Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature
© C. P. Niculescu and I. Rovenţa. 2009
Received: 3 June 2009
Accepted: 9 November 2009
Published: 18 November 2009
The Schauder fixed point theorem is extended to the context of metric spaces with global nonpositive curvature. Some applications are included.
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.
These spaces are also known as the Cat 0 spaces. See . 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
Every Hilbert space is a global NPC space. Its geodesics are the line segments.
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 . More information on global NPC spaces is available in [2, 3]. See also our papers [4–7].
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.
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
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.
In a locally compact global NPC space, the closed convex hull of each finite family of points has the fixed point property.
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 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 , 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 .
After this paper has been completed, Horvath has kindly informed us on his recent manuscript  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.
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.
and the proof is done.
We are now in a position to prove Theorem 1.3.
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.
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.
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, .
Theorem 2.3 easily yields the following criterion for the existence of fixed points.
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.
Definition 3.2 (J. Tits).
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.
An application is given in the next theorem.
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 .
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
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.
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