Common fixed point results in CAT(0) spaces

Let X be a complete CAT(0) space, T be a generalized multivalued nonexpansive mapping, and t be a single valued quasi-nonexpansive mapping. Under the assumption that T and t commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) [7] and [12], Shahzad and Markin (2008) [6], and Dhompongsa et al. (2005) [5].     

A note on fixed point sets in CAT(0) spaces

We show that the fixed point set of a quasi-nonexpansive selfmap of a nonempty convex subset of a CAT(0) space is always closed, convex and contractible. Moreover, we give a construction of a continuous selfmap of a CAT(0) space whose fixed point set is prescribed.     

The fixed point property and unbounded sets in CAT(0) spaces

In this work we study the fixed point property for nonexpansive self-mappings defined on convex and closed subsets of a CAT(0) space. We will show that a positive answer to this problem is very much linked with the Euclidean geometry of the space while the answer is more likely to be negative if the space is more hyperbolic. As a consequence we extend a very well known result of W.O. Ray on Hilbert spaces.     

Fixed point results for multimaps in CAT(0) spaces

Common fixed point results for families of single-valued nonexpansive or quasi-nonexpansive mappings and multivalued upper semicontinuous, almost lower semicontinuous or nonexpansive mappings are proved either in CAT(0) spaces or R-trees. It is also shown that the fixed point set of quasi-nonexpansive self-mapping of a nonempty closed convex subset of a CAT(0) space is always nonempty closed and convex.     

Iterative approaches to solving convex minimization problems and fixed point problems in complete CAT(0) spaces

In this paper, we propose a new modified proximal point algorithm for finding a common element of the set of common minimizers of a finite family of convex and lower semi-continuous functions and the set of common fixed points of a finite family of nonexpansive mappings in complete CAT(0) spaces, and prove some convergence theorems of the proposed algorithm under suitable conditions. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.     

Proximal point algorithms for solving convex minimization problem and common fixed points problem of asymptotically quasi-nonexpansive mappings in CAT(0) spaces with convergence analysis

In this paper, we introduce the modified proximal point algorithm for common fixed points of asymptotically quasi-nonexpansive mappings in CAT(0) spaces and also prove some convergence theorems of the proposed algorithm to a common fixed point of asymptotically quasi-nonexpansive mappings and a minimizer of a convex function. The main results in this paper improve and generalize the corresponding results given by some authors. Moreover, we then give numerical examples to illustrate and show efficiency of the proposed algorithm for supporting our main results.     

Representations of groups with CAT(0) fixed point property

We show that certain representations over fields with positive characteristic of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over \({\mathbb {Z}}\), \({\mathrm{SL}}_k({\mathbb {Z}})\), the special automorphism group of a free group, \(\mathrm{SAut}(F_k)\), the mapping class group of a closed orientable surface, \(\mathrm{Mod}(\Sigma _g)\), and many other groups. In the case of characteristic zero, we show that low dimensional complex representations of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image if they always have compact closure.     

CAT(0) spaces and expanders

We construct a CAT(0) space Y with Izeki–Nayatani invariant δ(Y) =?1. By a similar construction, we also prove that there exists a CAT(0) space which does not have Markov type p for every p?>?1.     

Invariant approximations in CAT(0) spaces

Some common fixed point and invariant approximation results for CAT(0) spaces are obtained. Our results improve and extend some results of Shahzad and Markin [N. Shahzad, J. Markin, Invariant approximation for commuting mappings in hyperconvex and CAT(0) spaces, J. Math. Anal. Appl. 337 (2008) 1457–1464] and Dhompongsa, Kaewkhao, and Panyanak [S. Dhompongsa, A. Kaewkhao, B. Panyanak, Lim’s theorem for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl. 312 (2005) 478–487].     

Alternating projections in CAT(0) spaces

By using recently developed theory which extends the idea of weak convergence into CAT(0) space we prove the convergence of the alternating projection method for convex closed subsets of a CAT(0) space. Given the right notion of weak convergence it turns out that the generalization of the well-known results in Hilbert spaces is straightforward and allows the use of the method in a nonlinear setting. As an application, we use the alternating projection method to minimize convex functionals on a CAT(0) space.     

The hybrid projection algorithm for finding the common fixed points of nonexpansive mappings and the zeroes of maximal monotone operators in Banach spaces

The proposal of this article is to construct a new modified block by using the hybrid projection method and prove the strong convergence theorem for this method, which include the fixed point set of an infinite family of weak relatively nonexpansive mappings and zeroes of a finite family of maximal monotone operators in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property. The results presented in this article improve and generalize some well-known results in the literature.     

The proximal point algorithm in metric spaces

The proximal point algorithm, which is a well-known tool for finding minima of convex functions, is generalized from the classical Hilbert space framework into a nonlinear setting, namely, geodesic metric spaces of non-positive curvature. We prove that the sequence generated by the proximal point algorithm weakly converges to a minimizer, and also discuss a related question: convergence of the gradient flow.     

Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces

Ricerche di Matematica - In this paper, we use the gate condition on two multivalued k-demicontractive mappings to approximate a common solution of a finite family of monotone inclusion problem and...     

Lim's theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀x∈E, ∀α∈[0,1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421-430]. The related result for unbounded R-trees is given.     

A residual algorithm for finding a fixed point of a nonexpansive mapping

This paper presents a weak convergence residual algorithm for finding a fixed point of a nonexpansive mapping in a real Hilbert space. To study the numerical behavior of the algorithm it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.     

A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz-type condition as well as Hölder continuity of the bifunctions involved.     

Boundaries of cocompact proper CAT(0) spaces

A proper CAT(0) metric space X is cocompact if it has a compact generating domain with respect to its full isometry group. Any proper CAT(0) space, cocompact or not, has a compact metrizable boundary at infinity ∂_∞X; indeed, up to homeomorphism, this boundary is arbitrary. However, cocompactness imposes restrictions on what the boundary can be. Swenson showed that the boundary of a cocompact X has to be finite-dimensional. Here we show more: the dimension of ∂_∞X has to be equal to the global ?ech cohomological dimension of ∂_∞X. For example: a compact manifold with non-empty boundary cannot be ∂_∞X with X cocompact. We include two consequences of this topological/geometric fact: (1) The dimension of the boundary is a quasi-isometry invariant of CAT(0) groups. (2) Geodesic segments in a cocompact X can “almost” be extended to geodesic rays, i.e. X is almost geodesically complete.     

Quadratic divergence of geodesics in CAT(0) spaces

A finite CAT(0) 2-complexX is produced whose universal cover possesses two geodesic rays which diverge quadratically and such that no pair of rays diverges faster than quadratically. This example shows that an aphorism in Riemannian geometry, that predicts that in nonpositive curvature nonasymptotic geodesic rays either diverge exponentially or diverge linearly, does not hold in the setting of CAT(0) complexes. The fundamental group ofX is that of a compact Riemannian manifold with totally geodesic boundary and nonpositive sectional curvature.Partially supported by NSF grant DMS-9200433     

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification ${ \overline{X} = X \cup \partial X}We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification [`(X)] = X è?X{ \overline{X} = X \cup \partial X} . Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.