Fixed point theorems in spaces and -trees
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We show that if U is a bounded open set in a complete CAT(0) space X, and if f : U → X is nonexpansive, then f always has a fixed point if there exists p ∈ U such that x ∈ / [p, f (x)) for all x ∈ ∂U. It is also shown that if K is a geodesically bounded closed convex subset of a complete R-tree with int(K) = ∅, and if f : K → X is a continuous mapping for which x ∈ / [p, f (x)) for some p ∈ int(K) and all x ∈ ∂K, then f has a fixed point. It is also noted that a geodesically bounded complete R-tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory. 1. Introduction A metric space X is said to be a CAT(0) space (the term is due to M. Gromov—see, e.g., [1, page 159]) if it is geodesically connected, and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include the classical hyperbolic spaces, Euclidean buildings (see [2]), the complex Hilbert ball with a hyperbolic metric (see [6]; also [12, inequality (4.3)] and subsequent comments), and many others. (On the other hand, if a Banach space is a CAT(κ) space for some κ ∈ R, then it is necessarily a Hilbert space and CAT(0).) For a thorough discussion of these spaces and of the fundamental role they play in geometry, see Bridson and Haefliger [1]. Burago et al. [3] present a somewhat more elementary treatment, and Gromov [8] a deeper study. In this paper, it is shown that if U is a bounded open set in a complete CAT(0) space X, and if f : U → X is nonexpansive, then f always has a fixed point if there exists p ∈ U such that x ∈ / [p, f (x)) for all x ∈ ∂U. (In a Banach space, this condition is equivalent to the classical Leray-Schauder boundary condition: f (x) − p = λ(x − p) for x ∈ ∂U and λ > 1.) It is then shown that boundedness of U can be replaced with convexity and geodesic boundedness if X is an R-tree. In fact this latter result holds for any continuous mapping. Three variants of the classical fixed edge theorem in graph theory are also obtained. Precise definitions are given below. Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:4 (2004) 309–316 2000 Mathematics Subject Classification: 54H25, 47H09, 05C05, 05C12 URL: http://dx.doi.org/10.1155/S1687182004406081
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Fixed point theory
2. Preliminary remarks Let (X,d) be a metric space. Recall that a geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c(0) = x, c(l) = y, and d(c(t),c(t )) = |t − t | for all t,t ∈ [0,l]. In particular, c is an isometry and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. When unique, this geodesic is denoted [x, y]. The space (X,d) is said to be a geodesic space if every two points of X are joined by a geo
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