ULTRA-METHODS IN METRIC FIXED POINT THEORY

M. A. KHAMSI AND B. SIMS

Contents

1. Preface

Over the last two decades ultrapower techniques have become major tools for the development and understanding of metric fixed point theory. In this short chapter we develop the Banach space ultrapower and initiate its use in studying the w-fpp. For a more extensive and detailed treatment than is given here the reader is referred to [A-K] and [S82].

2. Introduction to Ultrapowers of Banach spaces

Throughout the chapter I will denote an index set, usually the natural numbers N for most situations in metric fixed point theory.

Definition 2.1. A filter on I is a nonempty family of subsets F ⊆ 2^{I} satisfying (i) F is closed under taking supersets. That is, A ∈ F and A ⊆ B ⊆ I ⟹ B ∈ F. (ii) F is closed under finite intersections. A, B ∈ F ⟹ A ∩ B ∈ F.

\underline{Examples.}

(1) The entire set 2^{I} defines a filter.

(2) The Fréchet filter {A ⊆ I : I\A is finite }

(3) For i_{0} ∈ I, F_{i_{0}} := {A ⊆ I : i_{0} ∈ A}. Filters of the form F_{i_{0}} for some i_{0} ∈ I are termed trivial, or non-free filters.

(4) If (I, ⪯) is a lattice, then the family of supersets of sets of the form M_{i_{0}} = {i : i ⪰ i_{0}}, for i_{0} ∈ I, is a filter. To see this, note that M_{i_{0}} ∩ M_{j_{0}} = M_{i_{0}∨j_{0}}.

A filter F is proper if it does not equal the power set of I, 2^{I}. Equivalent conditions are: ∅ ∉ F, or F has the finite intersection property. Throughout this chapter, we will take filter to mean proper filter.

Definition 2.2. An ultrafilter U on I is a filter on I which is maximal with respect to ordering of filters on I by inclusion: that is, if U ⊆ F and F is a filter on I, then F = U. Zorn’s lemma ensures that every filter has an extension to an ultrafilter.

Lemma 2.3. A filter U ⊂ 2^{I} is an ultrafilter on I if and only if for every A ⊆ I precisely one of A or I\A is in U.

Proof. (⇒) We show that if I \ A ∉ U, then A ∈ U. If I \ A ∉ U, then I \ A has no subset which is an element of U; hence every element of U meets A. The family B = {A ∩ U : U ∈ U} therefore has the finite intersection property and so its supersets form a filter F_{B}. But U ⊆ F_{B}, because U ⊇ U ∩ A ∈ F_{B}, and so by the maximality F_{B} = U. Also, A = A ∩ I ∈ F_{B} by F(i), and so A ∈ U.

(⇐) Note: the condition automatically ensures U is proper because I ∈ U and so ∅ = I \I ∉ U. Now, let F be a filter on I with U ⊆ F, we show F = U. Assume not, then there exists A ∈ F with A ∉ U. However, we then have I \ A ∈ U ⊆ F. So both A and I \ A belong to F which , by F(i), implies that ∅ = A ∩ (I \ A) ∈ F, contradicting F proper.

As a consequence of this lemma: For an ultrafilter U on I if A_{1} ∪ A_{2} ∪ ⋯ ∪ A_{n} ∈ U then at least one of the sets A_{1}, A_{2}, ⋯, A_{n} is in U, and an ultrafilter is nontrivial (free) if and only if it contains no finite subsets. It will henceforth be a standing assumption that all the filters and ultrafilters with which we deal are nontrivial. We say U is countably complete if it is closed under countable intersections. Ultrafilters which are not contably complete are particularly useful for some purposes. It is readily seen that an ultrafilter U is countably incomplete if and only if there exist elements A_{0}, A_{1}, ⋯, A_{n}, ⋯ in U with

I = A_{0} ⊃ A_{1} ⊃ A_{2} ⊃ ⋯ ⊃ A_{n} ⊃ ⋯ and ⋂_{n=0}^{∞} A_{n} = ∅.

We shall see that this structure allows us to readily extended inductive and diagonal type arguments into ultrpowers. Every ultrafilter U over N is necessarily countably incomplete (A_{n} := {n, n + 1, n + 2, ⋯ } ∈ U).

One of the most exiting result about ultrafilters deal with compactness. Before we state this result, we will need to link ultrafilters with the concept of convergence in topological spaces.

Definition 2.4. For a Hausdorff topological space (Ω, T ), ultrafilter U on I, and (x_{i})_{i∈I} ⊆ Ω we say

lim_{U} x_{i} ≡ T − lim_{U} x_{i} = x_{0}

if for every neighbourhood N of x_{0} we have {i ∈ I : x_{i} ∈ N} ∈ U.

Limits along U are unique and satisfy all the usual limit theorems when Ω is a linear topological space. If U is on N and (x_{n}) is a bounded sequence in R then

lim inf_{n} x_{n} ≤ lim_{U} x_{n} ≤ lim sup_{n} x_{n}.

Moreover, if C is a closed subset of Ω (x_{i})_{i∈I} ⊆ C, then lim_{U} x_{i} belongs to C when it exists.

Remark 2.5. Let X be a metric space. If U is an ultrafilter and lim_{U} x_{n} = x, with (x_{n}) ⊂ X,

then there exists a subsequence of (x_{n}) which converges to x.

The next theorem is fundamanetal since it characterizes compactness by use of ultrafilters.

Theorem 2.6. Let K be a Hausdorff topological space. K is compact if and only if lim_{U} x_{i} exists

for all (x_{i})_{i∈I} ⊂ K and any ultrafilter U over I.

When the space in question is a linear topological vector space, the convergence over an ultrafilter has simialr behaviours as the traditional convergence. In particular, we have:

Proposition 2.7. Let X be a linear topological vector space, and U an ultrafilter over an index set I.

(i) Suppose that (x_{i})_{i∈I} and (y_{i})_{i∈I} are two subsets of X such that lim_{U} x_{i} and lim_{U} y_{i} exist. Then

lim_{U} (x_{i} + y_{i}) = lim_{U} x_{i} + lim_{U} y_{i} and lim_{U} α x_{i} = α lim_{U} x_{i}

for any scalar α ∈ R.

(ii) If X is a Banach lattice and (x_{i})_{i∈I} is a subset of positive elements of X, i.e. x_{i} ≥ 0, then lim_{U} x_{i} is also positive.

Now we are ready to define the ultrapower of a Banach space. Let X be a Banach space and U an ultrafilter over an index set I. We can form the substitution space

ℓ_{∞}(X) := {(x_{i})_{i∈I} : ∥(x_{i})∥_{∞} = sup_{i∈I} ∥x_{i}∥ < ∞}.

Then,

N_{U}(X) := {(x_{i})_{i∈I} ∈ ℓ_{∞}(X) : lim_{U} ∥x_{i}∥ = 0}

is a closed linear subspace of ℓ_{∞}(X).

Definition 2.8. The Banach space ultrapower of X over U is defined to be the Banach space quotient

(X)_{U} := ℓ_{∞}(X)/N_{U}(X),

with elements denoted by [x_{i}]_{U}, where (x_{i}) is a representative of the equivalence class. The quotient norm is cannonically given by

∥[x_{i}]_{U}∥ = lim_{U} ∥x_{i}∥.

Remark 2.9. The mapping J : X ⟶ (X)_{U} defined by

J (x) := [x] := [x_{i}]_{U}, where x_{i} = x, for all i ∈ I

is an isometric embedding of X into (X)_{U}. Using the map J, one may identify X with J (X) seen as a subspace of (X)_{U}. It may happen that we will omit to mention the map J and consider X as a subspace of (X)_{U}.

Some authors [****] have avoided the use of ultrafilters. Indeed, let X be a Banach space and let ℓ_{∞}(X) as before, i.e.

ℓ_{∞}(X) = {(x_{n}); x_{n} ∈ X and sup ||x_{n}|| < ∞} .

The space c_{0}(X) = {(x_{n}) ∈ ℓ_{∞}(X); lim_{n→∞} ||x_{n}|| = 0 } is a closed subspace of ℓ_{∞}(X). Set

X̃ := ℓ_{∞}(X)/c_{0}(X).

For any [(x_{n})] ∈ X̃, we have

∥[(x_{n})]∥X̃ = lim sup_{n→∞} ∥x_{n}∥ .

The Banach space X̃ enjoys many similar properties as (X)_{U}. In this chapter, we will continue to use the ultrafilter construction.

In what follows, we describe some of the fundamental results related to ultrapowers. We will not be exhaustive and let the reader read more on this by consulting [****].

Proposition 2.10. Let (X)_{U} be an ultrapower of a Banach space X. Then for any ε > 0 and any finite dimensional subspace Y_{0} of (X)_{U}, there exists a subspace X_{0} of X with the same dimension and a linear map T : X_{0} → Y_{0} susch that

(1 − ε)∥x∥ ≤ ∥T (x)∥ ≤ (1 + ε)∥x∥

holds for all x ∈ X_{0}.

Any Banach space which enjoys a similar property as the one described above for (X)_{U} is called finitely representable in X. Therefore, any ultrapower of X is finitely representable in X. Note that we may avoid using the map T by introducing the so-called Banach-Mazur distance between normed spaces.

Definition 2.11. Let X and Y be Banach spaces. The Banach-MAzur distance between X and Y is

d(X, Y) = inf{∥T ∥ ∥T^{−1}∥; where T is an isomorphism from X onto Y } .

When X and Y are not isomorphic we simply set d(X, Y) = ∞.

Therefore, X is finitely representable in Y if and only if for any ε > 0 and any finite dimensional subspace X_{0} of X, there exists a subspace Y_{0} of Y with the same dimension susch that d(X_{0}, Y_{0}) < 1+ε. It is quite amazing to find out that ultrapowers catches up the finitely representable spaces. Indeed, we have

Theorem 2.12. Let X and Y be Banach spaces. Assume that Y is finitely representable in X. Then there exist an index set I and an ultrafilter U on I such that Y is isometrically isomorphic to a subspace of (X)_{U}.

Example 2.13. Ultrapowers of the Hilbert space It is known that a Banach space X is a Hilbert space if and only if

∥x + y∥^{2} + ∥x − y∥^{2} = 2∥x∥^{2} + 2∥y∥^{2}

for any x, y ∈ X. Let (X)_{U} be an ultrapower of X. Let [(x_{i})] and [(y_{i})] be two elements in (X)_{U}. We have

∥[(x_{i})] + [(y_{i})]∥^{2} = ∥[(x_{i} + y_{i})]∥^{2} = lim_{U} ∥x_{i} + y_{i}∥^{2}

and

∥[(x_{i})] − [(y_{i})]∥^{2} = ∥[(x_{i} − y_{i})]∥^{2} = lim_{U} ∥x_{i} − y_{i}∥^{2} .

Since

lim_{U} ∥x_{i} + y_{i}∥^{2} + lim_{U} ∥x_{i} − y_{i}∥^{2} = lim_{U} ∥x_{i} + y_{i}∥^{2} + ∥x_{i} − y_{i}∥^{2} ,

and using the Hilbert structure of X, we get

lim_{U} ∥x_{i} + y_{i}∥^{2} + lim_{U} ∥x_{i} − y_{i}∥^{2} = lim_{U} 2∥x_{i}∥^{2} + 2∥y_{i}∥^{2} .

Hence

∥[(x_{i})] + [(y_{i})]∥^{2} + ∥[(x_{i})] − [(y_{i})]∥^{2} = 2∥[(x_{i})]∥^{2} + 2∥[(y_{i})]∥^{2}

which clearly implies (X)_{U} is a Hilbert space.

This example, though easy to prove, is extremelly rich in many ways. Indeed, what the reader should learn from it is that the ultrapower catches any property satisfied by the Banach space and described by finitely many vectors. May be one of the most useful is the lattice structure. Indeed, if X is a lattice Banach space, then any ultrapower (X)_{U} is also a Banach lattice. In order to see this, we will say that x̃ ∈ (X)_{U} is positive if and only if one can find a subset (x_{i}) of positive elements in X such that x̃ = [(x_{i})]. In this case, (X)_{U} enjoys most of the important lattice properties satisfied by X.

Also from the above example, we see that a nonreflexive Banach space may not be finitely representable in X, a Hilbert space. In other words, only reflexive Banach spaces may be finitely representable in X. This lead to a new concept, called super-property.

Definition 2.14. Let P be a property defined on a Banach space X. We say that X has the property ”super-P” if every finitely representable Banach space in X has P.

In particular the Hilber space is superreflexive. One may think that the Hilbert space is the only example of superreflexive Banach space. In the following example, we show that a rich family of Banach spaces are reflexive.

Example 2.15. Let X be a Banach space. For any ε > 0, set

δ_{X}(ε) = inf \{1 − \frac{1}{2}∥x + y∥; x, y ∈ X and ∥x∥ ≤ 1 , ∥x∥ ≤ 1\} .

The function δ_{X}(ε) is called the modulus of uniform convexity of X. Define the characteristic of uniform convexity of X as

ε_{0}(X) = sup{ε; δ_{X}(ε) = 0} .

A Banach space X is uniformly convex if and only if δ_{X}(ε) > 0 for any ε > 0. And it is said to be uniformly nonsquare if and only if ε_{0}(X) < 2. Let us next discuss the link between these concepts and ultrapowers. Let (X)_{U} be an ultrapoer of X. Then, for any ε > 0, we have

δ_{X}(ε) = δ_{(X)_{U}} (ε) .

So in particular, we also have ε_{0}(X) = ε_{0}((X)_{U}). Since uniformly nonsquare are reflexive, then we deduce that uniformly nonsquare Banach spaces are also superreflexive. In fact, Enflo [***] (see also Pisier [***]) has shown that X is supereflexive if and only if there exists an equivalent norm (in X) which is uniformly convex. More on this may be found in [****].

Next we discuss the ultraproducts of maps. Let X and Y be two Banach spaces. Let (X)_{U} and (Y)_{U} be their associated ultrapowers. Let T_{i} : D ⊂ X → Y be a family of maps indexed by the same set I (as the ultrapowers) defined on D. Consider

[D]_{U} = {x̃ ∈ (X)_{U} ; there exists a representative (d_{i}) of x̃ with d_{i} ∈ D} .

Define [(T_{i})]_{U} : [D]_{U} → (Y)_{U} by

[(T_{i})]_{U}([(d_{i})]_{U}) = [(T_{i}(d_{i}))]_{U} .

We have to make sure that (T_{i}) satisfy enough conditions which will allow this definition to make sense. In particular, we have the following

Proposition 2.16. Using the above notations, we have (i) [D]_{U} is convex if D is also convex; (ii) [D]_{U} is closed if D is closed; (iii) [D]_{U} is bounded if D is bounded; (iv) [(T_{i})]_{U} is Lipschitzian provided that each T_{i} is Lipschitzian and the family of their constant

of Lipschitz (|T_{i}|) is bounded;

(v) [(T_{i})]_{U} is a bounded linear operator provided the (T_{i}) are linear operators which are uniformly bounded, i.e. sup_{i∈I} ∥T_{i}∥ < ∞.

Recall that a map T : D ⊂ X → Y is said to be Lipschitzian with λ as a constant of Lipschitz if

∥T (x) − T (y)∥ ≤ λ∥x − y∥

for any x, y ∈ D. The above results may be useful in studying the dual space of the ultrapower for example. Indeed, let (x_{i}^{∗}) be a uniformly bounded family of linear functionals defined on X (i.e. elements of the dual space X^{∗}). Then from the above results, we generate a bounded linear functional [(x_{i}^{∗})]_{U} defined as before. This linear functional belongs to the dual of the ultrapower,

i.e. ((X)_{U})^{∗}. One may then ask whether this construction gives all the elements of the dual

space ((X)_{U})^{∗}. The answer is given by the following theorem

Theorem 2.17. Let X be a Banach space. Then

((X)_{U})^{∗} = (X^{∗})_{U}

if and only if X is superreflexive.

More on this and similar results may be found in [***].

We will close this section with the interesting example of L_{1}[0, 1].

Example 2.18. In this example, we will discuss the ultrapower of the Banach space L_{1}[0, 1].

3. Fixed Point Theory

In this chapter, we focus of the branch of fixed point theory considered as an outgrowth of the contraction mapping principle of Banach and Picard. The mappings involved are of the form T : K → K, where K is a subset of a Banach space, where T is nonexpansive, thus

∥T (x) − T (y)∥ ≤ ∥x − y∥

for any x, y ∈ K.

Because of its important linkages with the theory of monotone and accretive operators, fixed point theory for nonexpansive mappings has long been considered a fundamental part of nonlinear functional analysis. However, the attempt to classify those subsets of Banach spaces which have the fixed point property for such mappings has now become a study in its own right-one which has yielded many elegant results and led to numerous discoveries in Banach space geometry. Fixed point theory for nonexpanisve mappings has its origins in the 1965 existence theorems of F. Browder, D. Gohde, and W.A. Kirk. Although such mappings are natural extensions of the contraction mappings, it was clear from the outset that the study of nonexpansive mappings required techniques which go far beyond the purely metric approach. The traditional methods used in studying nonexpansive mappings have invovled an interwining of geometrical and topological arguments. The original theorems of Browder and Gohde exploited the special geometrical structure of uniformly convex Banach spaces. The somewhat more general theorem of Kirk asserts that K has the fixed point property for nonexpansive mappings if K is weakly-compact and convex and at the same time possesses a property called ”normal structure”. The early phases of the development of the theory centered around the identification of classes of spaces whose bounded convex sets possess normal structure, and it was soon discovered that certain weaknings and variants of normal structure also suffice. The turning point for this so-called classical fixed point theory for nonexpansive mappings was the discovery of D. Alspach in 1980 (published in 1981) of an example of a weakly-compact convex subset of L_{1}[0, 1] which fails to have the fixed point property for nonexpansive mappings. Alspach’s example set the stage for Maurey’s surprising discovery that all bounded closed convex subsets of reflexive subspaces of L_{1}[0, 1] do have the fixed point property for nonexpansive mappings. Maurey’s ideas were original and set the stage to a new direction that led to many new results. It is, in our opinion, the second revolution after the 1965 existence theorems.

Definition 3.1. Let X be a Banach space. A bounded, closed, convex subset K of X is said to have the fixed point property (f.p.p.) if every nonexpansive mapping T : K → K has a nonempty fixed point set F ix(T). Recall that x ∈ F ix(T) if and only if T (x) = x.

Let X be a Banch space and C a nonempty, weakly-compact, convex subset of X. Assume that C does not have the fixed point property; therefore there exists a nonexpansive mapping T : C → C with an empty fixed point set. Set

F = {K ⊂ C; K is nonempty, closed, convex, and invariant under T, i.e. T (K) ⊂ K} .

Clearly F is not empty, since C ∈ F. The weak-compactness of C ensures that F satisfies the assumptions for Zorn’s lemma. Thereofre F has minimal elements.

Definition 3.2. A convex set K is said to be minimal for T if K is a minimal element of F.

Clearly any set K which is minimal for T has more than one point, i.e.

diam(K) = sup{∥x − y∥; x, y ∈ K} > 0 ,

since otherwise T would have a fixed point. Next we discuss some of the basic results satisfied by minimal invariant sets.

Proposition 3.3. Let K be a minimal set for T. Then

\overline{conv}(T (K)) = K .

The next result gives an interesting property satisfied by minimal sets.

Proposition 3.4. Let K be a minimal set for T, and α : K → R_{+} be a lower semi-continuous convex function such that

α(T (x)) ≤ α(x) , for all x ∈ K.

Then α is a constant function.

Some of the direct consequences of this result are:

Proposition 3.5. Any minimal set K for T is diametral, i.e.

sup{∥x − y∥; y ∈ K} = diam(K)

for any x ∈ K.

It is this property that led to Kirk’s discovery of his fixed point theorem via normal structure property. First let us introduce some notations. For any bounded subset A of X, we define

r(x, A) = sup{∥x − y∥; y ∈ A},  for any x ∈ A,
R(A)  = inf{r(x, A); x ∈ A},
diam(A) = sup{r(x, A); x ∈ A},
C(A)  = {x ∈ A; r(x, A) = R(A)}.

The positive number R(A) is called the Chebyshev radius of A, and the set C(A) is called the Chebyshev center of A. It should be clear that for any x ∈ A, we have

R(A) ≤ r(x, A) ≤ diam(A) .

Note that A is diametral if and only if C(A) = A, which is also equivalent to R(A) = diam(A).

Definition 3.6. A convex subset K of X is said to have normal structure if each bounded convex subset S of K is not diametral. The Banach space X is said to have normal structure if each convex subset has normal structure.

On normal structure and related concepts in Banach spaces, one can consult [******]. Kirk’s fixed point theorem now may be stated:

Theorem 3.7. Let X be a Banach space. If C is a nonempty weakly-compact convex subset of X, and C has normal structure, then C has f.p.p.

For many years, it was unknown whether normal structure and the fixed point property are equivalent until 1975, when Karlovitz [***] proved that was not the case. He discovered a new property satisfied by the minimal invariant sets. This property was also discovered independently by Goebel [***]. Before we describe this new property, we will need some more facts about nonexpansive mappings. Indeed, let K be a nonempty, bounded, closed, convex subset of a Banach space X, and T : K → K be nonexpansive. Fix ε ∈ (0, 1) and x_{0} ∈ K, and consider the map T_{ε} : K → K defined by

T_{ε}(x) = εx_{0} + (1 − ε)T (x)

for all x ∈ K. T_{ε} is clearly a contraction mapping. Hence it has a unique fixed point x_{ε} ∈ K, i.e. T_{ε}(x_{ε}) = x_{ε}. We have

∥T (x_{ε}) − x_{ε}∥ ≤ ε diam(K) .

In other words, we have

inf{∥T (x) − x∥; x ∈ K} = 0 .

The evaluation of this infinimum is connected to the so-called minimal displacement. See [****].

Definition 3.8. A sequence (x_{n}) satisfying lim_{n→∞} ∥x_{n} − T (x_{n})∥ = 0, is called an approximate fixed point sequence.

We showed that nonexpansive mappings always have approximate fixed point sequence.

Goebel and Karlovitz has shown the following:

Theorem 3.9. Let K be a minimal set for T. Then for any approximate fixed point sequence (x_{n}) in K, we have

lim_{n→∞} ∥x_{n} − x∥ = diam(K)

for any x ∈ K.

Maurey’s new ideas were based on translating these properties in the ultrapower context. May be the natural way to explain his ideas is to notice that a nonexpansive mapping my fail to have a fixed point (as shown by Alspach) but these maps always have an approximate fixed point sequence. These sequences may be seen as points in the ultrapower space. Therefore for someone who is at ease with the use of ultrapowers, like Maurey, it was very natural to consider such spaces.

Let C be a nonempty, convex, weakly-compact subset of a Banach space X and T : C → C a nonexpansive mapping with no fixed point. As before, let K be a minimal set for T. Let U be an ultrafilter on the set of positive integers. We define in (X)_{U}

K̃ := {[x_{n}]_{U} : x_{n} ∈ K, for all n ∈ N}.

C̃ is a convex subset, with diam(C̃) = diam(C), containing the isometric copy J (K) of K, and on which T̃ : K̃ ⟶ K̃ defined by

T̃([x_{n}]_{U}) = [T (x_{n})]_{U},

where the representatives x_{n} are chosen from K, is a well defined nonexpansive mapping which leaves J (K) invariant. We now establish a number of basic results for K̃ and T̃ constructed as above.

Proposition 3.10. The set K̃ in (X)_{U} is closed. Hence, when X is a superreflexive space K̃ is weakly-compact.

Proof. Let N = A_{0} ⊃ A_{1} ⊃ A_{2} ⊃ ⋯ ⊃ A_{n} ⊃ ⋯, being a nested sequence of sets with each

A_{n} ∈ U, and ⋂_{n≥1} A_{n} = ∅.

Suppose [t_{i}^{1}]_{U}, [t_{i}^{2}]_{U}, ⋯ is a sequence of points in K̃, with each t_{i}^{j} ∈ K, which converges to [x_{i}]_{U} ∈ (X)_{U}. By passing to a subsequence if necessary we may without loss of generality assume that

∥[t_{i}^{m}] − [x_{i}]∥ = lim_{U} ∥t_{i}^{m} − x_{i}∥ < \frac{1}{m}.

For each m ∈ N let

B_{m} := \{i ∈ N : ∥t_{i}^{m} − x_{i}∥ < \frac{2}{m}\} ∩ A_{m} ∈ U.

and put B_{0} := N and t_{i}^{0} := 0, then

N = B_{0} ⊃ B_{1} ⊃ B_{2} ⊃ ⋯ ⊃ B_{m} ⊃ ⋯, and ∩_{m=0}^{∞} B_{m} = ∅.

From this it follows that for each i ∈ N there is a unique m such that i ∈ B_{m}\B_{m+1}. Define y_{i} := t_{i}^{m}, for this m, in particular then y_{i} ∈ K. Now, given any m ∈ N, for each i ∈ B_{m} there is a unique p ≥ m with i ∈ B_{p}\B_{p+1}. thus,

∥y_{i} − x_{i}∥ = ∥t_{i}^{p} − x_{i}∥ < \frac{2}{p} ≤ \frac{2}{m},

and so

\{i ∈ N : ∥y_{i} − x_{i}∥ < \frac{2}{m}\} ⊇ B_{m} ∈ U.

We therefore have that U − lim ∥y_{i} − x_{i}∥ = 0, which yields the desired conclusion that [x_{i}]_{U} ∈ K̃.

Proposition 3.11. If (x_{n}) is an approximate fixed point sequence for T, then [x_{n}]_{U} is a fixed point of T̃. So T̃ always has fixed points in K̃.

Conversely, from a fixed point (indeed an approximate fixed point sequence) for T̃ in K̃ we can readily extract an approximate fixed point sequence for T. If K is weakly-compact and a minimal invariant set for T, so that the Goebel-Karlovitz lemma applies, then in the above proposition we also have ∥[x_{n}]_{U} − J x∥ = diam(K), for all x ∈ K. Since, in this case we can always assume without loss of generality that diam K = 1 and that (x_{n}) converges weakly to 0 (so, 0 ∈ K), we may suppose that ∥[x_{n}]_{U}∥ = d([x_{n}]_{U}, J K) = 1. The following is a significant observation of B. Maurey [M].

Lemma 3.12. Given any two fixed points ã = [a_{n}]_{U} and b̃ = [b_{n}]_{U} of T̃ in K̃ there is a fixed point c̃ with

∥ã − c̃∥ = ∥c̃ − b̃∥ = \frac{1}{2}∥ã − b̃∥.

Proof. We may assume that λ := ∥ã − b̃∥ := lim_{U} ∥a_{n} − b_{n}∥ > 0. For each m ∈ N let

A_{m} := \{n ≥ m : ∥a_{n} − b_{n}∥ ≤ λ + \frac{2}{m^{2}} and ∥a_{n} − T a_{n}∥, ∥b_{n} − T b_{n}∥ ≤ \frac{1}{m^{2}} \} ,

then A_{m} ∈ U, N =: A_{0} ⊃ A_{1} ⊃ A_{2} ⊃ ⋯ ⊃ A_{n} ⊃ ⋯ and ⋂_{n≥1} A_{n} = ∅.

For each n ∈ N let K_{n} := \{c ∈ K : ∥a_{n} − c∥, ∥b_{n} − c∥ ≤ λ/2 + \frac{1}{m}\} where m is the unique

element of N for which n ∈ A_{m}\A_{m+1}. Then K_{n} is bounded, closed, convex and nonempty since

∥a_{n} − 1/2 (a_{n} + b_{n})∥ = \frac{1}{2} ∥a_{n} − b_{n}∥ ≤ \frac{λ}{2} + \frac{1}{m^{2}} ≤ \frac{λ}{2} + \frac{1}{m}

and similarly, ∥b_{n} − 1/2 (a_{n} + b_{n})∥ ≤ λ/2 + 1/m, so

\frac{1}{2} (a_{n} + b_{n}) ∈ K_{n}.

Now, define a strict contraction, T_{n} on K_{n} by,

T_{n}z := (1 − \frac{1}{m})T z + \frac{1}{2m}(a_{n} + b_{n}).

To see that K_{n} is T_{n}-invariant let z ∈ K_{n}, then

∥a_{n} − T_{n}z∥ = ∥a_{n} − ((1 − \frac{1}{m})T z + \frac{1}{2m}(a_{n} + b_{n})) ∥

≤ (1 − \frac{1}{m})∥a_{n} − T z∥ + \frac{1}{2m}∥a_{n} − b_{n}∥

≤ (1 − \frac{1}{m})∥a_{n} − T a_{n}∥ + (1 − \frac{1}{m})∥T a_{n} − T z∥ + \frac{1}{2m}∥a_{n} − b_{n}∥

≤ (1 − \frac{1}{m})∥a_{n} − T a_{n}∥ + (1 − \frac{1}{m})∥a_{n} − z∥ + \frac{1}{2m}∥a_{n} − b_{n}∥

≤ (1 − \frac{1}{m})\frac{1}{m^{2}} + (1 − \frac{1}{m})(λ/2 + \frac{1}{m}) + \frac{1}{2m}(λ + \frac{2}{m^{2}} )

= \frac{1}{m^{2}} − \frac{1}{m^{3}} + λ/2 + \frac{1}{m} − \frac{1}{2m}λ − \frac{1}{m^{2}} + \frac{1}{2m}λ + \frac{1}{m^{3}}

= λ/2 + \frac{1}{m}.

and similarly, ∥b_{n} − T_{n}z∥ ≤ λ/2 + \frac{1}{m}. Thus, T_{n} has a unique fixed point, c_{n} ∈ K_{n}. That is,

c_{n} = T_{n}c_{n} = (1 − \frac{1}{m})T c_{n} + \frac{1}{2m}(a_{n} + b_{n}).

and we have,

∥T c_{n} − c_{n}∥ = \frac{1}{m}∥T c_{n} − (a_{n} + b_{n})/2∥

≤ \frac{1}{2m}(∥T c_{n} − a_{n}∥ + ∥T c_{n} − b_{n}∥)

≤ \frac{1}{m}(λ/2 + \frac{1}{m}).

It therefore follows from the above construction that for each m ∈ N the set of n for which ∥T c_{n} − c_{n}∥ ≤ (1/m)(λ/2 + 1/m) contains A_{m} and so is in U. Consequently, for c̃ := [c_{n}]_{U} we have

∥c̃ − T̃c̃∥ := lim_{U} ∥c_{n} − T c_{n}∥ = 0

and so c̃ is a fixed point of T̃. Similarly, from ∥a_{n} − c_{n}∥, ∥b_{n} − c_{n}∥ ≤ λ/2 + 1/m for all n ∈ A_{m}\A_{m+1}, and consequently for all n ∈ A_{m}, we have ∥ã − c̃∥ and ∥b̃− c̃∥ are less than or equal to λ/2. Since λ = ∥ã − b̃∥, the triangle inequlity then ensures that ∥ã − c̃∥ = ∥b̃ − c̃∥ = ∥ã − b̃∥/2 and the result is established.

This Lemma states that the fixed point set of T̃ is metrically convex. An appeal to Menger’s theorem then ensures the existence of a continuous path of fixed points joining any two fixed points of T̃ and lying within the metric segment between them.

Remark 3.13. When C is weakly-compact and a minimal invariant set for T it is always possible to find two such fixed points ã and b̃ of T̃ with ∥ã−b̃∥ = diamC̃. To see this, we may without loss of generality suppose that diam C = 1 and that we have an approximate fixed point sequence (x_{n}) for T, with (x_{n}) weakly convergent to 0. Applying the Goebel-Karlovitz lemma we may extract a subsequence (x_{n_{i}}) such that ∥x_{n_{i}} − x_{n_{i+1}}∥ ⟶ diamC. Taking ã := [x_{n_{2i}}] and b̃:= [x_{n_{2i−1}}] yields two fixed points of T̃ with

∥ã∥ = ∥b̃∥ = ∥ã − b̃∥ = 1.

The following generalization of the Goebel-Karlovitz lemma, due to P. K. Lin [Lin] has proved basic for establishing the fixed point property using ultrapower methods.

Lemma 3.14. Suppose C is a weakly-compact minimal invariant set for T. If (ã_{n}) is an approximate fixed point sequence for T̃ in C̃ then

lim_{n} ∥ã_{n} − J x∥ = diam(C), for all x ∈ C.

Proof. Suppose this were not the case. Without loss of generality we may take diam(C̃) = diam(C) = 1, and by passing to a subsequence if necessary assume that ∥ã_{n} − T̃ã_{n}∥ < 1/n. Then there are ε_{0} > 0, x_{0} ∈ C, and n_{0} ∈ N with

∥ã_{n} − J x_{0}∥ < 1 − ε_{0}, for all n > n_{0}.

Let ã_{n} = [a_{m}^{n}]_{U}, with a_{m}^{n} ∈ C, and define

A_{n} := {m : ∥a_{m}^{n} − x_{0}∥ < 1 − ε_{0}/2},

and

B_{n} := {m : ∥a_{m}^{n} − T a_{m}^{n}∥ < 2/n}.

Then A_{n} and B_{n} are in U. Put m_{0} = 0 and for n ∈ N inductively choose m_{n} ∈ A_{n} ∩ B_{n} ∩ {m_{n−1} + 1, m_{n−1} + 2, ⋯ } ∈ U. Then the sequence (a_{m_{n}}^{n}) is such that

∥a_{m_{n}}^{n} − T a_{m_{n}}^{n}∥ < 2/n.

That is, (a_{m_{n}}^{n}) is an approximate fixed point sequence for T in C. But,

∥a_{m_{n}}^{n} − x_{0}∥ < 1 − ε_{0}/2,

an observation which is difficult to reconcile with the fact that (a_{m_{n}}^{n}) is, by the Goebel/Karlovitz lemma, diameterizing for C.

Remark 3.15. If W is any nonempty closed convex and T̃-invariant subset of C̃, then, by the standard construction using Banach’s contraction mapping principle, W contains an approximate fixed point sequence for T̃. So, by the above lemma, for every x ∈ C we have sup{∥w̃ − J x∥ : w̃ ∈ W } = diam C. In particular, if we have ‘normalized’ so that diam C = 1 and 0 ∈ C, then

sup_{w̃∈W} ∥w̃∥ = 1.

This leads to an important strategy for establishing the fixed point property in a class of spaces. Namely, try to construct a nonempty closed convex and T̃-invariant subset W of C̃ in such a way that the hypotheses on the spaces preclude the existence of elements in W with norms arbitrarily close to one; thereby contradicting the above lemma and hence the existence of a fixed point free nonexpansive self mapping of a nonempty weakly compact convex subset in the space. Indeed, we know of only one proof establishing the fixed point property for a class of spaces via ultraproduct methods that does not use this approach, and that is S. Prus’ proof [??] that uniformly non-creasey spaces have the fixed point property.

We illustrate the strategy outlined in the above remark with just one example, due to Garcia-Falset [GF95], others may be found scattered throughout this Handbook. See also the Notes and remarks section for references to the literature. Let U be a given ultra filter over N and for each Banach space X define a parameter R(X) by,

R(X) := sup{lim_{U} ∥x + x_{n}∥ : ∥x∥ ≤ 1, ∥x_{n}∥ ≤ 1, for n ∈ N and (x_{n}) → 0 weakly}.

Equivalently, R(X) is the ‘smallest’ number such that

lim_{U} ∥x + x_{n}∥ ≤ R(X)∥x∥ ∨ (lim_{U} ∥x_{n}∥),

for all x ∈ X and all weak null sequences (x_{n}). For example, R(c_{0}) = 1, while R(L_{1}) = 2, in general 1 ≤ R(X) ≤ 2.

Proposition 3.16. If X is a Banach space with R(X) < 2, then X has the weak-fixed point property.

Proof. Suppose X fails the weak-fixed point property. Then there exists a weakly-compact convex set C with diam(C) = 1 which is a minimal invariant set for some nonexpansive mapping T. Further we may assume that C contains a weakly-null approximate fixed point sequence (a_{n}) for T. Let C̃ and T̃ be defined as above and define

W := {[w_{n}]_{U} : w_{n} ∈ C, for n ∈ N, ∥[w_{n}]_{U} − [a_{n}]∥ ≤ 1/2, and D[w_{n}] ≤ 1/2},

where D[w_{n}] := lim_{Um,} lim_{U,n} ∥w_{m} − w_{n}∥. Then, W is readily seen to be a T̃-invariant, closed, convex, nonempty (as (1/2)[a_{n}] ∈ W) subset of C̃. Thus, by the above remark

sup{∥w̃∥ : w̃ ∈ W } = 1.

On the other hand, let w̃ = [w_{n}]_{U} be any element of W, where without loss of generality w_{n} ∈ C, for all n ∈ N, and let w_{0} be the weak-limit with respect to U of (w_{n}). Then,

∥w̃∥ = lim_{U} ∥w_{n}∥
= lim_{U} ∥(w_{n} − w_{0}) + w_{0}∥
≤ R(X)(lim_{U} ∥w_{n} − w_{0}∥) ∨ ∥w_{0}∥,

by definition of R(X), as (w_{n} − w_{0}) converges weakly to 0, hence

∥w̃∥ ≤ R(X) lim_{U,n} lim_{U,m} ∥w_{n} − w_{m}∥ ∨ ∥w_{n} − a_{n}∥,

by lower semi-continuity of the norm, since

lim_{U} w_{m} = lim_{U}(w_{n} − a_{n}) = w_{0} .

Hence

∥w̃∥ ≤ R(X) × \frac{1}{2} ∨ \frac{1}{2} = R(X)/2 < 1.

A contradiction which establishes the result.

Our choice of the above result to illustrate the strategy in the previous remark is based on its utility; the parameter involved is readily evaluated for many spaces and the criteria is satisfied in a large class of spaces. Since nearly uniformly smooth (NUS) Banach spaces are readily seen to have R(X) < 2 [GF95], the result answers in the affirmative the long standing question of whether or not NUS spaces have the weak-fixed point property. In a weakly orthogonal Banach lattice R(X) is less than or equal to the Riesz angle α(X) introduced in [B-S], thus, this proposition generalizes results of [B-S] [S88] and [S92a]. The above proof is typical of those for many of the more recent results in metric fixed point theory, starting with Maurey’s 1982 proof of the weak-fixed point property for c_{0}. Note that, since a numeric contradiction is arrived at, by carefully analysing the proof, the gap (here between R(X)/2 and 1) can be exploited to establish the weak-fixed point property for spaces whose Banach-Mazur distance from a space satisfying the assumptions is not too great. This is the basis for many of the results given in the chapter on Stability of the fixed point property for nonexpansive mappings, where many more existence results, in the more general setting of stability, and applications of the methods outlined here may be found.

4. Maurey’s fundamental theorems

5. More fixed point theorems

6. Notes and Remarks

The results developed in this chapter have been used to establish a number of results in metric fixed point theory, in particular the weakly-fixed point property from a variety of conditions on the space X. The notion of an ultrafilter dates back to work of Tarski in 1930 and that of an ultrapower to Skolem in 1934. A rich theory for ultrapowers and ultraproducts (of sets and models) has been built up by a succession of logicians: Los, Frayne, Morel, Scott, Tarski, Hanf, Chang, Keisler, Robinson, Luxemburg and Shelah to mention only a few. Banach space ultrapowers and ultraproducts were formally introduced by Dacunha-Castelle and Krivine in 1972 and were subsequently developed and applied by Stern, Heinrich and many others. They are now an important and widely used tool for probing the geometry and structure of Banach spaces. They have been particularly important in the study of local theory, superproperties, operator ideals and the isomorphic classification of Banach spaces. Of particular significance for us is the following result. If P is a separably determined Banach space property that is inherited by subspaces; for example, reflexivity, then a space X has super-P if and only if some (and hence every) ultrapower of X over a countably incomplete ultrafilter has P.

BRAILEY LET ME THINK ABOUT THE PROOF OF THIS ONE Ultrapower methods were first introduced into metric fixed point theory by B. Maurey [M] in 1982 when he used this technique to provide a positive resolution to the long standing question of whether or not c_{0} had the weakly-fixed point property. He took W to be the metric midpoint set for two fixed points of T̃ constructed as in remark ***. This was generalized in [B-S] to obtain the weakly-fixed point property for Banach lattices with a Riesz angle

α(X) := sup{∥ |x| ∨ |y| ∥ : x, y ∈ B_{X}} < 2

and for which

lim inf_{m} lim inf_{n} ∥ |x_{n}| ∧ |x_{m}| ∥ = 0, whenever (x_{n}) converges weakly to 0.

Lattices with this last property were referred to as weak orthogonal Banach lattices. A formally stronger version of weak orthogonality:

lim inf_{n} ∥ |x_{n}| ∧ |x| ∥ = 0, whenever (x_{n}) converges weakly to 0 and x ∈ X,

was shown to imply the weakly-fixed point property by Sims [S88], [S92a]. This was obtained using a W first defined by P. K. Lin in 1983 and used to establish the weakly-fixed point property for Banach spaces with a 1-unconditional basis [Lin]. The set W used in these proofs consisted of those points in Maurey’s W whose distance from J C is less than or equal to a half, where in addition the points ã and b̃ were chosen to be ‘orthogonal’ to one another so that ∥ã + b̃∥ = ∥ã − b̃∥ = 1. A class of spaces in which such a choice is always possible was considered in [S88]. Such spaces were said to have property WORTH. It remains an open question whether or not all spaces with WORTH have the weakly-fixed point property. Several more ‘geometric’ variants of these conditions have been introduced. For instance A. Jiménez-Melado and E. Lloréns-Fuster [JM-LF] considered the property of orthogonal convexity, gave examples of orthogonally convex spaces, and showed that it entails the weakly-fixed point property. A Banach space X is orthogonally convex if for every weak-null sequence (x_{n}) with

D(x_{n}) := lim sup_{m} lim sup_{n} ∥x_{m} − x_{n}∥ > 0

there exists β > 1/2 such that

lim sup_{m} lim sup_{n} sup{∥z∥ : ∥z − x_{m}∥, ∥z − x_{n}∥ ≤ β∥x_{m} − x_{n}∥} < D(x_{n}).

The characteristic of a sequence, D(w_{n}), first introduced in the above context and used in the proof of proposition 8, has played a crucial role in many of the more recent results. The proof that spaces with Kalton’s property (M) have the weakly-fixed point property [GF-S] it was found necessary to employ a W similar to that used in the proof of proposition 8, but defined by the asymmetric constraints ∥[w_{n}]_{U} − [a_{n}]∥ ≤ 1/2 − ε and D[w_{n}] ≤ 1/2 + ε, with ε > 0. Recall that X has property (M)if weak null types are constant on spheres about 0. That is, lim_{U} ∥x − x_{n}∥ = lim_{U} ∥y − x_{n}∥ whenever ∥x∥ = ∥y∥ and (x_{n}) weakly converges to 0. P. K. Lin used a W defined by a combination of all the constraints discussed above to establish what is currently the best known bound for the stability of ℓ_{2}’s f.p.p., for details see section ?? of the chapter on Stability of the fixed point property for nonexpansive mappings.

BRAILEY MORE SHOULD BE INCLUDED MAY BE IN DIFFERENT SECTION In the above results, and in many applications, a Banach space ultrapower (X)_{U} over N can be replaced by the space

ℓ_{∞}(X)/c_{0}(X),

where the quotient norm is cannonically given by ∥[x_{n}]∥ = lim sup_{n} ∥x_{n}∥, see for example: [B-S],

[GF95] and [GF-S]. However, calculations in this space usually entail an infestation of subsequence taking. In many instances it is possible to avoid the use of these larger ambient spaces altogether; for example, see [E] where an ultrapower free proof of Maurey’s result on the reflexive subspaces of L_{1} may be found. Such proof often obscure the essential steps in a veritable plague of epsilons and deltas. None-the-less, the disadvantages and advantages are largely cosmetic and it is up to the individual to choose which approach is most to their taste.