Chapter 1

METRIC ENVIROMENT OF THE TOPOLOGICAL FIXED POINT THEOREMS

Kazimierz Goebel

Maria Curie-Skłodowska University 20-031 Lublin, Poland

goebel@golem.umcs.lublin.pl

1. Introduction

In metric fixed point theory the term the fixed point property is usually related to a certain class of mappings described by some metric conditions. In topological part of the theory however, we use this term with respect to the wide class of spaces and families of continuous transformations. Let us begin with recalling the classical definition and facts.

Definition 1.1 A topological space X is said to have the (topological) fixed point property (t.f.p.p. for short) if each continuous mapping f : X → X has a fixed point.

It is easy to see that the above property is a topological invariant. Suppose X and Y are two homeomorphic spaces with X having the fixed point property. Let h : X → Y be a homeomorphism with h (X) = Y. Suppose f : Y → Y is continuous. Then g = h^{−1} ∘ f ∘ h : X → X and g is continuous. Thus there exists x ∈ X such that gx = x, and consequently, for y = hx we have y = fy.

Other operations which preserve the fixed point property are retractions. Recall that a subspace Y of X is a retract of X if there exists a continuous mapping (a retraction) r : X → Y such that rx = x for all x ∈ Y. For any given retract Y of X there can be many retractions on it. Suppose Y is a retract of X and let f : Y → Y be continuous. Then, with the use of any retraction r : X → Y, such f may be extended to a continuous mapping g = f ∘ r : X → X. Any fixed point of g must be a fixed point of f. Thus if X has the t.f.p.p. so have it all of its retracts.

The most famous and important theorem on the t.f.p.p., Brouwer’s Fixed Point Theorem, which was first proved in 1912 by L.E.J. Brouwer. In our terminology it can be formulated as

Theorem 1.2 The closed unit ball B^{n} in ℝ^{n} has the t.f.p.p.

Brouwer’s Theorem has many applications in analysis, differential equations and generally in proving all kinds of so called existence theorems for many types of equations. Its discovery has had a tremendous influence in the development of several branches of mathematics, especially algebraic topology. All known proofs of the Theorem require analytical, topological, combinatorial, “nonelementary” and “nonmetric” (whatever it means) arguments. The Theorem also has many equivalent formulations. The attempts (sometimes naive) to find more general versions, equivalents or an elementary proof of Brouwer’s Theorem have given rise to interesting questions of “metric type”.

We are not going to present here proofs of the Theorem but rather discuss the metric “environment” of it.

To follow let us define the standard simplex in ℝ^{n} as

T^{n−1} = {x ∈ ℝ^{n} : x_{i} ≥ 0 for i = 1, 2, ..., n and ∑_{i=1}^{n} x_{i} = 1}.  (1)

and formulate:

Theorem 1.3 For each n = 1, 2, …, T^{n−1} has the t.f.p.p.

In view of the initial remarks one can immediately formulate the “more general version”.

Theorem 1.4 Any nonempty, convex, closed and bounded subset K of ℝ^{n} has the t.f.p.p..

All three formulations are equivalent. It can be shown using combinations of some of the following facts:

a) All closed balls in ℝ^{n} (of positive radius and arbitrary center) are homeomorphic.

b) T^{n} ⊂ ℝ^{n+1} is homeomorphic with B^{n}.

c) Any nonempty convex and closed subset K of ℝ^{n} is a retract of ℝ^{n}. Thus it is a retract of any ball and any set containing it.

d) Any nonempty convex, bounded and closed subset K of ℝ^{n} is homeomorphic to a ball B^{k} for some k ≤ n.

There are many other equivalents of Brouwer’s Theorem. Let us recall two the most important. First is known as the Non-retraction Theorem.

Theorem 1.5 For n = 1, 2, ... the unit sphere S^{n−1} is not the retract of B^{n}.

The second concerns the contractibility of spheres. Let us recall that a topological space X is said to be contractible to a point z ∈ X if there exists a continuous function (a homotopy) H : [0, 1]×X → X such that for all x ∈ X, H (0, x) = x and H (1, x) = z. If X is contractible to a point z then it is contractible to any other point of X, so we can simply say X is contractible. In other (topological) terminology X is contractible if the identity mapping I on X is homotopic to a constant map.

The following is the so called No Contractibility Theorem.

Theorem 1.6 For n = 1, 2, ... the unit spheres S^{n−1} are not contractible.

The equivalence of Theorems 1.2, 1.5 and 1.6 (1.2, 1.5, 1.6 for short) can be shown in several ways. We present here arguments which will be useful later in the discussion of infinitely dimensional balls and spheres.

Proof of the equivalence. The unit sphere S^{n−1} does not have the t.f.p.p. since the antipodal map −I : x → −x is continuous (even isometric) on S^{n−1} without fixed points. Thus, since retractions preserve the t.f.p.p., 1.2implies 1.5.

To prove the contrary let us assume that 1.2 is false. Suppose for certain n there exists a continuous mapping T : B^{n} → B^{n} such that T x ≠ x for all x ∈ B^{n}. Extend this mapping to the doubled ball 2B^{n} by defining

T_{1}x = \{ \begin{matrix}T x & for ∥x∥ ≤ 1 \\ (2 − ∥x∥) T (\frac{x}{∥x∥}) & for1 < ∥x∥ ≤ 2 \end{matrix}  (2)

and the new mapping T_{2} : B^{n} → B^{n} as

T_{2}x = \frac{1}{2}T_{1} (2x) .

Observe that both mappings T_{1} and T_{2} are fixed point free and for all x ∈ S^{n−1} we have T_{2}x = 0. Now we can contradict1.5 by defining a retraction R : B^{n} → S^{n−1} as

Rx = \frac{x − T_{2}x}{∥x − T_{2}x∥}.  (3)

Thus 1.5 implies 1.2 and the equivalence of 1.2 and 1.5 is proved.

Suppose now that 1.6 does not hold and let H : [0, 1]×S^{n−1} → S^{n−1} be a homotopy with H (0, x) = x and H (1, x) = z = const. Take any radius 0 ≤ r < 1 and define

Rx = \{ \begin{matrix}z & for ∥x∥ ≤ r, \\ H (\frac{1−∥x∥}{1−r}, \frac{x}{∥x∥}) & for r < ∥x∥ ≤ 1. \end{matrix}  (4)

The consistency of the above formula for r = 0 comes from the uniform continuity of H. Observe that for each r, R is a retraction of B^{n} onto S^{n−1}. In view of this1.5implies 1.6.

On the other hand, if R is a retraction onto S^{n−1} then S^{n−1} would be contractible to R (0) by the homotopy defined as

H (t, x) = R ((1 − t) x) .  (5)

This concludes the proof of equivalence of all formulations 1.2, 1.3, 1.4, 1.5, 1.6. ■

To end this section we turn the readers attention to the fact that all the facts presented here are nonmetric in the sense that they do not depend on the norm ∥ ∥ used in ℝ^{n}. Geometric properties of balls will be important however, in the infinitely dimensional case we are going to discuss.

2. Schauder’s Theorem

From now on let X denote an infinitely dimensional Banach space with norm ∥ ∥ the closed unit ball B_{X} and the unit sphere S_{X} (B and S for short) . The crucial difference between infinitely dimensional spaces and ℝ^{n} is caused by the fact that closed, bounded and convex sets do not need to be compact. Recall that X is of finite dimension if and only if B_{X} is compact.

An important generalization of Brouwer’s Theorem was discovered in 1930 by J. Schauder and is known as Schauder’s Fixed Point Theorem.

Theorem 2.1 Any nonempty, compact, convex subset K of a Banach space has the t.f.p.p..

Proof. Let K ⊂ X be convex and compact. For any ε > 0 there exists a finite ε−net {a_{1}, a_{2}, ..., a_{p}} ⊂ K. Define a collection {α_{i}} of real valued nonnegative functions on K by taking

α_{i} (x) = \{ \begin{matrix}0 & if ∥x − a_{i}∥ ≥ ε, \\ ε − ∥x − a_{i}∥ & if ∥x − a_{i}∥ ≤ ε, \end{matrix} i = 1, 2, .., p.

Put

β_{i} (x) = \frac{α_{i} (x)}{∑_{i=1}^{p} α_{i} (x)}

and observe that ∑_{i=1}^{p} β_{i} (x) = 1. Now define the function Φ : K → K_{0} = K ∩ span {a_{1}, a_{2}, ..., a_{p}} by taking

Φ (x) = ∑_{i=1}^{p} β_{i} (x) a_{i}.

Obviously Φ is continuous and represents on K an ε−approximation of identity, which means that for x ∈ K we have ∥Φ (x) − x∥ ≤ ε. Let T : K → K be a continuous mapping. Let us modify it, defining T_{0} : K → K by T_{0} = Φ ∘ T. Thus T_{0} : K_{0} → K_{0} and since K_{0} is homeomrphic to a compact convex subset of finite dimensional space, T_{0} has a fixed point x ∈ K_{0}. From this we have

∥x − T x∥ ≤ ∥x − T_{0}x∥ + ∥T_{0}x − T x∥ = ∥Φ ∘ T x − T x∥ ≤ ε.

Therefore ,inf {∥x − T x∥ : x ∈ K} = 0, and since K is compact F ixT ≠ ∅. ■

It is in general not possible, even in limited ways, to extend Brouwer and Schauder theorems to noncompact settings. Let us recall, in more or less historical order, facts explaining the cause of the failure of Schauder theorem on noncompact sets. The most commonly known are Kakutani’s Examples (1943).

Example 1. Consider l^{2} (ℕ) as a model of a Hilbert space. For the unit ball, the mapping T : B → B

T (x_{1}, x_{2}, ...) = ((\sqrt{1 − ∥x∥^{2}}) , x_{1}, x_{2}, ...)

is clearly continuous with F ixT = ∅. Moreover it is easy to notice that T is uniformly continuous on B and T (B) ⊂ S. Also for any 0 < ε < 1 similar mappings

T_{ε} (x_{1}, x_{2}, ...) = (ε (1 − ∥x∥) , x_{1}, x_{2}, ...)

map B into B (but not into S) are fixed point free and are even lipschitzian. Indeed we have

∥T_{ε}x − T_{ε}y∥ ≤ \sqrt{(1 + ε^{2})} ∥x − y∥ .

The next of Kakutani’s constructions shows even more. Consider another model on Hilbert space namely l^{2} (ℤ) and an the isometric shift operator

A : x = {x_{i}}_{i∈ℤ} → {x_{i+1}}_{i∈ℤ} .

Thus the mapping T : B → B defined by

T x = \frac{1}{2} (1 − ∥x∥) e_{0} + Ax

where e_{0} = {δ_{i,0}}, is a lipschitzian homeomorphism of B onto B with lipschitzian inverse, yet F ixT = ∅.

All Kakutani’s examples have very simple structure. Each one represents a one dimensional perturbation of an isometry. They fail to have fixed points but share a somewhat weaker property. For all the above mappings we have

inf_{x∈B} ∥x − T x∥ = 0.  (6)

Since it is known that all nonexpansive self-mappings of B in l^{2} must have fixed points, Kakutani’s constructions show just how fragile is this property.

Balls in less regular spaces than l^{2} can accept even nonexpansive mappings without fixed points.

Example 2. Consider the space C [a, b] (e.g. C [−1, 1]) of continuous functions and its unit ball B. Let us recall that there exists a nonexpansive retraction Q : C [a, b] → B defined with the use of the function

α (t) = min \{1, max [−1, t]\} = \{ \begin{matrix}−1 & for t ≤ −1 \\ t & for − 1 ≤ t ≤ 1 \\ 1 & for t ≥ 1 \end{matrix}

as

Qf (t) = α ∘ f (t) = α (f (t)) .  (7)

Take any function g ∈ C [a, b] satisfying g (a) ≤ −2, g (b) ≥ 2 (for example g (t) = 2t for C [−1, 1]). Define the mapping T : B → B by the formula

T f (t) = Q (f (t) + g (t)) .

For each such mapping we have T f (a) = −1, T f (b) = 1 so T : B → S. Moreover T is nonexpansive, fixed point free and again satisfies (6).

Example 3. There exist isometries mapping the unit ball in c_{0} into its unit sphere without fixed points. The simplest formula for such a mapping is:

T (x_{1}, x_{2}, ...) = (1, x_{1}, x_{2}, ...) .

Our next example shows another type of singularity. This one is connected not with the metric regularity of the mapping but with its structure.

Example 4. One of the properties distinguishing infinitely dimensional Banach spaces from spaces of finite dimension is the fact that if dim X = ∞ then X is homeomorphic with the punctured space X⧹ {0}. More, there exists a homeomorphism h : X → X⧹ {0} such that hx = x for ∥x∥ ≥ 1. Now the mapping T : B → B,

T x = h^{−1} (−hx)

lacks fixed points but represents an involution which means T^{2} = I. In other words all point are fixed under T^{2} (T^{2}x = x).

The first general theorem on the extent of the topological fixed point property has been presented by V. Klee in 1950.

Theorem 2.2 A closed bounded convex subset C of a Banach space has the t.f.p.p. if and only if it is compact.

Recall two commonly known facts. First states that if C is noncompact then there exist a number r > 0 and a bounded sequence of points {x_{n}} , n = 0, 1, 2, .. in C such that

dist (x_{n+1,} span (x_{1}, x_{2}, ..., x_{n})) ≥ r > 0  (8)

The second is the well known Tietze’s Extension Theorem.

Theorem 2.3 Let (M, ρ) be a metric space and let Zbe a closed subset of M. Then for any continuous function f : Z → R there exists a continuous function (extension) f̃ : M → R such that F (x) = f (x) for all x ∈ Z. Moreover if f (Z) ⊂ [a, b] then f̃(M) ⊂ [a, b].

With the above we can prove Klee’s result.

Proof of Theorem 2.2. Suppose C ⊂ X is a closed bounded and convex. Let for {x_{n}} , n = 0, 1, 2, ... be a sequence in C satisfying (8). Without loss of generality we may assume that 0 ∈ C and that ∥x_{0}∥ ≥ r > 0. Let us construct a piecewise linear curve joining consecutive points x_{0}, x_{1}, x_{2}, ... by putting

Γ = ⋃_{n=0}^{∞} Γ_{n} where Γ_{n} are segments Γ_{n} = co (x_{n}, x_{n+1}) = [x_{n}, x_{n+1}]  (9)

Thus Γ is a closed subset of C and can be parametrized by a function γ : [0, ∞) → Γ. For t ∈ [0, ∞), let t = n + s where n is the integer part of t and 0 ≤ s < 1. Set

γ (t) = γ (n + s) = (1 − s) x_{n} + sx_{n+1}.

The function γ^{−1} : Γ → [0, ∞) is continuous and hence (in view of Tietze’s Theorem) has a continuous extension g : C → [0, ∞). Now a fixed point free mapping T : C → Γ ⊂ C can be defined as

T x = γ (g (x) + 1) .  (10)

Together with Schauder’s Theorem this completes the proof. ■

The assumption of boundedness of C is not crucial to our theorem. Unbounded convex sets either contain a halfline and are invariant under translations along it or contain a closed homeomorphic image of the half line and our arguments can be repeated.

The method of the construction used by Klee has a certain disadvantage. The curve Γ is obviously the retract of C via the retraction r (x) = γ (g (x)). However Γ is of infinite length but C is bounded, thus none of continuous mappings satisfying T (C) = Γ can be uniformly continuous. In view of our initial examples it is natural to ask if there is another method of construction leading to uniformly continuous fixed point free mappings?

Before we pass to the discussion of this problem let us notice that formulas (2), (3), (4), (5) used in the proof of equivalence in the introductory section allow us to formulate:

Theorem 2.4 If X is an infinitely dimensional Banach space then:

a the unit ball B does not have the t.f.p.p.,

b the unit sphere S is the retract of B,

c the unit sphere S is contractible.

Since we used explicit formulas all items a, b, c are equivalent in the sense that each one can be derived from another. The search of uniformly continuous extension of Theorem 2.4 has been initiated by B. Nowak (1979) who proved that for some Banach spaces there exists a uniformly continuous (even lipschitzian) retraction of B onto S. The result has been extended to all Banach spaces by Y. Benyamini and Y. Sternfeld in (1983). And finally Y. Sternfeld and P. K. Lin presented the following result.

Theorem 2.5 If C is a closed bounded convex but noncompact subset of a Banach space then for any k > 1 there exists mapping T : C → C satisfying the Lipschitz condition

∥T x − T y∥ ≤ k ∥x − y∥  (11)

and such that

d (T) = inf [∥x − T x∥ : x ∈ C] > 0.  (12)

The proof is based on an idea similar to those we used showing Theorem 2.2. We only need a modification of the extension theorem.

Theorem 2.6 Let (M, ρ) be a metric space and let Z be a closed subset of M. Suppose f : Z → ℝ is a function satisfying the Lipschitz condition

|fx − fy| ≤ kρ (x, y)  for x, y ∈ Z.  (13)

Then f has an extension f̃ : M → ℝ (f̃|_{Z} = f) which satisfies the same Lipschitz condition. Moreover, if a ≤ fx ≤ b for all x ∈ Z, then f̃ can be chosen so that a ≤ f̃x ≤ b for all x ∈ M.

Proof. It suffices to observe that the function defined for x ∈ M by

f̃_{1}x = sup {fy − kϱ (x, y) : y ∈ Z} ,

has the first property. The second holds upon taking

f̃x = max {a, min {b, f̃_{1}x}} .
■

Corollary 2.7 Analogous theorem holds for mappings F = {f_{1}, f_{2}, ...} : Z → l^{∞} satisfying

∥F x − F y∥_{l^{∞}} ≤ kρ (x, y) .

Since l^{∞} consists of bounded sequences with the supremum norm, the function F is represented by a sequence of “coordinate” functions {f_{i}} each of which satisfies (13). Extending each one f_{i}, according to the above theorem we can obtain the extension F̃ of F. Moreover, if the “coordinate functions” f_{i} are bounded all their extensions have the same bounds.

We are now in the position to prove the main result of this section.

Proof of Theorem 2.5. Let C be a bounded, closed and convex but noncompact subset of a Banach space X. As in the proof of Theorem 2.2 assume that 0 ∈ C, ∥x_{0}∥ ≥ r and a sequence {x_{n}} , n = 0, 1, 2, ... satisfies (8). Construct as before the piecewise linear curve Γ = Γ_{C} and define in C a sequence of triangles

Δ_{n} (C) = co {0, x_{n}, x_{n+1}} for n = 0, 1, 2, ...

and a “fan”

Δ (C) = ⋃_{n=1}^{∞} Δ_{n} (C) .

Each element x ∈ Δ (C) is of the form x = λγ_{C} (t) where λ ∈ [0, 1] and γ_{C} represents the already used parametrization of Γ_{C} by t = n + s ∈ [0, ∞). We use here C in the index position only to indicate that all objects are in the set C.

Let us pass now to the space l^{∞} (ℕ_{0}). For each n ∈ ℕ_{0} let e_{n} denote the standard unit vector {δ_{in}} = (0, ..., 0, 1, 0, ...) (where 1 is in the nth position). Define the sequence {Δ_{n}} of standard triangles as

Δ_{n} = co {0, e_{n}, e_{n+1}}
= {(0, ..., 0, ξ_{n}, ξ_{n+1}, 0, ...) : ξ_{n} ≥ 0, ξ_{n+1} ≥ 0, ξ_{n} + ξ_{n+1} ≤ 1}

and the “standard fan” as

Δ = ⋃_{n=0}^{∞} Δ_{n}.

Obviously again each point x of Δ can be represented as x = λγ (t) where λ and t are natural, for Δ and corresponding “frame” curve Γ generated by unit vectors, parameters. Both “fans” Δ (C) and Δ are locally finite dimensional (two dimensional) except for the neighborhood of zero. We leave to the reader a justification of the fact that both are homeomorphic and that the natural invertible mapping H : Δ → Δ (C) defined by

H (ξ_{n}e_{n} + ξ_{n+1}e_{n+1}) = ξ_{n}x_{n} + ξ_{n+1}x_{n+1}  (14)

is onto and moreover both H and H^{−1} are lipschitzian.

The next crucial step is to show that there exists a lipschitzian retraction of l^{∞} (ℕ_{0}) onto Δ. Let e be a vector whose coordinates are all equal to one. For x ∈ l^{∞} (ℕ_{0}) , set

E (x) = {ε ≥ 0 : (x − εe) ∨ 0 = {max {x_{i} − ε, 0}} ∈ Δ},

observe that since ∥x∥ ∈ E (x) , E (x) ≠ ∅ and put

ε (x) = inf E (x) .

Observe that, in the natural coordinatewise order, for x, y ∈ l^{∞} we have

x ≤ y + ∥x − y∥ e.

Thus ε (y) + ∥x − y∥ ∈ E (x) implying ε (x) ≤ ε (y) + ∥x − y∥. By symmetry we obtain

|ε (x) − ε (y)| ≤ ∥x − y∥ .

Therefore the retraction R : l^{∞} (ℕ_{0}) → Δ defined as Rx = (x − ε (x)) ∨ 0 fulfils the Lipschitz condition

∥Rx − Ry∥ ≤ 2 ∥x − y∥ .

In consequence, Δ (C) is a lipschitzian retract of C. Indeed, let H^{−1} : Δ (C) → Δ be the inverse homeomorphism to H defined by (14). By Corollary H^{−1} has a lipschitzian extension H̃^{−1} : C → l^{∞}. The desired lipschitzian retraction R_{C} : C → Δ (C) can be now defined as R_{C} = H ∘ R ∘ H̃^{−1}. Only we do not know an exact Lipschitz constant for it.

To complete the proof observe that we can start the construction of a lipschitzian mapping T : C → C with positive minimal displacement,

d (T) = inf {∥x − T x∥ : x ∈ C} > 0,

building first, for our standard “fan” Δ, a lipschtzian mapping T_{Δ} : Δ → Δ satisfying d (T_{Δ}) = inf {∥x − T_{Δ}x∥ : x ∈ Δ} > 0.

Once we have it, we can define the next mapping T_{Δ(C)} : Δ (C) → Δ (C) , also satisfying d (T_{Δ(C)}) > 0 by putting T_{Δ(C)} = H ∘ T_{Δ} ∘ H^{−1}. And we can finish with taking T = T_{Δ(C)} ∘ R_{C}.

Here is a formula for T_{Δ} in terms of (λ, t) coordinates. For the initial triangle Δ_{0} of Δ (i.e.,for 0 ≤ t ≤ 1) set

T_{Δ}x = T_{Δ} (λγ (t)) = \{ \begin{matrix}(1 − 3λt) e_{0} \\ (1 − t) e_{0} + (3λ − 1) t (\frac{e_{0}+e_{1}}{2} ) \\ γ (\frac{t}{2} + (3λ − 2)) \end{matrix}  for λ ∈ [0, \frac{1}{3}] , for λ ∈ (\frac{1}{3}, \frac{2}{3}] , for λ ∈ (\frac{2}{3}, 1] .

The above mapping maps Δ_{0} into Δ_{0} ∪ Δ_{1} without having fixed points. In view of compactness of Δ_{0} we have inf {∥x − T_{Δ}x∥ : x ∈ Δ_{0}} > 0.

Now we can extend T_{Δ} to all of Δ by putting for x ∈ ⋃_{n=1}^{∞} Δ_{n}, x = λγ (t) (t ≥ 1) :

T_{Δ}x = T_{Δ} (λγ (t)) = \{ \begin{matrix}(1 − 3λ) e_{0} \\ (3λ − 1) γ (t − \frac{1}{2}) \\ γ ((t − \frac{1}{2}) + (3λ − 2)) \end{matrix}  for λ ∈ [0, \frac{1}{3}] , for λ ∈ (\frac{1}{3}, \frac{2}{3}] , for λ ∈ (\frac{2}{3}, 1] .

We leave to the reader checking that T_{Δ} is lipschitzian and that

d(T_{Δ}) = inf {∥x − T_{Δ}x∥ : x ∈ Δ} = inf {∥x − T_{Δ}x∥ : x ∈ Δ_{0} ∪ Δ_{1}} > 0.

So we proved that there exists a mapping T : C → C satisfying the Lipschitz condition (11) and having positive minimal displacement, which means satisfying (12). We neither evaluated its Lipschitz constant k nor its minimal displacement d (T).

To show that such mappings do exists for any Lipschitz constant k > 1 observe that if T satisfies (11) and (12) then for any α ∈ (0, 1] the mapping T_{α} = (1 − α) I + αT : C → C is lipschitzian with constant (1 − α) + αk and d (T_{α}) = αd (T). This remark completes the proof of Theorem 2.5. ■

Theorem 2.5 has strong consequences concerning balls. Let T : B → B be a lipschitzian mapping having d (T) > 0. Close analysis of all the reasonings used in the proof of the equivalence of 1.2, 1.5, 1.6 leads to the following result which is stronger than Theorem 2.4

Theorem 2.8 Let X be an infinitely dimensional Banach space with the unit ball B and the unit sphere S. Then:

a For any k > 1 there exists a mapping T : B → B satisfying (11) such that d (T) > 0.

b There exists a retraction R : B → S satisfying (11) with certain (sufficiently large) k > 1.

c S is contractible to a point by a lipschitzian homotopy.

The last statement means that there is a homotopy H : [0, 1] × S → S with H (0, x) = x, H (1, x) = x_{0} for which there are two constants A > 0, B > 0 such that for any t, s ∈ [0, 1] and x, y ∈ S,

∥H (t, x) − H (s, y)∥ ≤ A |t − s| + B ∥x − y∥ .

All three statements are formally equivalent in the sense that each one can be derived from any other.

This theorem completes the qualitative part of our considerations. In the sections to follow we will discuss some quantitative problems connected with concrete estimates for d (T) or a Lipschitz constant for retractions of a ball onto a sphere.

3. Minimal displacement problem

As we already observed, all the fixed point free mappings defined by Kakutani have d (T) = 0. This property is natural for certain classes of selfmappings of convex and bounded sets. For example this is true for all nonexpansive mappings, their compact perturbations or the so called k-set contractions. On the other hand for k-lipschitzian mappings we have Theorem 2.5.

The term minimal displacement problem was probably first used by the present author in 1973. It came long before the Sternfeld and Lin result, from the observation that while looking for the fixed points of a mapping T : C → C we are often led to evaluate of the quantity

d (T) = inf_{x∈C} ∥x − T x∥

called the minimal displacement of T.

Sometimes we are able to establish uniform estimates for all T belonging to a certain class T . The minimal displacement for the whole class is defined as

d (T ) = sup_{T ∈T} d (T) = sup_{T ∈T} inf_{x∈C} ∥x − T x∥ .

The class T under consideration is often divided into subclasses T (k) indexed by some parameter k. In this case we can also investigate the function of this parameter defined by

φ (k) = d (T (k)) .

Under minimal displacement problems we place all questions related to finding or evaluating the above quantities. We restrict ourselves to investigations by the whole class or subclasses of Lipschitz maps defined on the unit ball B, or more generally on a closed convex and bounded subset C of a Banach space X or to its unit ball B.

We shall denote by L the class of all Lipschitz maps. L is naturally scaled by the increasing system of subclasses L (k),

L = ⋃_{k>0} L (k) ,  (15)

where for k ∈ [0, ∞) , L(k) consists of all mappings satisfying the Lipschitz condition with constant k

∥T x − T y∥ ≤ k ∥x − y∥ .

This notation will be used regardless of the domain of T and other restrictions. Subclasses of interest and their natural scaling by the Lipschitz constant k will be written as L_{0}, L_{1}... and L_{0} (k) , L_{1} (k) ... .

Let T : C → C be of class L (k). If k < 1 then due to the Banach Contraction Principle it has a fixed point. If k = 1 then T is nonexpansive with d (T) = 0. So we suppose k > 1. Take any point x ∈ C, any a > 0 and consider the mapping

T_{x} : y → (1 − \frac{1}{k + a}) x + \frac{1}{k + a}T y.  (16)

Since T_{x} is a contraction there exists exactly one solution of the equation T_{x}(y) = y, say y_{a}, and we have

∥y_{a} − T y_{a}∥ = (1 − \frac{1}{k + a}) ∥x − T y_{a}∥ ≤ (1 − \frac{1}{k + a}) r(x, C),

where r (x, C) = sup {∥x − z∥ : z ∈ C} is the radius of C about x. Consequently by choosing appropriate x′s and letting a → 0 we get

d (T) ≤ (1 − \frac{1}{k} ) r (C) ,

where r (C) = inf {r (x, C) : x ∈ C} is the Chebyshev radius of C (with respect to itself). Since this is the basic estimate let us formulate

Theorem 3.1 For any bounded, closed and convex set C and any T : C → C, T ∈ L (k) , we have

d(T) ≤ (1 − \frac{1}{k} ) r (C) .  (17)

It is known that the estimate (17), is the best possible for certain sets, even balls in some spaces, and certain transformations T .

Example 5. Let S^{+} be the positive part of the unit sphere in L^{1} (0, 1) ,

S^{+} = {f ∈ L^{1} (0, 1) : f ≥ 0 and ∫_{0}^{1} f = 1} .

Then r (S^{+}) = 2. For k > 1, put

t_{f} = sup \{t : ∫_{0}^{t} f (s) ds = 1 − \frac{1}{k} \}

and define the mapping T : S^{+} → S^{+} as

T f (t) = \{ \begin{matrix}0 & for t ≤ t_{f} \\ kf (t) & for t > t_{f} \end{matrix} .

We leave to the reader to check that T ∈ L (k) and that for any f ∈ S^{+}

∥f − T f∥ = 2 (1 − \frac{1}{k} ) = (1 − \frac{1}{k} ) r (S^{+}) = d (T) .

Thus T moves all the points of S^{+} by the same extremal distance. A small modification of the above construction shows that a similar situation occurs for the positive part of the unit sphere in the sequence space l^{1}.

Example 6. In the space of continuous functions C [0, 1] consider the set

K = {x : 0 = x (0) ≤ x (t) ≤ x (1) = 1)} .

Thus K is closed bounded convex and for all x ∈ K we have r (x, C) = r (C) = diam(C) = 1. Obviously the identity function e, (e (t) ≡ t) belongs to K. Take any function α ≠ e, α ∈ K and define the mapping T_{α} generated by α as

T_{α}x (t) = (α ∘ x) (t) = α (x (t)) .

Then T_{α} inherits the Lipschitz constant of α: if α satisfies |α (t) − α (s)| ≤ k |t − s| then T_{α} is also k-lipschitzian. Since each x ∈ K takes all values in [0, 1] we have

∥x − T_{α}x∥ = max |x (t) − α (x (t))|
= max |s − α (s)| = ∥e − α∥ = const. = d_{1} > 0.

The iterations of T_{α} are of the same type, T_{α}^{2} = T_{α∘α}, T_{α}^{3} = T_{α∘α∘α},... . Hence for any n = 1, 2, ... , we have

∥x − T_{α}^{n}x∥ = const. = d_{n} > 0.

All the iterations of any T_{α} have a constant positive displacement. Take the family of special α′s given by

α_{k} (t) = min {kt, 1} , k > 1.

Corresponding mappings T_{k} = T_{α_{k}} belong to the class L (k) with d (T_{k}) = 1 − \frac{1}{k}. Moreover since for any k > 0, l > 0 α_{k} ∘ α_{l} = α_{kl}, we have T_{k} ∘ T_{l} = T_{kl}. Thus the family {T_{k} : k > 1} is a semigroup, extremal with respect to minimal displacements on K.

Example 7. The same behavior of mappings can be observed even on balls. Let B be a unit ball in C [a, b] . Take k > 1 and modify the mapping from Example 2 by putting

T_{k}f (t) = Q (k (f (t) + g (t))) ,

for which we get

∥f − T_{k}f∥ > (1 − \frac{1}{k} )

for all f ∈ B. Similar examples can be constructed in many spaces with uniform (max) norms such as, balls in c_{0} and the spaces of differentiable functions C^{n} [a, b] with all kinds of standard norms. An interesting result has been obtained by K. Bolibok. It states that not only C [a, b] but all its subspaces of finite codimension allow a construction with the above property.

Thus, the above examples show that in some extremal situations the inequality (17) is sharp. It is natural to expected that some geometrical properties of the space X or the set C itself can prohibit such extremal behavior. Before giving examples let us return to the general scheme.

For a closed bounded and convex set C let L_{C} (k) denote the whole family of klipschitzian maps T : C → C. Define the function φ_{C} : [1, ∞) → [0, r(C)] by

φ_{C} (k) = sup {d (T) : T ∈ L_{C} (k)}
= sup {inf {∥x − T x∥ : x ∈ C} : T ∈ L_{C} (k)} .

Similarity arguments make it possible to normalize the size of C and without loss of generality consider only those sets C for which r (C) = 1. From now we will take this to be a standing assumption.

For the whole space X put

φ_{X} (k) = sup {φ_{C} (k) : C ⊂ X, r (C) = 1}

and to distinguish the very interesting case, when C is the unit ball B we will write

ψ (k) = ψ_{X} (k) = φ_{B} (k) .

where it will not lead to any misunderstanding we will omit the subscripts and write φ, ψ instead of φ_{C}, φ_{X,} ψ_{X}. With this notation Theorem 2.5 gives the following,

Corollary 3.2 If C is convex, closed bounded but noncompact then φ_{C} (k) > 0 for all k > 1.

Theorem 3.1 (evaluation (17)) implies

φ_{C} (k) ≤ φ_{X} (k) ≤ 1 − \frac{1}{k} ,  (18)

for any set C (with r (C) = 1). And so, in particular for the unit ball

ψ (k) ≤ 1 − \frac{1}{k} .  (19)

There are sets (and balls) for which

φ_{C} (k) = 1 − \frac{1}{k} ,

let us call them extremal.

The conclusions of our examples can be written in the following form. For the spaces L^{1} (0, 1) and l^{1}

φ_{\frac{1}{2} S^{+}} (k) = φ_{L^{1}(0,1)} (k) = φ_{l^{1}} (k) = 1 − \frac{1}{k} .

and for X = C [a, b]

ψ_{C[a,b]} (k) = φ_{C[a,b]} (k) = 1 − \frac{1}{k} ,

for all k > 1. Hence the above sets are extremal. However, there are also sets and balls for which the estimation (17) is not sharp. We shall return to these later.

We now discuss some basic properties of the functions φ_{C}, φ_{X}, ψ_{X}. Let T : C → C be of class L (k) with d (T) = d > 0. For α ∈ [0, 1] the mapping T_{α} = (1 − α) I + αT ∈ L (1 − α + αk) and satisfies d (T_{α}) = αd (T) = αd. Consequently,

Property 1. For any α ∈ [0, 1] we have

φ (1 − α + αk) ≥ αφ (k) .

In other words φ is concave with respect to k = 1, indeed the function

\frac{φ (k)}{k − 1}

is nonincreasing for k ≥ 1.

The next property comes from the following, previously used, construction. Take A > k and consider the mapping

T_{A}y = (1 − \frac{1}{A}) x + \frac{1}{A}T y.  (20)

Since the right hand side of (20) is a contraction with respect to y, it has exactly one fixed point depending on x and A, say F x = F_{A}x. Thus we have an implicit formula

F x = (1 − \frac{1}{A}) x + \frac{1}{A}T F x,  (21)

from which we get

F ∈ L (\frac{A − 1}{A − k} ) ,  (22)
T F ∈ L (k \frac{A − 1}{A − k} ) ,  (23)
∥x − F x∥ = \frac{∥F x − T F x∥}{A − 1} ≥ \frac{d}{A − 1},  (24)
∥x − T F x∥ = \frac{A}{A − 1} ∥F x − T F x∥ ≥ \frac{Ad}{A − 1}.  (25)

Thus for any A > k the observations (23) and (25) imply

φ (k \frac{A − 1}{A − k} ) ≥ \frac{A}{A − 1}φ(k).  (26)

Denoting k \frac{A−1}{A−k} by l we have l > k and (26) takes form

\frac{φ (l) l}{l − 1} ≥ \frac{φ (k) k}{k − 1} ,  (27)

and we have Property 2. The function

\frac{φ (k) k}{k − 1} ,

is nondecreasing for k ≥ 1.

The next consequences of the above are Property 3. The right hand side derivative

φ′(1) = lim_{k→1} \frac{φ (k)}{k − 1}  (28)

always exists and φ′ (1) > 0. Property 4 For all k ≥ 1

φ (k) ≥ φ′ (1) (1 − \frac{1}{k} ) .  (29)

Let us add one more property concerning ψ_{X} = φ_{B}. According to Theorem 2.8 there exists a lipschitzian retraction R : B → S with Lipschitz constant k_{0} say. For 0 < ε < 1, let

T_{ε}x = \{ \begin{matrix}−R (\frac{x}{ε}) & for ∥x∥ ≤ ε \\ −\frac{x}{∥x∥} & for ∥x∥ > ε \end{matrix} .

Thus T_{ε} ∈ L (\frac{k_{0}}{ε} ) and ∥x − T x∥ ≥ 1 − ε for all x ∈ B. Setting k = \frac{k_{0}}{ε} we obtain Property 5. For any Banach space X, lim_{k→∞} ψ (k) = 1. More,

ψ_{X} (k) ≥ 1 − \frac{k_{0}}{k} ,

where k_{0} is the Lipschitz constant of a retraction of B onto S.

However, this does not mean that in general lim_{k→∞} φ_{C} (k) = r (C) .

Example 8. Let X = l^{2}, let e_{i} = {δ_{ji}} be the sequence of standard unit vectors and let

C = \overline{co} \{e_{i} : i = 1, 2, ...\}

= {x = {x_{n}} : x_{n} ≥ 0 for n = 1, 2, ..., and ∑_{n=1}^{∞} x_{n} ≤ 1} .

We have r (C) = 1 but it is a nice exercise to prove that lim_{k→∞} φ_{C} (k) ≤ \frac{1}{\sqrt{2}}. Thus the above set is not extremal.

There are spaces which do not contain extremal sets. This follows from some technical evaluations. For example, it is known that for any Banach space X and the function φ = φ_{X} we have

2φ (2) + δ_{X} (2φ (2)) ≤ 1

and consequently

φ′ (1) + δ_{X} (φ′ (1)) ≤ 1,

where δ_{X} is the modulus of convexity of X. This implies that there are no extremal sets in uniformly convex spaces or more generally, in spaces with characteristic of convexity ε_{0} < 1 (i.e. for spaces satisfying δ_{X} (1) > 0 ).

An interesting behavior is shown by l^{1} (a very “square” space).

Example 9. Let B be the unit ball in l^{1}. Suppose T : B → B, T ∈ L (k) , is a mapping with d (T) = d > 0. Consider equation (20) and for A > k construct the mappings F and T F defined by (21). Suppose, for a moment, that T F (0) is finitely supported, i.e., that T F (0) lies in a subspace of l^{1} spanned by n basis vectors. Then if P_{n} denotes the natural projection of l^{1} onto this subspace, the mapping P_{n}T F has finite dimensional range and by Brouwer’s Theorem there is a point y ∈ P_{n} (B) such that y = P_{n}T F y. Thus in view of (22), (23), (24), (25),

k (\frac{A − 1}{A − k} ) ∥y∥ ≥ ∥T F y − T F (0)∥ = ∥T F y − y∥ + ∥y − T F (0)∥ ≥

≥ ∥T F y − y∥ + ∥T F (0)∥ − ∥y∥ ≥ 2d (\frac{A}{A − 1}) − ∥y∥

implying

∥y∥ ≥ 2 (\frac{A}{A − 1}) (\frac{A − k}{A (k + 1) − 2k} ) d.

On the other hand,

d (\frac{A}{A − 1}) ≤ ∥T F y − y∥ = ∥T F y − P_{n}T F y∥ = ∥T F y∥ − ∥y∥ ≤ 1 − ∥y∥ .

Combining the above we get

d ≤ (\frac{A − 1}{A}) (\frac{A (k + 1) − 2k}{A (k + 3) − 4k} )  (30)

for any A > k.

If T F (0) is not “finite dimensional” then the above evaluation can be obtained by taking small ε > 0, choosing n so large that ∥T F (0) − P_{n}T F (0)∥ < εd\frac{A}{A−1}, and repeating the above arguments with ε-proximity.

Finally, minimizing the right hand side of (30) with respect to A and taking d close to ψ (k), we obtain

ψ_{l^{1}} (k) ≤ \{ \begin{matrix}\frac{2+\sqrt{3}}{4} (1 − \frac{1}{k}) & for 1 ≤ k ≤ 3 + 2\sqrt{3}, \\ \frac{k+1}{k+3} & for k > 3 + 2\sqrt{3}. \end{matrix} .  (31)

Thus l^{1} space admits extremal sets, for example S^{+}, but the unit ball B is not extremal. We have

ψ_{l^{1}} (k) < φ_{l^{1}} (k) = 1 − \frac{1}{k}

and

ψ′_{l^{1}} (1) ≤ \frac{2 + \sqrt{3}}{4}.

To finish this topic let us mention another observation by K. Bolibok. Let B be the unit ball in C [0, 1] and let a sequence {t_{i}} be dense in [0, 1]. If we furnish C[0, 1] with a new (W. Zizler) norm

∥f∥_{Z} = ∥f∥ + (∑_{i=1}^{∞} (\frac{f (t_{i})}{2^{i}})^{2})^{\frac{1}{2}} ,

we get a space that is not only strictly convex but uniformly convex in every direction. Such a space has normal structure , but in this case is extremal and has extremal unit ball.

The investigation of minimal displacement can also be restricted to some subclasses of L. Let us conclude with some examples. The Subclasses we are going to present here are selected because they have relevance to topics in the next two sections and to other questions in the metric fixed point theory.

Example 10. Let us consider the subclass L_{1} of L consisting of mappings transforming B into S, T : B → S. Obviously,

L_{1} = ⋃_{k≥1} L_{1} (k) ,

where

L_{1} (k) = {T : B → S : T ∈ L (k)} .

For this family of mappings let us define,

ψ_{1,X} (k) = d (L_{1} (k)) = sup {d (T) : T ∈ L_{1} (k)} .

As before we shall sometimes omit the space; writing ψ_{1} (k) instead of ψ_{1,X} (k) . Obviously, ψ_{1,X} is nondecreasing, and of course

ψ_{1,X} (k) ≤ ψ_{X} (k) ≤ 1 − \frac{1}{k} .

The construction justifying Property 5 of ψ_{X} shows also that

lim_{k→∞} ψ_{1} (k) = 1,

while the construction from Example 7 indicates that for some spaces (e.g. C [a, b]) we may have

ψ_{1,X} (k) = ψ_{0,X} (k) = 1 − \frac{1}{k} .

This is not true in general (see the section on Hilbert space).

Example 11. This time let us look at the subclass L_{2} of L consisting of all mappings T : B → X (the image is not necessarily contained in B), sending all the points x ∈ S to the origin. This means that ∥x∥ = 1 implies ∥T x∥ = 0 or, in other words, T (S) = {0} . This class is also naturally divided into subclasses L_{2} (k) .

Our interest in this class is connected with the optimal retraction problem (see the next section).

Define, analogously to before, the function

ψ_{2} (k) = sup {d (T) : T ∈ L_{2} (k)} = d(L_{2} (k)).

Obviously, ψ_{2} (k) ≤ 1 for k ≥ 1. Let R : B → S be a lipschitzian retraction with Lipschitz constant k_{0}. Then the mapping T = I − R is of class L_{2} (k_{0} + 1) and since for all x ∈ B, ∥x − T x∥ = ∥Rx∥ = 1, we have

k ≥ k_{0} + 1 implies ψ_{2} (k) = 1.

Let us find the first evaluation of ψ_{2} (k) for smaller k’s. Without loss of generality we can assume that our mappings are defined on the whole space X and take value zero outside of B, (∥x∥ ≥ 1 implies T x = 0). Observe that for all x ∈ B, x ≠ 0,

∥T x∥ = ∥T x − T (\frac{x}{∥x∥})∥ ≤ k (1 − ∥x∥)

and

∥T x − T (0)∥ ≤ k ∥x∥ .

Consequently, ∥T (0)∥ ≤ k and for all x ∈ X we have

∥T x − \frac{1}{2}T (0)∥ ≤ \frac{1}{2} ∥T x∥ + \frac{1}{2} ∥T x − T (0)∥

≤ \frac{k}{2} (1 − ∥x∥) + \frac{k}{2} ∥x∥ = \frac{k}{2} .

This means that T transforms the ball B(\frac{1}{2}T (0), \frac{k}{2}) into itself. We leave to the reader a justification of the simple observation that

d (T) ≤ \frac{k}{2} ψ (k) ≤ \frac{k − 1}{2}.

Finally,

ψ_{2} (k) ≤ min \{1, \frac{k}{2} ψ (k)\} ≤ min \{1, \frac{k − 1}{2}\}

and in consequence

ψ′_{2} (1) ≤ \frac{1}{2}ψ′ (1) ≤ \frac{1}{2}.

If we restrict ourselves to the even narrower class L_{3} ⊂ L_{2} consisting of mappings for which T (B) ⊂ B, we obtain the next function ψ_{3} (k) with

ψ_{3} (k) ≤ min {ψ (k) , ψ_{2} (k)} .

We leave a proof that L_{3} ≠ ∅ and that lim_{k→∞} ψ_{3} (k) = 1 as an exercise for the reader.

The next three examples are of a slightly different nature and are less connected to the optimal retraction problem (see the next section) but are close to the classical theory of nonexpansive mappings and geometry of Banach spaces.

Recall that a mapping T is said to be nonexpansive if T ∈ L (1) or, in other words, if

∥T x − T y∥ ≤ ∥x − y∥ .

If T is nonexpansive, so are all of its iterates T^{n}, n = 0, 1, 2, 3, ... (T^{0} = Id). There is a class of mappings sharing a similar property. A mapping T is said to be uniformly lipschitzian if there exists k ≥ 1 such that

∥T^{n}x − T^{n}y∥ ≤ k ∥x − y∥

for n = 0, 1, 2, ... . The class UL of uniformly lipschitzian mappings is naturally divided into subclasses UL (k) . It is known that the class UL can be viewed as the class of all mappings nonexpansive with respect to equivalent metrics.

Indeed, if T ∈ UL then d(x, y) = sup_{n=0}^{∞} ∥ T^{n}x − T^{n}y ∥ is a metric, equivalent to the norm metric on B, with respect to which T is nonexpansive.

A basic fact about uniformly lipschitzian mappings is the following,

Theorem 3.3 If the space X is uniformly convex, then there exists a constant γ > 1 such that each mapping T : B → B of class UL (k) with k < γ has a fixed point.

This fact was first observed by W. A. Kirk and the present author in 1973 not only for balls but for all convex, bounded and closed subsets of X. Since then it has been investigated and extended to both more general and special cases by many authors. For example it is known that in the case of Hilbert space, all mappings T ∈ UL (k) with k < \sqrt{2} have fixed points while there exists a mapping T : B → B, T ∈ UL (\frac{π}{2} ) with d (T) = 0 but without fixed points (the example has been given by J.-B. Baillon).

Example 12. Let us consider the class L_{4} divided into natural subclasses L_{4} (k) consisting of all UL mappings T : B → B. For this setting define as before the function ψ_{4} (k) . There are two natural constants connected with ψ_{4}

γ_{0} (X) = sup {k : all mappings T ∈ L_{4} (k) have fixed point}

and

γ_{1} (X) = sup {k : all mappings T ∈ L_{4} (k) or that with d (T) = 0}
= sup {k : ψ_{4} (k) = 0} .

Obviously, γ_{0} (X) ≤ γ_{1} (X) and ψ_{4} (k) = 0 for k ≤ γ_{1} (X) . Moreover, γ_{1} (X) < ∞. Again the construction justifying Property 5 actually shows that lim_{k→∞} ψ_{4} (k) = 1.

The results mentioned above say that for uniformly convex (and some other) spaces, γ_{0} (X) > 1. In particular for Hilbert space, H, we have \sqrt{2} ≤ γ_{0} (H) ≤ \frac{π}{2} . Practically nothing is known about γ_{1}. Here are some questions. Are there spaces with γ_{0} < γ_{1}? For which spaces is γ_{1} = 1? In other words, does there exist a space with ψ_{4} (k) > 0 for k > 1? We hope the answer to this last question is affirmative, but an example is unknown. Finally, can one find a formula for ψ_{4,H} (k), or at least give some good estimations?

Example 13. Our next class L_{5} consists of all T : B → B that are 2-periodic (i.e. involutions), which means they satisfy the condition T^{2}x = x (or, T^{2} = Id). Obviously L_{5} ⊂ L_{4} and for each k, L_{5} (k) ⊂ L_{4} (k) .

Take any x ∈ B and let y = \frac{1}{2} (x + T x). If T ∈ L_{5} (k) , we have

∥T y − T x∥ ≤ k ∥x − y∥ = \frac{k}{2} ∥x − T x∥ ,

∥T y − x∥ = ∥T y − T^{2}x∥ ≤ k ∥y − T x∥ = \frac{k}{2} ∥x − T x∥ .

The above implies

∥y − T y∥ ≤ \frac{k}{2} ∥x − T x∥ .

Now, it is a routine observation that if k < 2 then the sequence of consecutive iterates x_{0} = x, x_{n+1} = \frac{1}{2} (x_{n} + T x_{n}) converges to a fixed point of T. In other words, all involutions of class L_{5} (k) with k < 2 have fixed points. For spaces with more regular geometrical structure, uniformly convex spaces or Hilbert spaces, this estimate can be made even larger. For example it is known that for Hilbert space the same holds for k < \sqrt{π^{2} − 3} = 2.62... .

Define now our last function ψ_{5,X} (k) = ψ_{5} (k) in the usual manner. The above remarks say that ψ_{5} (k) = 0 for k < 2 and even larger k in some spaces. Nothing more is known. The question of whether there are lipschitzian involutions of B without fixed points is still unsolved (examples are known of continuous fixed point free involutions, but none of these are uniformly continuous). The question;

Does there exists an involution T of B onto B with d(T) > 0

also remains unanswered.

This last question is connected with the well known problem of geometric nonlinear functional analysis concerning the uniform classification of spheres. It is known that for any infinitely dimensional Banach space X its unit ball B and its unit sphere S are homeomorphic. However the question whether B and S are Lipschitz equivalent is open.

Assume that there exists a homeomorphism h : B → S such that h (B) = S with h ∈ L (k) and h^{−1} ∈ L (k) for some k. Then

∥h^{−1}x − h^{−1}y∥ ≥ \frac{1}{k} ∥x − y∥

for all x, y ∈ S. Putting

T x = h^{−1} (−hx)

we obtain an involution of class L (k^{2}) for which

∥x − T x∥ = ∥x − h^{−1} (−hx)∥ = ∥h^{−1} (hx) − h^{−1} (−hx)∥

≥ \frac{1}{k} ∥2hx∥ = \frac{2}{k} .

Thus d (T) ≥ \frac{2}{k} and we would have

kψ_{5,X} (k^{2}) ≥ 2.

Since ψ_{5} (k) ≤ ψ (k) ≤ 1 − \frac{1}{k}, we can conclude that if such a homeomorphism exists then its Lipschitz constant k ≥ 1 + \sqrt{2}. Probably this is a very imprecise evaluation. Thus, finding better estimates for ψ_{5} (k) seems to be an interesting problem.

Example 14. Let us recall that a mapping T : C → C is said to be asymptotically nonexpansive if

∥T^{n}x − T^{n}y∥ ≤ k_{n} ∥x − y∥ ,

for n = 1, 2, ... , where the sequence of Lipschitz constants satisfies lim_{n→∞} k_{n} = 1. It is known that the existence of a fixed point for such mapping depends strongly on the geometrical regularity of C. However, the basic question of whether this condition implies d (T) = 0 remains open for general spaces.

To finish this section let us remark that to our knowledge no formula has been found for ψ_{X} (k) except for the extremal cases. The natural candidates to work with seem to be, Hilbert space and l^{1}. Other interesting questions are:

Does there exists a Banach space X extremal in the sense that for all Banach spaces Y and k ≥ 1

ψ_{X} (k) ≤ ψ_{Y} (k) ?

Is Hilbert space or l^{1} such a space?

4. Optimal retraction problem

The optimal retraction problem is the question connected with item b) of Theorem 2.8. As we already noticed, the value of the Lipschitz constant for a retraction R : B → S appears in some estimations of the various functions ψ. Close analysis of Theorems 2.6 and 2.8 leads to the conclusion that there exists an universal constant k_{0} such that in any infinite dimensional Banach space there is a retraction R : B → S belonging to L (k_{0}) . Until now there is no known evaluation concerning how large this constant might be. To formalize the question let us assign to each space X the number

k_{0} (X) = inf {k : there exists a retraction R : B_{X} → S_{X}, R ∈ L (k)} .

The claim we mentioned above means that for all Banach spaces, the coefficients k_{0} (X) are bounded from above by k_{0}. It is expected that k_{0} (X) may vary depending on the geometry of X.

The optimal retraction problem means finding, or estimating, the characteristic k_{0} (X) of X or the upper bound k_{0}.

To date there is no space X for which an exact value of k_{0} (X) is known. We have only some estimates from above and below. The gap is wide.

First let us recall that in any space X the radial projection P = P_{X} : X → B defined by

P x = \{ \begin{matrix}x & for ∥x∥ ≤ 1, \\ \frac{x}{∥x∥} & for ∥x∥ > 1, \end{matrix}

is lipschitzian. This projection is of class L (2) in arbitrary space X while for some spaces its Lipschitz constant can be smaller (e.g. P_{H} ∈ L (1) for Hilbert space). In consequence, for any x, y ∈ X with ∥x∥ ≥ r > 0, ∥y∥ ≥ r, we have

∥\frac{x}{∥x∥} − \frac{y}{∥y∥}∥ = ∥P \frac{x}{r} − P \frac{y}{r} ∥ ≤ \frac{2}{r} ∥x − y∥ .

This allows us to extend any retraction R : B → S to the whole space by assuming Rx = P x for x ∈ X \ B. Since as we will see, R has (on B) a Lipschitz constant larger then 2 this operation does not increase the constant.

The first simple and rough estimates come from concrete examples.

Example 15. Let X = C [0, 1]. For f ∈ B define the mapping A : B → X by

Af (t) = |f (t) + 1 − 2 (1 − ∥f∥) t| − 1 + 2 (1 − ∥f∥) t.

An analysis of this mapping leads to the observations that Af = f for all f ∈ S and inf \{∥Af∥ : f ∈ B\} = \frac{1}{7} (exercise). Now define the retraction

Rf = \frac{Af}{∥AF ∥} = P (7Af).

Since A ∈ L (5) , P ∈ L (2) it follows that R ∈ L (70) and thus k_{0} (C [0, 1]) ≤ 70.

Some spaces are “resistant” to producing good examples, moreover examples rarely lead to optimal constants. Let us pass to more general reasoning.

Let R : B → S be a retraction of class L (k) (in the terminology of the previous section, R ∈ L_{1}(k)). Then the mapping T = −R also belongs to L_{1} (k) and moreover T^{2} = R. Let x ∈ B and let ∥x − T x∥ = d > 0 (T does not have fixed points). The rectifiable (lipschitzian) curve γ : [0, 1] → S defined by

γ (t) = T ((1 − t) x + tT x) ,

joins two antipodal points T x = −Rx and T^{2}x = Rx. Its length exceeds the so called girth of the sphere g (X) defined as the infimum of lengths of curves lying on S and joining antipodal points. For any space g(X) ≥ 2, for some “flat” spaces equality holds but for example in Hilbert space g (H) = π. Let the length of γ be l. Thus l ≤ kd. Selecting, x we can get d ≤ ψ_{1} (k) + ε and consequently

k (1 − \frac{1}{k} ) ≥ kψ_{1X} (k) ≥ g (X) .  (32)

This implies

k_{0} (X) ≥ g (X) + 1 ≥ 3,  (33)

for all spaces. A better estimate can be obtained for spaces for which g (X) > 2 or if ψ_{1,X} (k) < 1 − \frac{1}{k} (see the section on Hilbert space).

There are other “tricks” leading to some estimations of k_{0} (X) from below. For example it is known that if ψ_{X} (k) ≤ 1 − \frac{a}{k+b} for large k with certain a > 1, b > 0 then k_{0} (X) ≥ a + 2. This, in view of (31, implies k_{0}(l^{1}) ≥ 4. However, an effective method of searching for such estimates is still to be found.

Let us present here two ways of finding estimations from above.

Example 16. Let T : B → B be such that ∥x − T x∥ ≥ d > \frac{1}{2} for all x ∈ B. If T ∈ L (k) then obviously k > ψ_{X}^{−1} (\frac{1}{2}) > 2. Consider the mapping A : B → X defined by

Ax = x − (1 − ∥x∥)T x.  (34)

Observe that A ∈ L (k + 2) and that Ax = x for x ∈ S. The mapping A is bounded away from zero. We have

∥Ax∥ ≥ ∥x∥ − (1 − ∥x∥) ∥T x∥ ≥ 2 ∥x∥ − 1

and

∥Ax∥ ≥ ∥x − T x∥ − ∥x∥ ∥T x∥ ≥ d − ∥x∥ .

Thus

∥Ax∥ ≥ max \{2 ∥x∥ − 1, d − ∥x∥\} ≥ min_{t∈[0,1]} max \{2t − 1, d − t\} = \frac{1}{3} (2d − 1) .

Now putting

Rx = \frac{Ax}{∥Ax∥} = P (\frac{3}{2d − 1}Ax)

and denoting by k(P) and k (R) the Lipschitz constants of P and R respectively, we obtain a retraction R : B → S with

k (R) ≤ \frac{3k (P) (k + 2)}{2d − 1}.

Since d can be chosen arbitrarily close to ψ (k) = ψ_{X} (k) we obtain

k_{0} (X) ≤ 3k (P) inf \{\frac{k + 2}{2ψ (k) − 1} : k > ψ^{−1} (\frac{1}{2})\} .  (35)

For extremal spaces with ψ_{X} (k) = 1 − \frac{1}{k} this gives

k_{0} (X) ≤ 3k (P) inf_{k>2} \frac{k (k + 2)}{k − 2} = 3k (P) (2 + \sqrt{2})^{2} ≈ (34.97...)k (P) < 70.

This is not the best possible estimate. Modifying (34)

Ax = x − (1 − α (∥x∥)) T x,  (36)

where α : [0, 1] → [0, 1] is an increasing function with α (0) = 0, α (1) = 1 and playing a similar game we get better estimates. For example using α (t) = t^{2} leads to k_{0} (X) ≤ 44.85.... Also, some computer experiments performed with functions α_{n} (t) = t^{n} for n ≤ 20 indicate that in the case k (P) = 2, ψ_{X} (k) = 1 − \frac{1}{k}, we get the best result for n = 4, k ≈ 4.02.... The experimental evaluation gives k_{0} (X) ≤ 37.74.... Further use of the formula (34) is limited by the lack of good estimates of ψ_{X} for balls which are not extremal.

The next procedure is based on properties of the classes L_{2} (k) , L_{3} (k) introduced in Example 11 and the corresponding functions ψ_{2,X} (k) , ψ_{3,X} (k).

Example 17. Let T : B → X, T ∈ L_{2} (k) with d (T) = d > 0. Thus since T (S) = 0, the mapping R : B → S,

Rx = \frac{x − T x}{∥x − T x∥} = P (\frac{x − T x}{d})  (37)

is a lipschitzian retraction. It follows that

R ∈ L_{1} (k (P) \frac{k + 1}{d (T)}) ,

and consequently

k_{0} (X) ≤ k (P) inf \frac{k + 1}{ψ_{2,X} (k)}.  (38)

Since we have only that

ψ_{2} (k) ≤ min \{1, \frac{k}{2} ψ (k)\} ≤ min \{1, \frac{k − 1}{2}\}

and we do not know either the strict value of ψ_{2} (k) nor its estimates from below, we can not proceed further. However it may be worthwhile to notice that if there is a space X extremal in the sense that ψ_{2,X} (k) = min \{1, \frac{k−1}{2}\} then (37) leads to k_{0} (X) ≤ 8. So far such a space has not been found.

Some evaluations can be obtained by means specific to the space under consideration (see also the Hilbert space section).

Example 18. Let X = C [0, 1] . Let Q be the nonexpansive retraction Q : X → B defined by (7). For any r > 0 put Q_{r}f = rQ (\frac{1}{r}f) . It is not difficult to check that for any r_{1} > 0, r_{2} > 0 and f, g ∈ C [0, 1] ,

∥Q_{r_{1}}f − Q_{r_{2}}g∥ ≤ max {∥f − g∥ , |r_{1} − r_{2}|} .

As we have already shown there exists a mapping T_{1} : B → B of class L (k) such that d (T_{1}) = 1 − \frac{1}{k}. Consider the ball of radius 2 (B_{2} = 2B) and extend T_{1} to the mapping T_{2} : 2B → B by putting

T_{2}f = \{ \begin{matrix}T_{1}f \\ T_{1} (Qf) \\ Q_{k(2−∥f∥)} (T (Qf)) \end{matrix}  for ∥f∥ ≤ 1, for 1 < ∥f∥ ≤ 2 − \frac{1}{k}, for 2 − \frac{1}{k} < ∥f∥ ≤ 2.

Again T_{2} ∈ L (k) and for all f ∈ S_{2} (∥f∥ = 2) we have T_{2}f = 0. Moreover, it can be observed that ∥f − T_{2}f∥ ≥ 1 − \frac{1}{k} for all f ∈ B_{2}. Now defining

T f = \frac{1}{2}T_{2} (2f)

we get a mapping T ∈ L_{3} (k) with d (T) = \frac{1}{2} (1 − \frac{1}{k}) . This leads to the observation that

\frac{1}{2} (1 − \frac{1}{k} ) ≤ ψ_{3,C[0,1]} (k) ≤ ψ_{2,C[0,1]} (k) ≤ min\{1, \frac{k − 1}{2}\}.  (39)

If we use the left inequality of (37) to generate the retraction

Rf = \frac{f − T f}{∥f − T f∥} = P ( \frac{f − T f}{\frac{1}{2} (1 − \frac{1}{k})}) ,

we get R ∈ L (4\frac{k(k+1)}{k−1}) and therefore

k_{0} (C [0, 1]) ≤ 4 min_{k>1} \frac{k (k + 1)}{k − 1} = 4 (1 + \sqrt{2})^{2} = 23.31... .

Since lim_{k→∞} ψ_{3,X} (k) = 1 the left inequality of (39) is not sharp. Thus the above estimate is far from being sharp but to our knowledge it is so far the best known.

As a by-product of the last construction, from (39) we get

ψ′_{3,C[0,1]} (1) = ψ′_{2,C[0,1]} (1) = \frac{1}{2}.

The following examples and estimations are connected with an analysis of item c) of Theorem 2.8. Let us begin with

Theorem 4.1 Let H : [0, 1]×S → S be a homotopy such that for each x ∈ S, H(0, x) = x and H(1, x) = x_{0} ∈ S. Suppose that

∥H (c, x) − H (d, y)∥ ≤ A |c − d| + α ∥x − y∥ ,

holds for x, y ∈ S and c, d ∈ [0, 1] where A, α are nonnegative constants. Then there is a retraction R : B → S satisfying for x, y ∈ B :

∥Rx − Ry∥ ≤ \frac{2α}{r} ∥x − y∥ ,

where r is the solution of the equation

\frac{2α}{r} = \frac{A − 2α ln r}{1 − r} .  (40)

Proof. Take any 0 < r < 1 and a continuous convex function a : [r, 1] → [0, 1] such that a (r) = 0, a (1) = 1. Define the retraction

Rf = \{ \begin{matrix}x_{0} & if ∥x∥ ≤ r, \\ H (1 − a (∥x∥) , \frac{x}{∥x∥}) & if r < ∥x∥ ≤ 1. \end{matrix}

The Lipschitz constant of R depends strongly on the selection of the radius r and the function a. To obtain an optimal estimate, let x, y ∈ B with ∥x∥ ≥ r, ∥y∥ ≥ r. Then

∥Rx − Ry∥ = ∥H (1 − a (∥x∥) , \frac{x}{∥x∥}) − H (1 − a (∥y∥) , \frac{y}{∥y∥})∥
≤ A |a (∥x∥) − a (∥y∥)| + α ∥\frac{x}{∥x∥} − \frac{y}{∥y∥}∥
≤ Aa′ (max \{∥x∥ , ∥y∥\}) |∥x∥ − ∥y∥| + \frac{2α}{max \{∥x∥ , ∥y∥\}} ∥x − y∥
≤ max_{s∈[r,1]} \{Aa′ (s) + \frac{2α}{s}\} ∥f − g∥ .

In order to make optimal choices for a and r, observe that since

∫_{r}^{1} (Aa′ (t) + \frac{2α}{t}) dt = A − 2Br

does not depend on the function a, the Lipschitz constant of R is minimal if a is chosen to satisfy

Aa′ (t) + \frac{2α}{t} = \frac{A − 2α ln r}{1 − r} = c (r) .

Then we have

∥Rx − Ry∥ ≤ c (r) ∥x − y∥ .

Also, c takes on its minimal value when c (r) = \frac{2α}{r} (calculus). ■

We give see two applications of this theorem.

Example 19. Consider the subspace C_{0} [0, 1] consisting of those functions f for which f (0) = 0. For f ∈ S and c ∈ [0, 1] , set

H_{1}(c, f) (t) = min {1, |(1 + 2c) f (t) + 1| − 1};
H_{2} (c, f) (t) = max {c − 1 + ct, f (t)};
H_{3} (c, f) (t) = (1 − c) f (t) + ct.

Notice that since ∥f∥ = 1, f must assume the value 1 or −1 (or both). Check that H_{1} joins any function f = H_{1} (0, f) along the path on the sphere with the final function H_{1} (1, f) assuming value 1. Then the second homotopy ends with the function H_{2} (1, f) (t) ≥ t. Finally H_{3} (1, f) (t) ≡ t.

Now observe that for f, g ∈ S and c, d ∈ [0, 1] ,

∥H_{1} (c, f) − H_{1} (d, g)∥ ≤ 2 |c − d| + 3 ∥f − g∥;
∥H_{2} (c, f) − H_{2} (d, g)∥ ≤ 2 |c − d| + ∥f − g∥;
∥H_{3} (c, f) − H_{3} (d, g)∥ ≤ |c − d| + ∥f − g∥ .

Let us piece these homotopies together by defining

H (c, f) = \{ \begin{matrix}H_{1} (\frac{5c}{2}, f) \\ H_{2} (\frac{5c}{2} − 1, H_{1} (1, f)) \\ H_{3} (5c − 4, H_{2} (1, H_{1}(1, f))) \end{matrix}  for c ∈ [0, \frac{2}{5}] , for c ∈ (\frac{2}{5}, \frac{4}{5}] , for c ∈ (\frac{4}{5}, 1] .

Thus H joins the functions f ∈ S to the identity function e (t) ≡ t on [0, 1] , and

∥H (c, f) − H (d, g)∥ ≤ 5 |c − d| + 3 ∥f − g∥ .

Using our theorem and solving (40) numerically for r we obtain r ≈ 0.345 and c (r) ≈ 17.38. In particular we have the estimate

k_{0} (C_{0} [0, 1]) < 17.38.

With the use of Theorem 4.1 and some modifications of it, evaluations for other spaces have been obtained (for references see the Notes section). For example k_{0} (c_{0}) ≤ 35.18, k_{0} (l^{1}) ≤ 31.64, k_{0} (H) ≤ 31.45. For all the subspaces X of C [0, 1] with finite codimension, k_{0}(X) ≤ 35.18. It is also known that our last estimate is not sharp, k_{0} (C_{0} [0, 1]) ≤ 15.82.

So far, the space with the smallest estimate for k_{0} is L^{1} (0, 1) .

Example 20. Here let us consider the unit sphere S in L^{1} (0, 1) . For any f ∈ S and any c ∈ [0, 1] , define

t_{f} (c) = sup {t : ∫_{0}^{t} |f (t)| dt = c} .

Set

H (c, f) = \{ \begin{matrix}|f (t)| & if t ≤ t_{f} (c) , \\ f (t) & if t > t_{f} (c) . \end{matrix}

Obviously, H (0, f) = f and H (1, f) = |f| . Thus H contracts S not to a point but to S^{+}. It requires some tricky calculations to show that

∥H (c, f) − H (d, g)∥ ≤ 2 |c − d| + 2 ∥f − g∥ .

Now let R be the retraction given by

Rf = \{ \begin{matrix}\frac{1}{r} (r − ∥f∥ + |f|) & if ∥f∥ ≤ r, \\ H (1 − a (∥f∥) , \frac{f}{∥f∥}) & if ∥f∥ > r. \end{matrix}

Then for ∥f∥ ≤ r and ∥g∥ ≤ r,

∥Rf − Rg∥ ≤ \frac{2}{r} ∥f − g∥

while as in the previous example for ∥f∥ ≥ r, ∥g∥ ≥ r we get

∥Rf − Rg∥ ≤ \frac{4}{r} ∥f − g∥ ,

where r is chosen to satisfy

\frac{2 − 4 ln r}{1 − r} = \frac{4}{r}.

Thus r ≈ 0.424, and

k_{0} (L^{1} (0, 1)) < 9.43.

Observe that this example also shows some limitations on the proposed method. The Lipschitz constant 2 for the numerical argument of H can not be improved. Indeed if f < 0 then H joins two antipodal points f and |f| along the curve on S of length 2.

5. The case of Hilbert space

In spite of geometrical regularity Hilbert space remains very resistant for the investigation of minimal displacement problems and optimal retraction in Hilbert space. Let C be a closed bounded and convex subset of a Hilbert space H. Two facts of a geometrical nature are useful. For any x ∈ H there is a unique point y ∈ C satisfying

∥x − y∥ = inf {∥x − z∥ : z ∈ C} .

Denoting y = P_{C}x (or shortly y = P x) we obtain a retraction P = P_{C} : H → C. The retraction P is nonexpansive,

∥P x − P y∥ ≤ ∥x − y∥ ,

for all x, y ∈ H and moreover P and I − P are monotone which means that for all x, y ∈ H

\begin{matrix}(P x − P y, x − y) ≥ 0, \\ ((x − P x) − (y − P y) , x − y) = ∥x − y∥^{2} − (P x − P y, x − y) ≥ 0. \end{matrix}

In consequence of the above, for any x ∈ H and any z ∈ C we have

∥x − z∥^{2} ≥ ∥x − P x∥^{2} + ∥P x − z∥^{2} ,

implying

∥x − z∥ ≥ ∥x − P x∥ and ∥x − z∥ ≥ ∥P x − z∥ .

The mapping P = P_{C} is called the nearest point projection on C. For the unit ball B the nearest point projection P_{B} coincides with the radial projection P_{H} discussed earlier.

The second fact is that if r (C) is the Chebyshev radius of C then there exists a unique point z ∈ C, the Chebyshev center of C, with the property that the ball centered at z of radius r (C) contains C, C ⊂ B (z, r (C)).

Let r (C) = 1 and assume (without loss of generality) that z = 0 is the Chebyshev center of C. Then C ⊂ B. Suppose T : C → C is a mapping of class L (k) with d = d (T) > 0. The mapping T can be extended to the mapping T̃ : H → C by putting

T̃x = \{ \begin{matrix}T x & for x ∈ C, \\ T P_{C}x & for x ∉ C. \end{matrix}

In view of the above facts, T̃ ∈ L (k) , T : B → B and d \binom{}{T} = d (T) . Consequently we get

Theorem 5.1 For Hilbert space H The unit ball B is extremal in the sense that for any C with r (C) = 1

φ_{C} (k) ≤ φ_{B} (k) ,

and in consequence

φ_{H} (k) = ψ_{H} (k) .

The next advantage of Hilbert space geometry is connected with the construction, described by equations (20) and (21). Let T : B → B belongs to L (k) and for A > k let the mapping F : B → B be implicitly defined by

F x = (1 − \frac{1}{A}) x + \frac{1}{A}T F x

or in another form,

T F x = AF x − (A − 1) x.

All the properties (22), (23), (24) and (25) remain valid and in addition we have

k^{2} ∥F x − F y∥^{2} ≥ A^{2} ∥F x − F y∥^{2} − 2A (A − 1) (F x − F y, x − y) + (A − 1)^{2} ∥x − y∥^{2}

implying

2 (F x − F y, x − y) ≥ \frac{A^{2} − k^{2}}{A (A − 1)} ∥F x − F y∥^{2} + \frac{A − 1}{A} ∥x − y∥^{2} ,  (41)

or in a weaker form

2 (F x − F y, x − y) ≥ \frac{A − 1}{A} ∥x − y∥^{2} .  (41’)

This shows that F is (strongly) monotone and allows us to find a first evaluation for φ_{H} (k) .

Theorem 5.2 For Hilbert space H

φ (k) = ψ (k) ≤ (1 − \frac{1}{k} ) \sqrt{\frac{k}{k + 1}}  (42)

and consequently

φ′ (1) = ψ′ (1) ≤ \frac{1}{\sqrt{2}}.  (43)

Proof. Let T, F be as above and let d (T) = d. In view of (41) , for each x ∈ B we have

∥T F^{2}x − x^{2}∥ = ∥(T F^{2}x − F x) + (F x − x)∥^{2}

= ∥A (F^{2}x − F x) + (F x − x)∥^{2}

= A^{2} ∥F^{2}x − F x∥^{2} + 2A (F^{2}x − F x, F x − x) + ∥F x − x∥^{2}

≥ A^{2} ∥F^{2}x − F x∥^{2} + (A − 1) ∥F x − x∥^{2} + ∥F x − x∥^{2}

≥ (A^{2} + A) \frac{d^{2}}{(A − 1)^{2}} ,

For x = 0, 1 ≥ ∥T F^{2}(0)∥ and we get

d ≤ \frac{A − 1}{\sqrt{A^{2} + A}} = (1 − \frac{1}{A}) \sqrt{\frac{A}{A + 1}}.

Choosing d close to φ (k) and letting A → k we get (42) , then (43) easily follows. ■

Actually the above inequality also serves to obtain other evaluations. Suppose T ∈ L_{1} (k) and T : B → S. Then, for x = 0 we have ∥T F^{2}(0)∥ = ∥T F (0)∥ = A ∥F (0)∥ = 1 and we obtain

ψ_{1} (k) ≤ (1 − \frac{1}{k} )^{\frac{3}{2}} .  (44)

It follows that ψ′_{1} (1) = 0 and that for Hilbert space ψ_{1} (k) < ψ (k) at least for small k. In both cases we used only the weaker inequality (41′) .The stronger, (41) can be used to obtain, estimates from below that one better than (29) ,. Observes that

∥F^{2}x − x∥^{2}

= ∥F^{2}x − F x∥^{2} + 2 (F^{2}x − F x, F x − x) + ∥F x − x∥^{2}

≥ ∥F^{2}x − F x∥^{2} + \frac{A^{2} − k^{2}}{A (A − 1)} ∥F^{2}x − F x∥^{2} + \frac{A − 1}{A} ∥F x − x∥^{2} + ∥F x − x∥^{2}

≥ \frac{d^{2}}{(A − 1)^{2}} [2 + \frac{A^{2} − k^{2}}{A (A − 1)} + \frac{A − 1}{A}] .

Since d can be assumed to be close to ψ(k) , F^{2} ∈ L (((A − 1)/(A − k))^{2}) and x can be arbitrarily selected, standard arguments then lead to

ψ ((\frac{A − 1}{A − k} )^{2}) ≥ (\frac{ψ (k)}{A − 1})^{2} [2 + \frac{A^{2} − k^{2}}{A (A − 1)} + \frac{A − 1}{A}] .

For each fixed l > 1 select A so that l = ((A − 1)/(A − k))^{2} . We leave it to the reader to justify that after making the substitutions and letting k → 1 we get

ψ (l) ≥ \begin{matrix}√ & \overline{2} \\ √ & \overline{1 + \frac{1}{\sqrt{l}}} \end{matrix} ψ′ (1) (1 − \frac{1}{l}) .  (45)

All the above results seem to be imprecise.. The exact formula for ψ_{H} (k) as well as the exact value of ψ′_{H} (1) remain unknown. If we pass to the optimal retraction problem not much is known either. Estimates for k_{0} (H) are far from satisfactory. Let us present some sample results.

Example 21. The girth g (H) = π. Thus, in view of (44) , if k is the Lipschitz constant for a retraction R : B → S, the evaluation (32) takes the form

k (1 − \frac{1}{k} )^{\frac{3}{2}} ≥ kψ_{1,H} (k) ≥ π.

This implies k_{0}(H) ≥ 4.51... .

The next follows the pattern of Example 18.

Example 22. Let T : B → B be of class L (k) with d(T) = d. Consider the ball B_{1+d} = (1 + d) B and a natural extension T_{1} : B_{1+d} → B defined as

T_{1}x = \{ \begin{matrix}T x & for ∥x∥ ≤ 1, \\ T P x & for 1 < ∥x∥ ≤ 1 + d. \end{matrix}

In view of known properties of P, T_{1} ∈ L (k) and d (T_{1}) = d (T) = d. Now consider the sphere S_{1+d+1/k} = (1 + d + \frac{1}{k})S and extend T_{1} by putting

T_{1}x = 0 if ∥x∥ = 1 + d + \frac{1}{k} .

Observe that T_{1} ∈ L (k) on B_{1+d} ∪ S_{1+d+1/k} and as such, can be (via the well known Kirzbraun Extension Theorem) be extended to the whole ball B_{1+d+1/k} without increasing the Lipschitz constant. Finally, then, we have a mapping

T_{1} : B_{1+d+ \frac{1}{k}} → B

of class L (k) satisfying T_{1}(S_{1+d+1/k}) = 0 and d (T_{1}) = d. To finish put

T̃ (x) = \frac{1}{1 + d + \frac{1}{k}} T_{1} ((1 + d + \frac{1}{k} ) x)

and observe that T̃ : B → B, T̃ ∈ L (k) , T (S) = 0 and d(T̃) = d(1+d+ \frac{1}{k})^{−1}. Construct the retraction R : B → S by

Rx = \frac{x − T̃x}{∥x − T̃x∥} = P (\frac{x − T̃x}{d} (1 + d + \frac{1}{k} )) .

Obviously,

R ∈ L (\frac{k + 1}{d} (1 + d + \frac{1}{k} ))

and since d can be chosen close to Ψ (k) , we get the estimate

k_{0} (H) ≤ min_{k>1} \frac{k + 1}{ψ (k)} (1 + ψ (k) + \frac{1}{k} ) .

Using (29) and (42) we can derive from this the simple (but not sharp) estimate

k_{0} (H) ≤ min_{k>1} \frac{k (k + 1)}{ψ′ (1) (k − 1)} (1 + (1 − \frac{1}{k} ) \sqrt{\frac{k}{k + 1}} + \frac{1}{k} )

< \frac{2}{ψ′ (1)} min_{k>1} \frac{k (k + 1)}{k − 1} = \frac{2}{ψ′ (1)} (1 + \sqrt{2})^{2} .

In other words

k_{0} (H) ψ′_{H} (1) < 2 (1 + \sqrt{2})^{2} = 11.65... .

The above connects two unknown values. We suggest that the reader try to find an improvement using (45) instead of (29) .

Finally let us check how homotopy arguments work in the case of Hilbert space.

Example 23. Let us follow the steps used to estimate k_{0} (L^{1}) . Let L^{2} (0, 1) serve as a model of Hilbert space. For any function f ∈ S and any c ∈ [0, 1] define

t_{f} (c) = sup {t : ∫_{0}^{t} |f|^{2} dt = c}

and

H (c, f) = \{ \begin{matrix}|f| & if t ≤ t_{f} (c) \\ f & if t > t_{f} (c) \end{matrix} .

Obviously H (0, f) = f and H (1, f) = |f| . It is a technicality to prove that H : [0, 1] × S → S is a uniformly continuous homotopy (but not lipschitzian!). Verification of this is left as an exercise. It can be done, for example, by showing that

∥H (c, f) − H (d, g)∥^{2} ≤ ∥f − g∥^{2} + 4 ∥f − g∥ + 4 ∥c − d∥ .

For any 0 < r < 1, a uniformly continuous retraction R : B → S can be defined by

Rf = \{ \begin{matrix}P (\sqrt{2}\frac{r−∥f∥+|f|}{r}) & if ∥f∥ ≤ r \\ H (\frac{1−∥f∥}{1−r}, \frac{f}{∥f∥}) & if ∥f∥ > r \end{matrix} .

Such retractions are not lipschitzian. However their moduli of continuity can be evaluated. This is probably the simplest way to show that in Hilbert space S is uniformly contractible to its positive part and consequently to a point, S is a uniform retract of B, and finally that there are uniformly continuous mappings T : B → B with positive minimal displacement (e.g. T = −R).

There have been attempts to use the above (and similar) constructions to obtain some evaluations for k_{0} (H) . First T. Komorowski and J. Wosko found the way to modified the above retractions to make them lipschitzian. This let them show that k_{0} (H) ≤ 64.25 . . .. Then K. Bolibok has used other homotopies to get k_{0} (H) ≤ 31.45 . . .. Still all the evaluations known are obtained via individual technical “tricks” and seem to be very imprecise.

6. Notes and remarks

The results, examples, comments and ideas presented above do not form a unified theory. They represent rather a collection of scattered questions, remarks, observations and partial solutions to problems connected with classical fixed point theorems. We believe that some of the questions presented here are intriguing and are waiting for new ideas to obtain final solutions.

As mentioned, there is already some literature devoted to the subject.

The most celebrated Brouwer’s Fixed point Theorem was first published in 1912 in [9]. There are many proofs scattered in books on topology or mathematical analysis.

The theorem is usually proved by means of either combinatorial arguments, homology theory, differential forms, degree theory or other methods of differential topology. The analytical proof which seems to be the most “metric” is that presented by J. Milnor [33]. There is also an interesting “elementary” proof of the Nonretraction Theorem by Y. Kannai [23]. Schauder’s Theorem appeared in 1930 (see [35]). Very soon after that S. Ulam raised the question concerning the existence of a retraction of the unit ball onto its sphere in infinitely dimensional Hilbert space. This problem was placed in the famous collection of mathematical problems known as The Scottish Book (see D. Mauldin [32]). Spread over time, subsequent publications by S. Kakutani 1943 [24], V. Klee 1955 [26], B.Nowak 1979 [34], Y. Benyamini and Y. Sternfeld 1983 [2], P. K. Lin and Y. Sternfeld 1985 [30] gave the complete answer to the “qualitative” questions concerning the failure of Brouwer and Schauder’s theorems in noncompact settings.

The “quantitative” part of the field, namely, the investigations of the minimal displacement problem and the optimal retraction problem was initiated by K. Goebel in 1973 [17]. The first examples of mappings with d (T) > 0 come from this paper. The first rough estimates of k_{0} (X) are due to C. Franchetti, 1985, 1986 [11, 12].

Since then both problems have been intensively discussed and investigated in the mathematical analysis seminar run by K. Goebel at Maria Curie-Sklodowska University, Lublin, Poland. It resulted in a number of M.Sc. and Ph.D. theses containing many partial results, remarks and observations (see T. Komorowski [27], J. Wośko [39], K. Bolibok [3]). Some were published: K. Bolibok [4],[5],[6], K. Bolibok and K. Goebel [7], [8],K. Goebel [18],[19], K. Goebel and W. Kaczor [20], K. Goebel and T. Komorowski [22],T. Komorowski and J.Wośko [28].

It is worth mentioning that the minimal displacement problem was also discussed for other classes of mappings than subclasses of L.

M. Furi and M. Martelli [13],[14],[15] came up with several results concerning kset-contractions. These are mappings satisfying the Darbo (α-Lipschitz) condition α (T (E)) ≤ kα (E) for all sets E on which T acts (α denotes the Kuratowski’s measure of noncompactness). S. Reich [36],[37],[38] studied pseudolipschitzian mappings and k-set contractions which are not necessarily self-mappings but satisfy a so called inwardness (or weak inwardness) condition. W. A. Kirk [25] presented some facts concerning the class of Hőlder maps. J. Wośko in his thesis [39] discussed the minimal displacement problem for the whole class of continuous mappings C. For a given set K, d (C) turned out to be a number characterizing the “degree” of noncompactness of K.

The evaluation of minimal displacement is often an initial step to obtain fixed point results. Let us mention some papers which follow this line: J. Bryszewski, M. Serbinowski [10], J. Garcia-Falset, E. Llorens-Fuster, B. Sims [16], T. Kuczumow, S. Reich, A. Stachura [29], S. Massa, D. Roux [31].

There are not many papers concerning other optimality criterion for the retraction of B onto S, for example (the value k_{0} (X)). An interesting observation has been presented by J. Wosko [39]; it is proved, via a concrete example, that in the space of continuous function C [0, 1] , for any ε > 0, there exists a retraction R : B → S that is a (1 + ε)-set-contraction.

Finally let us mention that special chapters on the subjects discussed can be found in the book by K. Goebel and W. A. Kirk [21]. Also some facts are discussed in the recent monograph by Y. Benyamini and J. Lindenstrauss [1].

References

[1] Benyamini Y., Lindenstrauss J. Geometric nonlinear functional analysis. Vol 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, [2] Benyamini Y., Sternfeld Y. Spheres in infinitely dimensional normed spaces are Lipschitz con-

tractible. Proc. Amer. Math. Soc. 88 (1983), 439-445.

[3] Bolibok K. Minimal displacement and retraction problems for balls in Banach spaces (Polish), Mariae Curie-Sklodowska University,Thesis (1999)

[4] Bolibok K. Constructions of lipschitzian mappings with non zero minimal displacement in spaces L^{1}(0, 1) and L^{2} (0, 1) . Annal. Univ. Mariae Curie-Sklodowska University, 50 (1996), 25-31.

[5] Bolibok K. Construction of a lipschitzian retraction in the space c_{0}. Annal. Mariae Curie-Sklodowska University, 51 (1997), 43-46

[6] Bolibok K. Minimal displacement and retraction problems in the space l^{1}. Nonlinear Analysis Forum, 3 (1998), 13-23.

[7] Bolibok K., Goebel K. A note on minimal displacement and retraction problems. J. Math. Anal. Appl., 207 (1997), 308-314.

[8] Bolibok K., Goebel K. A minimal displacement problem and related topics. Proceedings of the 1-st Polish Symposium on Nonlinear Analysis, Łódź (1997), 61-76.

[9] Brouwer L.E.J. Über Abbildungen von Mannigfaltigkeiten. Math. Ann. 71 (1912), 97-115.

[10] Bryszewski J., Serbinowski M. On the minimal displacement under admissible maps. Bull. Acad. Polon. Sci. Sér. Sci. Math. 271979), 201-207.

[11] Franchetti C. A new geometric property of the unit ball of a normed space. (Italian) Rend. Sem. Mat. Fis. Milano 55 (1985), 113-122.

[12] Franchetti C. Lipschitz maps and the geometry of the unit ball in normed spaces. Arch. Math. (Basel) 46 (1986), 76-84.

[13] Furi M., Martelli M. On the minimal displacement of point under α− Lipschitz maps in normed spaces. Boll. Un. Mat. Ital. 9 (1974), 791-799.

[14] Furi M. Martelli M. A Lefshetz type theorem for the minimal displacement of points under maps defined on a class on ANR’s. Boll.Un. Mat. Ital. 10 (1974), 174-181.

[15] Furi M. Martelli M. On minimal displacement under acyclic valued maps defined on a class of ANR’s. Boll.Un. Mat. Ital. 11 (1975), 238-246.

[16] Garcia-Falset J., Llorens-Fuster E., Sims B. Fixed point theory for almost convex functions. Nonlinear Anal. 32 (1998), 601-605.

[17] Goebel K. On the minimal displacement of points under lipschitzian mappings. Pacific J. Math. 48 (1973), 151-163.

[18] Goebel K. A way to retract balls onto spheres. Nonlinear Analysis and Applications, Marcel Dekker, New York, (to appear).

[19] Goebel K. On minimal displacement problem and retractions of balls onto spheres, Proceedings of ICMAA 2000, (to appear).

[20] Goebel K., Kaczor W. Remarks on failure of Schauder’s theorem in noncompact setting. Annal. Mariae Curie-Sklodowska University, 51 (1997), 99-108.

[21] Goebel K., Kirk W.A. Topics in metric fixed point theory. Cambridge University Press, London (1990).

[22] Goebel K., Komorowski T. Retracting balls onto spheres and minimal displacement problem. Fixed point theory and applications (Marseille 1989), Pitman Res. Notes Math. Ser., 252 (1991), 155-172.

[23] Kannai Y. An elementary proof of the no-retraction theorem. Amer. Math. Monthly,88 (1981) 264-268.

[24] Kakutani S. Topological properties of the unit sphere of a Hilbert space. Proc. Imp. Acad. Tokyo 19 (1943), 269-271.

[25] Kirk W.A. Hőlder continuity and minimal displacement. Numer Funct. Anal. Optim. 19 (1998), 71-79.

[26] Klee V. Some Topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30-45.

[27] Komorowski T. Selected topics on lipschitzian mappings (Polish). Maria Curie-Sklodowska University, Thesis, Lublin (1987).

[28] Komorowski T., Wośko J. A remark on the retracting of a ball onto a sphere in an infinite dimensional Hilbert space. Math. Scand. 67 (1990), 223-226.

REFERENCES 33

[29] Kuczumow T., Reich S., Stachura A. Minimal displacement of points under holomorphic mappings and fixed point properties for union of convex sets. Trans. Amer. Math. Soc. 343 (1994), 575-586.

[30] Lin P.K., Sternfeld Y. Convex sets with the fixed point property are compact. Proc. Amer. Math. Soc. 93 (1985), 633-639.

[31] Massa S., Roux D. On the minimal displacement under generalized nonexpansive multivalued mappings. Rend. Mat. 12 (1980), 577-585.

[32] Mauldin D. (ed) The Scottish Book: Mathematical Problems from the Scottish Cafe, Birkhauser, Boston (1981).

[33] Milnor J. Analytic proof of the “Hairy Ball Theorem” and Brouwer Fixed Point Theorem. Amer. Math. Monthly 85 (1978), 521-524.

[34] Nowak B. On the Lipschitz retraction of the unit ball in infinite dimensional Banach space onto boundary. Bull. Acad. Polon. Sci. 27 (1979),861-864.

[35] Schauder J. Der Fixpuntsatz in Funktionalräumen. Studia Math 2 (1930), 171-180.

[36] Reich S. Minimal displacement of points under weakly inward pseudo-lipschitzian mappings I. Atti Accad. Naz. Lincei 59 (1975), 40-44.

[37] Reich S. Minimal displacement of points under weakly inward pseudo-lipschitzian mappings II. Atti Accad. Naz. Lincei 60 (1976), 95-96.

[38] Reich S. A minimal displacement problem. Comment. Math. Univ. St. Pauli, 26 (1977), 131-135.

[39] Wośko J. Minimal displacement problem (Polish). Mariae Curie-Sklodowska University, Thesis (1995)

[40] Wośko J. An example related to the retraction problem. Ann. Univ. Mariae Curie-Sklodowska, Sect. A 45 (1991), 127-130.

Index

Chebyshev center, 27 Chebyshev radius, 11 Contraction

k-set-, 31 Fixed point property

topological, 1 Involution

2-periodic, 19 Kakutani’s Examples, 4 Lipschitz extension, 6 Mapping

asymptotically nonexpansive, 20 uniformly lipschitzian, 18 Minimal displacement, 10 Non-Contractability of spheres, 2 Optimal retraction in Hilbert space, 26 Problem

minimal displacement, 10, 26 optimal retraction, 20

Projection

nearest point, 26 Theorem

Brouwer’s fixed point, 30 Brouwer’s, 1 non-retraction, 2 nonretraction, 31 Schauder’s fixed point, 3 Schauder’s, 31