Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature

Abstract

The Schauder fixed point theorem is extended to the context of metric spaces with global nonpositive curvature. Some applications are included.

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 for which the following inequality holds true: for each pair of points there exists a point such that for all points

(1.1)

These spaces are also known as the Cat 0 spaces. See [1]. In a global NPC space, each pair of points can be connected by a geodesic (i.e., by a rectifiable curve such that the length of is for all . Moreover, this geodesic is unique. The point that appears in Definition 1.1 is the midpoint of and and has the property

(1.2)

Every Hilbert space is a global NPC space. Its geodesics are the line segments.

The upper half-plane H, endowed with the Poincaré metric,

(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 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 will denote a global NPC space.

Definition 1.2.

A set is called convex if for each geodesic joining .

A function is called convex if the function is convex for each geodesic that is,

(1.4)

for all

One can prove that the distance function is convex with respect to both variables, a fact which implies that every ball in a global NPC space is a convex set. See [8, Corollary ].

All closed convex subsets of a global NPC space are in turn spaces of the same nature.

The aim of the present paper is to discuss the extension of the Schauder fixed point theorem to the framework of global NPC spaces. The main result is as follows.

Theorem 1.3.

Suppose that is a closed convex subset of a global NPC space with the property that the closed convex hull of every finite subset of is compact. Then every continuous map whose image is relatively compact has a fixed point.

The convex hull is defined in the context of global NPC spaces via the formula

(1.5)

where and for the set consists of all points in the ambient global NPC space which lie on geodesics which starts and end in . The convex hull of a finite subset is not necessarily closed, but we can mention two important cases when this happens. The first one is that of Hilbert spaces. In fact, in any locally convex Hausdorff space, if are compact convex subsets, then the convex hull of their union is compact too. See the monograph of Day [9,page ]. The second case is provided by the following result.

Lemma 1.4.

In a locally compact global NPC space, the closed convex hull of each finite family of points has the fixed point property.

Recall that a topological space has the fixed point property if every continuous map from to itself has a fixed point.

Proof.

In a locally compact global NPC space each bounded and closed convex subset is compact. In particular, the closed convex hull of each finite family of points is compact.

On the other hand, each NPC space is an absolute retract. In fact, each space with an upper curvature bound is an absolute neighborhood retract and an absolute neighborhood retract that is contractible is an absolute retract. To see this, recall that all the balls are convex in and all intersections of balls are empty or contractible. See [10, page 12]. Thus has arbitrary fine coverings with the property that all the intersections of the sets from each covering are empty or contractible. This implies that is an absolute neighborhood retract. See [11,Theorem (b)].

In order to end the proof we have to show that every compact absolute retract has the fixed point property. In fact, can be embeded topologically into the Hilbert cube . According to the classical Schauder fixed point theorem (see [12,page 25]) has the fixed point property, hence also each retract of , in particular .

An immediate consequence of Lemma 1.4 is the following generalization of the Brouwer fixed point theorem.

Theorem 1.5.

In a global NPC space, every compact convex set has the fixed point property.

Theorem 1.5 proves to be instrumental in our extension of the Schauder fixed point theorem. Our argument combines this result with the existence of some nicely behaved projections onto the compact convex subsets of a global NPC space. The details are presented in the next section.

In [7], the statement of Theorem 1.5 was mentioned without any reference and used to extend two others important results in convex analysis: the Knaster-Kuratowski-Mazurkievicz lemma and Fan's minimax inequality. The elegant proof indicated above was suggested to us by A. Lytchak.

The present paper ends with an application of Theorem 1.3 to the graph theory, by indicating a sufficient condition under which an edge-preserving map leaves fixed some vertex or an edge. Our result was inspired by previous work due to Kirk [13].

After this paper has been completed, Horvath has kindly informed us on his recent manuscript [14] that contains related results in the slightly more general context of complete quasi-Busemann spaces.

2. Proof of Theorem 1.3

We start noticing the existence of an analogue of the orthogonal projection in the framework of global NPC spaces.

Proposition 2.1 (see [8,Proposition ]).

• (i) For each closed convex set in a global NPC space there exists a unique map (projection onto such that

(2.1)

 (ii) is orthogonal in the sense that

(2.2)

for every

(iii) is nonexpansive,

(2.3)

Based on Proposition 2.1 we can now prove that the compact subsets of a global NPC space are surprisingly special: they are "almost" finite-dimensional.

Corollary 2.2.

Let be a compact subset of a global NPC space . Then for each there exists a finite subset of and a function such that for all .

Proof.

Since is compact, for arbitrarily fixed there is a finite subset such that

(2.4)

Let be the orthogonal projection of onto the closed convex set According to Proposition 2.1(ii), for each we have

(2.5)

and the proof is done.

We are now in a position to prove Theorem 1.3.

For this, let us denote by the closure of By our hypotheses, is a compact set and thus Corollary 2.2 applies. Hence for each natural number we may choose a finite subset of and a map such that

(2.6)

Since we may consider the maps

(2.7)

By Theorem 1.5, each of these maps has a fixed point, say . Since is compact, we may assume (by passing to a subsequence if necessary) that the sequence is convergent to a point .

We will prove that is a fixed point for the function . In fact, by applying (2.6) for we get , and so the sequences and both have the limit . Since is continuous we conclude that . This ends the proof of Theorem 1.3.

As a consequence we can state the following analogue of the Leray-Schauder principle.

Theorem 2.3.

Let be a global NPC space with the property that the closed convex hull of every finite subset is compact and let be a map that verifies the following two conditions:

(i) maps the bounded sets into relatively compact sets;

(ii)there exists a closed ball such that every point with is outside the geodesic segment of endpoints and

Then has a fixed point.

Proof.

Let be the orthogonal projection onto According to our hypothesis (i), the image of the map is relatively compact, and by Theorem 1.3 there must be a point such that . If then and thus has a fixed point.

Suppose that . Because we infer that , and is a point on the geodesic joining the points and . This contradicts (ii). The proof is done.

The classical Leray-Schauder principle refers to the maps from a Banach space into itself that verifies the condition (i) above and the following substitute of the condition (ii): There exists a number  such that    implies    for  all  

Applications of the classical Leray-Schauder principle to partial differential equations can be found in many books. See, for example, [12].

Theorem 2.3 easily yields the following criterion for the existence of fixed points.

Corollary 2.4.

Let be a global NPC space with the property that the closed convex hull of every finite subset is compact and let be a map that verifies the following two conditions:

(i) maps the bounded sets into relatively compact sets;

(ii)

Then has a fixed point.

In recent years much activity was done to build -analogues of classical function spaces. However, few things are known in connection to partial differential equations. See [15, 16].

Instead, we will mention here that exactly the same argument we used for Theorem 1.3 yields also the classical fixed point theorem of Browder, Gohde, and Kirk concerning the nonexpansive maps.

Theorem 2.5.

Let be a closed, bounded, convex set in a real Hilbert space. Then each nonexpansive map has at least one fixed point.

See [17, 18], for other interesting results concerning the existence of fixed points in uniformly convex metric spaces.

3. An Application to -Trees

A graph is an ordered pair , where is a set and is a binary relation on (). The elements of are called vertices, and the elements of are called edges.

Definition 3.1.

Let be a graph. A map is said to be edge-preserving if implies .

The map   leaves an edge fixed if there exists an edge which coincides with the edge .

Definition 3.2 (J. Tits).

An -tree is a metric space such that

(i)there is a unique geodesic segment for each pair of points

(ii)if , then .

A complete -tree is a global NPC space with the property that the closed convex hull of every finite subset is compact. As a consequence we can specialize the Leray-Schauder principle (Theorem 2.3 above) to the context of complete -trees.

Lemma 3.3.

Let be a complete -tree, and let be a map that verifies the following two conditions:

(i) maps bounded sets into relatively compact sets;

(ii)there exists a closed convex subset of and a vertex such that , for every .

Then has a fixed point.

An application is given in the next theorem.

Theorem 3.4.

Let be a connected reflexive graph with no cycle. We attach to the graph an -tree by identifying each edge by a unit interval of the real line and assigning the shortest path distance to any pair of points of . With respect to this metric, is complete. Let be an edge-preserving map which verifies the conditions (i) and (ii) of Lemma 3.3. Then leaves fixed either an vertex of or an edge of .

Proof.

By definition, if is a boundary point of . According to Lemma 3.3, admits a fixed point . Two possibilities may occur: either is a vertex of , or lies properly on the unit interval joining the vertices of some edge . If fails to leave some vertex of fixed, then necessarily is the midpoint of the metric interval with and . But in this case is a fixed edge of .

4. An Open Question

In what follows we discuss an example built on the global NPC space (of all dimensional positive definite matrices with real coefficients), when endowed with the trace metric,

(4.1)

here are the eigenvalues of . Since similarities preserve eigenvalues, this metric is invariant under similarities, that is,

(4.2)

Notice that is similar with

(4.3)

and this fact assures the positivity of the eigenvalues of .

A proof that is a Cartan-Hadamard manifold (and thus a global NPC space) may be found in [19]. See also [20].

Given two matrices and in , we may consider the map

(4.4)

whose unique fixed point is the geometric mean of and , that is,

(4.5)

Moreover is the unique matrix in such that

(4.6)

The existence of seems to escape both to Theorems 1.3 and 2.5 above. We leave open the question of proving a stronger fixed point result that encompasses the existence of and the two aforementioned theorems.