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. A geodesic triangle (x1,x2,x3) in a geodesic metric space (X, d) consists of three points in X (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle (x1,x2,x3)in(X, d) is a triangle (x1,x2,x3):=(¯x1, x¯2, x¯3) in the Euclidean 2 plane E such that dE2 (¯xi, x¯j)=d(xi,xj)fori, j ∈{1, 2, 3}. A geodesic metric space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. Let be a geodesic triangle in X and let be a comparison triangle for . Then is said to satisfy the CAT(0) inequality if for all x, y ∈and all compar- ison pointsx, ¯ y¯ ∈ ,d(x, y) ≤ dE2 (¯x, y¯). The following theorem yields a characterization of hyperconvex CAT(0) spaces.
Theorem 1 ([23]). For a metric space M the following are equivalent: (i) M is a complete R-tree; (ii) M is hyperconvex and has unique metric segments.
It is known that a complete R-tree is a complete CAT(0) space ([9, p. 167]). On the other hand, a CAT(0) space has unique metric segments. If it is also hyperconvex then by Theorem 1 it must be a complete R-tree. Thus we have:
Theorem 2. A CAT(0) space is hyperconvex if and only if it is a complete R-tree. Vol. 2 (2007) Recent results in metric fixed point theory 199
A proof that a complete R-tree is injective is given in [30]. Since injective spaces are known to be hyperconvex ([5]), this also gives (i)⇒(ii). Another proof that (i)⇒(ii) is given in Aksoy and Maurizi [4]. Their proof is based on an inter- esting four-point property of metric trees. Definition 3. A metric space (X, d) is said to satisfy the four-point property if for each set of four points x, y, z, w ∈ X the following holds: d(x, y)+d(u, w) ≤ max{d(x, u)+d(y,w),d(x, w)+d(y,u)}. Since one obtains the triangle inequality by taking u = v, the four-point property is a stronger condition. In [13] it is shown that a metric space is a complete R-tree if and only if it is complete, connected, and satisfies the four-point property. We now turn to the Lifshits character of these spaces. Balls in X are said to be c-regular if the following holds: For each k
Gated sets Many of the ideas discussed above, especially those in R-trees, can be couched in a more abstract framework. A subset Y of a metric space X is said to be gated ([14]) if for any point x/∈ Y there exists a unique point xY ∈ Y (called the gate of x in Y ) such that for any z ∈ Y,
d(x, z)=d(x, xY )+d(xY ,z). Obviously gated sets in a complete geodesic space are always closed and convex. It is known ([14]) that gated subsets of a complete geodesic space X are proximinal nonexpansive retracts of X. Specifically, if A is a gated subset of X, then the mapping that associates with each point x in X its gate in A (i.e., the gate map, or “nearest point map”) is nonexpansive. Several other properties of gated sets can be found, for example, in [44, p. 98]) In particular it can be easily shown by induction that the family of gated sets in a complete geodesic space X has the Helly property.ThusifS1,...,Sn is a collection of pairwise intersecting n ∅ gated sets in X then i=1 Si = . The gated subsets of an R-tree are precisely its closed and convex subsets. Thus the following results apply to R-trees.
Proposition 1 ([17]). Let (X, d) be a complete geodesic space, and let {Hα}α∈Λ be a collection of nonempty gated subsets of X which is downward directed by set inclusion. If X (or more generally, some Hα) does not contain a geodesic ray, then ∅ α∈Λ Hα = .
Proposition 2 ([17]). Let (X, d) be a complete geodesic space, and let {Hn} be { } a descending sequence of nonempty gated subsets of X. If Hn has a bounded ∞ ∅ selection, then n=1 Hn = .
3. The fixed point property for R-trees G. S. Young, Jr. obtained the following result in 1946. He notes explicitly in [47] that compactness is not needed. Theorem 6 ([46]). Let M be an arcwise connected Hausdorff space which is such that every monotone increasing sequence of arcs is contained in an arc. Then M has the fixed point property (for continuous maps). In [33], J. C. Mayer and L. G. Oversteegen proved that for a separable metric space (X, d) the following are equivalent: 1. X is an R-tree. 2. X is a locally arcwise connected and uniquely arcwise connected metric space. If a complete R-tree is geodesically bounded it is easy to see that every monotone increasing sequence of arcs is contained in an arc. In view of this, we have the following. Vol. 2 (2007) Recent results in metric fixed point theory 201
Theorem 7. A complete geodesically bounded R-tree has the fixed point property for continuous maps.
Although the validity of Theorem 7 goes back to Young’s 1946 result, a more constructive metric approach might be of interest. The following proof is taken from [26].
Proof of Theorem 7. For u, v ∈ X we let [u, v] denote the (unique) metric segment joining u and v and let [u, v)=[u, v]\{v}. We associate with each point x ∈ X a point ϕ(x) as follows. For each t ∈ [x, f(x)], let ξ(t) be the point of X for which
[x, f(x)] ∩ [x, f(t)] = [x, ξ(t)].
(It follows from the definition of an R-tree that such a point always exists.) If ξ(f(x)) = f(x) take ϕ(x)=f(x). Otherwise it must be the case that ξ(f(x)) ∈ [x, f(x)). Let
A = {t ∈ [x, f(x)] : ξ(t) ∈ [x, t]},B= {t ∈ [x, f(x)] : ξ(t) ∈ [t, f(x)]}.
Clearly A ∪ B =[x, f(x)]. Since ξ is continuous, both A and B are closed. Also A = ∅ as f(x) ∈ A. However, the fact that f(t) → f(x)ast → x implies B = ∅ (because t ∈ A implies d(f(t),f(x)) ≥ d(t, x).) Therefore there exists a point ϕ(x) ∈ A ∩ B. If ϕ(x)=x then f(x)=x and we are done. Otherwise x = ϕ(x) and [x, f(x)] ∩ [x, f(ϕ(x))] = [x, ϕ(x)].
n Now let x0 ∈ X, and let xn = ϕ (x0). Assuming the process does not termi- nate upon reaching a fixed point of f, by construction the points {x0,x1,x2,...} are linear and thus lie on a subset of X which is isometric with a subset of the real line, i.e., on a geodesic. Since X does not contain a geodesic of infinite length it must be the case that ∞ d(xi,xi+1) < ∞, i=0 and hence that {xn} is a Cauchy sequence. Suppose limn→∞ xn = z. Then by continuity
lim f(xn)=f(z), n→∞ and in particular {f(xn)} is a Cauchy sequence. However, by construction,
d(f(xn),f(xn+1)) = d(f(xn),xn+1)+d(xn+1,f(xn+1)).
Since limn→∞ d(f(xn),f(xn+1)) = 0 it follows that limn→∞ d(f(xn),xn+1)= d(f(z),z)=0andf(z)=z. 202 W. A. Kirk JFPTA
4. Best approximation in R-trees Ky Fan’s classical best approximation theorem (see [18]) asserts that if C is a nonempty compact convex subset of a normed linear space E and if f : C → E is continuous, then there exists a point z ∈ C such that z − f(z) =inf{x − f(z) : x ∈ C}. Over the years this theorem has been extended in various ways; see e.g., Singh et al. [41] for a discussion. Also [36] and [37] provide extensions of Fan’s theorem to set-valued mappings and to noncompact sets, respectively. There have been two recent approaches to best approximation for set-valued mappings in R-trees. In [27] Fan’s best approximation theorem is extended to upper semicontinuous mappings in an R-tree. The proof given in [27] is constructive—a modification of the proof of Theorem 7—although as we note below there is a nice topological approach. A second approach is found in [32] where it is shown that a lower semicontinuity assumption also suffices. We begin with the approach of [27]. Once again we assume that the space X is geodesically bounded, that is, we assume that X does not contain a geodesic of infinite length. Theorem 8 ([27]). Suppose X is a closed convex subset of a complete R-tree Y ,and suppose X is geodesically bounded. Let T : X → 2Y be an upper semicontinuous mapping whose values are nonempty closed convex subsets of Y . Then there exists apointz ∈ X such that dist(z,T(z)) = inf dist(x, T (z)). x∈X Let X be a connected Hausdorff space. A point p ∈ X separates u, v ∈ X if u and v are contained in disjoint open subsets of X\{p}. If e ∈ X it is possible to define a relation Γe on X × X in the following way:
Γe =({e × X}) ∪ Δ(X × X) ∪{(x, y):x separates e from y}.
It is known (see Ward [45]) that Γe is a partial order. A connected Hausdorff space X is said to satisfy property D(3) if the following condition holds: If A and B are disjoint closed connected subsets of X, then there exists z ∈ X such that z separates A and B. L. E. Ward, Jr. obtained the following result in 1974. Theorem 9 ([45]). Suppose X is a connected Hausdorff space that satisfies property D(3). Suppose also that there exists e ∈ X such that, relative to Γe, each chain in X has a maximal element and a minimal element. Let f : X → 2X be an upper semicontinuous mapping whose values are nonempty closed connected subsets of X. Then f has a fixed point. Theorem 8 is an easy consequence of Ward’s theorem. ProofofTheorem8([27]). It is known that the nearest point projection of a CAT(0) space onto a closed convex subset of the space is nonexpansive ([9, p. 176]), Vol. 2 (2007) Recent results in metric fixed point theory 203 and since an R-tree is a CAT(0) space ([9, p. 167]), the nearest point map R of Y onto X is nonexpansive. Hence the map R ◦ T : X → 2X is upper semicontinuous and has a fixed point z by Theorem 9. Thus there exists y ∈ T (z) such that R(y)=z. However, since R is the nearest point map, it must be the case that R(y)=z for all y ∈ T (z). If z ∈ T (z) we are finished. Otherwise, choose y1 ∈ T (z) such that d(z,y1) = dist(z,T(z)). Then if x ∈ X and x = z,
dist(z,T(z)) = d(z,y1)
Theorem 13 ([32]). Let X be a paracompact topological space, (Y,d) a complete R- tree, and T : X → 2Y an almost lower semicontinuous mapping whose values are nonempty bounded closed convex subsets of Y. Then T has a continuous selection.
5. Applications to graph theory A graph is an ordered pair (V,E) where V is a set and E is a binary relation on V (E ⊆ V × V ). Elements of E are called edges. We are concerned here with (undirected) graphs that have a “loop” at every vertex (i.e., (a, a) ∈ E for each a ∈ V ) and no “multiple” edges. Such graphs are called reflexive. In this case E ⊆ V × V corresponds to a reflexive (and symmetric) binary relation on V. For a graph G =(V,E) a map f : V → V is edge-preserving if (a, b) ∈ E ⇒ (f(a),f(b)) ∈ E. For such a mapping we simply write f : G → G. There is a standard way of metrizing connected graphs: let each edge have length one and take distance d(a, b) between two vertices a and b to be the length of the shortest path joining them. With this metric, edge-preserving mappings become precisely the nonexpansive mappings. (Keep in mind that in a reflexive graph an edge-preserving map may collapse edges between distinct points since loops are allowed.) We now turn to the classical Fixed Edge Theorem and show how it is a consequence of results of the preceding section. Theorem 14 ([34]). Let G be a reflexive graph that is connected, contains no cycles, and contains no infinite paths. Then every edge-preserving map of G into itself fixes an edge. Proof. Suppose f : G → G is edge-preserving. Since a connected graph with no cycles is a tree, one can construct from the graph G an R-tree T by identifying each (nontrivial) edge with a unit interval of the real line and assigning the shortest path distance to any two points of T. It is easy to see that with this metric T is complete. One can now extend f affinely on each edge to the corresponding unit interval, and the resulting mapping f¯ is a nonexpansive (hence continuous) mapping of T → T. Thus f¯ has a fixed point z by Theorem 7. Moreover, since T has unique metric segments and f¯ is nonexpansive, the fixed point set F of f¯ is convex (and closed). It follows that either F contains a vertex of G, or z is the midpoint of a unit interval of T , in which case f must leave the corresponding edge fixed. An application of Baillon’s theorem about commuting families of nonexpan- sive mappings in hyperconvex metric space tells us even more. For details, see [17]. Theorem 15 ([17]). Let G be a reflexive graph that is connected, contains no cycles, and contains no infinite paths. Suppose F is a commuting family of edge-preserving mappings of G into itself. Then either: (a) there is a unique edge in G that is left fixed by each member of F;or (b) some vertex of G is left fixed by each member of F. Vol. 2 (2007) Recent results in metric fixed point theory 205
It is likely that the above result is known in a more abstract framework. This seems to be a natural formulation resulting from the metric context.
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W. A. Kirk Department of Mathematics University of Iowa Iowa City, IA 52242, USA e-mail: [email protected]
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