Khatibzadeh and Ranjbar [14] and Chaipunya and Kumam [8]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida app…