Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature
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Research Article Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature Constantin P. Niculescu and Ionel Rovent¸a Department of Mathematics, University of Craiova, 200585 Craiova, Romania Correspondence should be addressed to Ionel Rovent¸a, [email protected] Received 3 June 2009; Accepted 9 November 2009 Recommended by Anthony To Ming Lau The Schauder fixed point theorem is extended to the context of metric spaces with global nonpositive curvature. Some applications are included. Copyright q 2009 C. P. Niculescu and I. Rovent¸a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction The aim of our paper is to discuss the extension of the Schauder fixed point theorem to the framework of spaces with global nonpositive curvature abbreviated, global NPC spaces. A formal definition of these spaces is as follows. Definition 1.1. A global NPC space is a complete metric space E E, d for which the following inequality holds true: for each pair of points x0 , x1 ∈ E there exists a point y ∈ E such that for all points z ∈ E, 1 1 1 d2 z, y ≤ d2 z, x0 d2 z, x1 − d2 x0 , x1 . 2 2 4
1.1
These spaces are also known as the Cat 0 spaces. See 1. In a global NPC space, each pair of points x0 , x1 ∈ E can be connected by a geodesic i.e., by a rectifiable curve γ : 0, 1 → E such that the length of γ|s,t is dγs, γt for all 0 ≤ s ≤ t ≤ 1. Moreover, this geodesic is unique. The point y that appears in Definition 1.1 is the midpoint of x0 and x1 and has the property 1 d x0 , y d y, x1 dx0 , x1 . 2
1.2
2
Fixed Point Theory and Applications Every Hilbert space is a global NPC space. Its geodesics are the line segments. The upper half-plane H {z ∈ C : Imz > 0}, endowed with the Poincar´e metric, ds2
dx2 dy2 , y2
1.3
constitutes another example of a global NPC space. In this case the geodesics are the semicircles in H perpendicular to the real axis and the straight vertical lines ending on the real axis. A Riemannian manifold M, g is a global NPC space if and only if it is complete, simply connected, and of nonpositive sectional curvature. Besides manifolds, other important examples of global NPC spaces are the Bruhat-Tits buildings in particular, the trees. See 1. More information on global NPC spaces is available in 2, 3. See also our papers 4–7. In what follows E will denote a global NPC space. Definition 1.2. A set C ⊂ E is called convex if γ0, 1 ⊂ C for each geodesic γ : 0, 1 → E joining γ0, γ1 ∈ C. A function ϕ : C → R is called convex if the function ϕ ◦ γ : 0, 1 → R is convex for each geodesic γ : 0, 1 → C, γt γt , that is, ϕ γt ≤ 1 − tϕ γ0 tϕ γ1
1.4
for all t ∈ 0, 1. One can prove that the distance function d is convex with respect to both variables, a fact which implies that every ball in a global NPC space is a convex
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