J. fixed point theory appl. 2 (2007), 195–207 c 2007 Birkh¨auser Verlag Basel/Switzerland 1661-7738/020195-13, published online 10.10.2007 Journal of Fixed Point Theory DOI 10.1007/s11784-007-0031-8 and Applications Some recent results in metric fixed point theory W. A. Kirk Dedicated to Edward Fadell and Albrecht Dold on the occasion of their 80th birthdays Abstract. This is a survey of recent results on best approximation and fixed point theory in certain geodesic spaces. Some of these results are related to fundamental fixed point theorems in topology that have been known for many years. However the metric approach is emphasized here. Mathematics Subject Classification (2000). Primary 05C12, 54H25, 47H09. Keywords. Best approximation, fixed points, CAT(0) spaces, metric trees, hyperconvex spaces. 1. Introduction Fixed point theory for nonexpansive and related mappings has played a funda- mental role in many aspects of nonlinear functional analysis for many years. The theory has traditionally involved an intertwining of geometrical and topological arguments in a Banach space setting. However, because the theory is fundamen- tally metric in nature, there has been a trend in recent years to seek applications in settings where the underlying algebraic linear structure of a Banach space is not present. This trend perhaps has its origins in the concept of a hyperconvex metric space due to Aronszajn and Panitchpakdi [5]. (Definitions of terms dis- cussed in the Introduction will be given later.) The fact that bounded hyperconvex spaces have the fixed point property for nonexpansive mappings has been known for many years and is basically due, independently, to Sine [40] and Soardi [42]. Subsequently J.-B. Baillon [6] extended this result to commuting families of non- expansive mappings, and since then a flourishing theory has evolved, as evidenced by the Esp´ınola–Khamsi survey in [16]. More recently, many of the standard ideas of nonlinear analysis have been extended to the class of so-called CAT(0) spaces (see [24], [25]). While many of the Banach space ideas carry over to a complete CAT(0) setting without essential 196 W. A. Kirk JFPTA change, often a more geometrical approach is required, with less emphasis on topo- logical concepts caused by, among other things, the absence of a weak topology. There is an interesting class of spaces which are both complete CAT(0) spaces and hyperconvex metric spaces. These are the complete R-trees (or metric trees). Indeed, a CAT(0) space is hyperconvex if and only if it is a complete R-tree. In this survey we discuss some recent metric fixed point results in each of the settings just described, with emphasis on results in R-trees. In fact, the results in R-trees have some interesting connections with classical fixed point results in topology. We begin by describing each of these settings in more detail. We stress the metric approach here, although some of these results may be derived from more abstract theory; see, e.g., Horvath [20]. 2. Preliminaries Hyperconvex spaces A metric space Y is said to be hyperconvex if every family {B(yα; rα)}α∈A of closed balls centered at yα ∈ Y with radii rα ≥ 0 has nonempty intersection whenever d(yα,yβ) ≤ rα + rβ ∀α, β ∈ A. Such spaces include, among others, the classical L∞ spaces ([29]). It is known that compact hyperconvex spaces (often called Helly spaces) are contractible and locally contractible; hence they have the fixed point property for continuous mappings (see [35]). We now list some other properties of hyperconvex spaces. 1. Hyperconvex spaces are metrically convex in the sense of Menger. This means that given any two points x and y in a hyperconvex space M there is a point z ∈ M, x = z = y, such that d(x, z)+d(z,y)=d(x, y). 2. Hyperconvex spaces are complete. This is an easy consequence of the def- inition. As a consequence of a classical theorem of K. Menger, there is a metric segment (an isometric image of a real line interval) joining any two points of a hyperconvex space whose length is equal to the distance between the points. In view of 1 and 2 the following are equivalent: (i) M is hyperconvex; (ii) M is metrically convex and has the binary ball intersection property, that is, any family of closed balls in M has nonempty intersection whenever any two of its members intersect. 3. Hyperconvex spaces are injective [5]. This means that if M is hyperconvex and if f : X → M is a nonexpansive mapping defined on a metric space X, then f has a nonexpansive extension f : Y → M to any metric space Y ⊃ X. 4. If M is hyperconvex, and if M is a subspace of a metric space Y, then there is a nonexpansive retraction of Y onto M. This follows from 3. Vol. 2 (2007) Recent results in metric fixed point theory 197 5. Every metric space M has a hyperconvex hull; that is, there is a hyper- convex metric space ε(M) which contains an isometric copy of M, and which has the property that no proper subset of ε(M) which contains M metrically is hyperconvex. This is a result of Isbell [21]. 6. ([42], [40], [6]) If M is a bounded hyperconvex metric space, and if f : M → M is a nonexpansive mapping, then f has a fixed point. In fact, a commuting family of nonexpansive mappings of M → M always has a common fixed point. 7. ([22], [28]) If M is a compact hyperconvex metric space and if f : M → M is continuous (or condensing), then f has a fixed point. A discussion of the preceding facts and many more can be found in the survey [16]. R-trees (metric trees) There are many equivalent definitions of R-tree. Here are two of them. Definition 1. An R-tree is a metric space M such that for every x and y in M there is a unique arc between x and y and this arc is isometric to an interval in R (i.e., is a geodesic segment). Definition 2. An R-tree is a metric space M such that (i) there is a unique geodesic segment denoted by [x, y] joining each pair of points x and y in M; (ii) [y,x] ∩ [x, z]={x}⇒[y,x] ∪ [x, z]=[y,z]. The following is an immediate consequence of (ii). (iii) If x, y, z ∈ M there exists a point w ∈ M such that [x, y] ∩ [x, z]=[x, w] (whence by (i), [x, w] ∩ [z,w]={w}). Standard examples of R-trees include the “radial” and “river” metrics on R2. For the radial metric, consider all rays emanating from the origin in R2. Define 2 the radial distance dr between x, y ∈ R as follows: dr(x, y)=d(x, 0) + d(0,y). (Here d denotes the usual Euclidean distance and 0 denotes the origin.) For the river metric ρ, if two points x, y are on the same vertical line, define ρ(x, y)= d(x, y). Otherwise define ρ(x, y)=|x2| + |y2|+ |x1 − y1|, where x =(x1,x2)and y =(y1,y2). Much more subtle examples exist; e.g., the real tree of Dress and Terhalle [15]. The concept of an R-tree goes back to a 1977 article of J. Tits [43]. The idea has also been attributed to A. Dress [13], who first studied the concept in 1984 and called it T -theory. Bestvina [8] observes that much of the importance of R-trees stems from the fact that in many situations a sequence of negatively curved objects (manifolds, groups) gives rise (in some sense “converges”) to an R-tree together with a group acting on it by isometries. There are applications in biology and computer science 198 W. A. Kirk JFPTA as well. The relationship with biology stems from the construction of phylogenetic trees [39]. Concepts of “string matching” are also closely related with the structure of R-trees [7]. CAT(0) spaces A metric space is a CAT(0) space (the term is due to M. Gromov—see, e.g., [9, p. 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. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [9] or Burago et al. [11]. We note in particular that a complex Hilbert ball with the hyperbolic metric (see [19]; also inequality (4.3) of [38] and subsequent comments) is a CAT(0) space. Let (X, d) be a metric space. 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 it is unique this geodesic is denoted [x, y]. The space (X, d) is said to be a geodesic space if any two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points.