Chapter 1

STABILITY OF THE FIXED POINT PROPERTY FOR NONEXPANSIVE MAPPINGS

Jesús Garcia-Falset

Antonio Jiménez-Melado

Enrique Llorens-Fuster

1. Introduction

In 1971 Zidler ([Zi 71]) showed that every separable Banach space (X, ∥·∥) admits an equivalent renorming, (X, ∥ . ∥_{0}), which is uniformly convex in every direction (UCED), and consequently it has weak normal structure and so the weak fixed point property (WFPP) [D-J-S 71].

Later, in 1981, Alspach [A 82] showed that the separable Banach space (L^{1}[0, 1], ∥.∥_{1}) lacks the WFPP. As a consequence of both Zidler’s and Alspach’s results it was known that, in general, the WFPP is not preserved under topological isomorphisms.

Following on from this observation, came the well known, yet still open, problem in metric fixed point theory (see [E-L-O-S 83]): To find a nontrivial class of isomorphic Banach spaces such that each one of its members has the WFPP. A trivial example of such a class is the family of all Banach spaces isomorphic to ℓ_{1} or to any other Schur space.

On the positive side of this problem there are some well known results that show that the WFPP is inherited from Banach spaces with nice geometric properties. This suggests that we may concentrate our attention on what is nowadays known as the stability problem, that is: Given a Banach space (X, ∥ · ∥) with the WFPP, does this property extend to isomorphic spaces close enough to (X, ∥ · ∥) in the Banach Mazur sense. Of course, the question we would like to answer is how far from X is the WFPP preserved. However, with the techniques currently available this is over ambitious and we must content ourselves with lower estimates on how far from X the space may be.

Notice that we may try to answer the above question, without appealing to equivalent renormings, by showing that the existence of a fixed point for the mapping T is guaranteed by some property of T inherited by those mappings that are nonexpansive with respect to an equivalent norm. From the early 1970’s it was known that every nonexpansive mapping in a Banach space is uniformly lipschitzian with respect to any equivalent norm. So we may start by considering uniformly lipschitzian mappings. To be more precise, we have the following elementary result.

Theorem 1.1 Let (X, ∥ · ∥) be a Banach space such that every weakly compact convex subset of X has the fixed point property for uniformly k-lipschitzian mappings. If d(X, Y) < k then Y has the WFPP.

Proof. Since d(X, Y) < k there exists a bicontinuous isomorphism U : Y → X such that ∥U∥∥U^{−1}∥ < k. Suppose that C is a weakly compact convex subset of Y and T : C → C is ∥.∥_{Y} -nonexpansive. Then the mapping T̃ : U(C) → U(C) given by

T̃(U(x)) = U(T (x))

is well defined and it is straightforward to see that for all positive integer n and all x ∈ C,

T̃^{n}(U(x)) = U(T^{n}(x)).

Thus, for all x, y ∈ C,

∥T̃^{n}(U(x)) − T̃^{n}(U(y))∥_{X} ≤ ∥U∥∥T^{n}(x) − T^{n}(y)∥_{Y}
≤ ∥U∥∥x − y∥_{Y}
≤ ∥U∥∥U^{−1}∥∥U(x) − U(y)∥_{X}

for any positive integer n. This shows that T̃ is uniformly λ-lipschitzian on the weakly compact convex subset U(C) with λ = ∥U∥∥U^{−1}∥ < k. Consequently there exists U(x) ∈ U(C) such that T̃(U(x)) = U(x). That is there exists x ∈ C such that U(T (x)) = U(x), which implies that T (x) = x. ■

This theorem shows the close relationship between fixed point theorems for uniformly lipschitzian mappings and bounds of stability for the WFPP. In fact, any fixed point theorem for uniformly k-lipschitzian mappings on weakly compact convex sets, yields k as a “radius of stability” for the WFPP.

In [G-K-T 74] Goebel, Kirk and Thele showed that weakly compact convex subsets of any Banach space (X, ∥·∥) with characteristic of convexity ε_{0}(X) < 1, have the FPP for uniformly k-lipschitzian mappings where k is a constant depending on the modulus of convexity of (X, ∥ · ∥) . More precisely,

Corollary 1.2 Let (X, ∥ · ∥) be a Banach space with ε_{0}(X) < 1. Let k > 1 be the unique solution of the equation

k (1 − δ_{X}(k^{−1})) = 1.  (1.1)

If d(X, Y) < k, then Y has the WFPP.

For a Hilbert space the solution of equation (1.1) is k = \sqrt{5}/2.

Theorem 1.1 and the corresponding Corollary allow us to say that the WFPP is stable in spaces with ε_{0}(X) < 1, in the sense that it is retained when the norm is changed slightly.

In this same sense, some sufficient conditions for the WFPP are also stable. A typical instance of such a condition would be ε_{0}(X) < 1.

Theorem 1.3 [D-T 83] Let (X, ∥·∥) a Banach space with ε_{0}(X) < 1. If k is the unique solution of the equation (1.1), and d(X, Y) < k, then ε_{0}(Y) < 1.

Proof. There exists a bicontinuous isomorphism U : X → Y such that ∥U∥∥U^{−1}∥ < k. Without loss of generality we may suppose that ∥U^{−1}∥ = 1. Choose y_{1}, y_{2} ∈ S_{Y} such that

∥y_{1} − y_{2}∥_{Y} ≥ \frac{∥U∥}{k}

and take x_{1} := U^{−1}(y_{1}), x_{2} := U^{−1}(y_{2}). It is immediate that ∥x_{1}∥_{X} ≤ 1 and ∥x_{2}∥_{X} ≤ 1, and

\frac{∥U∥}{k} ≤ ∥y_{1} − y_{2}∥_{Y} = ∥UU^{−1}y_{1} − UU^{−1}y_{2}∥_{Y} ≤ ∥U∥∥x_{1} − x_{2}∥_{X}.

Thus, \frac{1}{k} ≤ ∥x_{1} − x_{2}∥_{X}, and by the definition of δ_{X},

∥\frac{1}{2}(x_{1} + x_{2})∥ ≤ 1 − δ_{X} (\frac{1}{k} ) .

Therefore

∥\frac{1}{2}(y_{1} + y_{2})∥ ≤ ∥U∥ ∥\frac{1}{2}(x_{1} + x_{2})∥ ≤ ∥U∥ (1 − δ_{X} (\frac{1}{k} )) < k (1 − δ_{X} (\frac{1}{k} )) = 1.

This implies that

δ_{Y} (\frac{∥U∥}{k}) ≥ 1 − ∥U∥ (1 − δ_{X} (\frac{1}{k} )) > 0.

Thus ε_{0}(Y) ≤ \frac{∥U∥}{k} < 1 and the proof is complete. ■

Although both Corollary 1.2 and Theorem 1.3 yield the same stability bound for the WFPP, we can see that they are results of different kind and that the second one is stronger.

The aim of this chapter is to give the best known stability bounds for the WFPP. Nevertheless, as the above results illustrates, one can find in the literature two main types of stability theorems, depending on one or other of the following schemes.

A) If (X, ∥ · ∥) has property (P) and (Y, | · |) is a Banach space isomorphic to X, then (Y, |.|) has (P) whenever d(X, Y) < k(X).

B) If (X, ∥ · ∥) has property (P) and (Y, | · |) is a Banach space isomorphic to X, then (Y, | · |) has the WFPP whenever d(X, Y) < k(X).

Here (P) is a geometric property that implies the WFPP.

2. Stability of normal structure

Although weak normal structure (WNS) is the most fruitful sufficient condition for the WFPP, it is not a property that is easy to check. It is perhaps for this reason that many sufficient conditions for WNS have been studied. We examine in this section the stability of some of them.

2.1. Behaviour of Bynum’s coefficients ([By 80])

ΛStability, normal structure Bynum’s paper [By 80], p. 432 reads: When any of the normal structure coefficients is greater than one, this condition is contagious.

The following definitions are stated according to Bynum’s paper [By 80].

Definition 2.1 Let (x_{n}) be a bounded sequence in the Banach space (X, ∥ · ∥) . The real number

diam_{a} ((x_{n})) := lim_{n→∞} (sup ∥x_{i} − x_{j}∥ : i, j ≥ n)

ΛAsymptotic diameter of a sequence is called the asymptotic diameter of the sequence (x_{n}). In the same way, the real number

r_{a}((x_{n})) := inf\{lim sup ∥y − x_{n}∥ : y ∈ \overline{co}((x_{n}))\}

ΛAsymptotic radius of a sequence is called the asymptotic radius of the sequence (x_{n}).

The normal structure coefficient of X is defined by

N(X) := inf \{\frac{diam(C)}{r(C)} : C bounded, convex, C ⊂ X, diam(C) > 0\}

ΛNormal structure coefficient and we say that X has uniform normal structure (UNS) if N(X) > 1.

Definition 2.2 The weakly convergent sequence coefficient of X, denoted by W CS(X) is defined by

W CS(X) := sup \{M : ∀(x_{n}) ∈ WCS(X) ∃y ∈ \overline{co}\{x_{n}\} s.t. M lim sup_{n} ∥x_{n} − y∥ ≤ diam_{a} ((x_{n}))\} .

ΛWeakly convergent sequence coefficient Here WCS(X) is the set of all the weakly convergent sequences in X.

Bynum’s theorems below seem to be the oldest results on stability of normal structure.

The normal structure coefficient of X : N(X).

Theorem 2.3 For X, Y isomorphic Banach spaces, N(X) ≤ d(X, Y)N(Y).

Corollary 2.4 If N(X) > 1 and d(X, Y) < N(X) then Y has UNS.

Proof. From the above theorem

\frac{N(X)}{d(X, Y)} ≤ N(Y).

Thus, N(Y) > 1 and Y has uniform normal structure. ■

The weakly convergent sequence coefficient of X : W CS(X). Bynum’s weakly convergent sequence coefficient, W CS(X), is perhaps one of most widely studied in metric fixed point theory. In particular one can find in the literature many different equivalent definitions for it. Among others, we have the following.

Definition 2.5 For each bounded sequence (x_{n}) in the Banach space (X, ∥ · ∥) we define the following constants.

The diameter of (x_{n}):

diam((x_{n})) := sup{∥x_{m} − x_{n}∥ : m, n ∈ ℕ}.

The separation of (x_{n}) :

sep((x_{n})) := inf{∥x_{m} − x_{n}∥ : m ≠ n}.

We will use the following notation

D[(x_{n})] := lim sup_{m} [lim sup_{n}{∥x_{m} − x_{n}∥}] .

ΛD[(x_{n})] It is easy to see that, for each bounded sequence (x_{n}) in X,

r_{a}((x_{n})) ≤ D[(x_{n})] ≤ diam_{a}((x_{n})) ≤ diam((x_{n})).

We recall the following combinatorial result.

Ramsey’s Theorem Suppose n ∈ N and C is an infinite set. If P_{n}(C) is the collection of all the subsets of C with n elements, and

P_{n}(C) = A ∪ B,

then there exists an infinite subset D of C such that P_{n}(D) ⊂ A or P_{n}(D) ⊂ B.

Lemma 2.6 Let C a bounded infinite set and r ≥ 0. Then there exist a sequence (y_{n}) with infinite range in C so that either diam((y_{n})) ≤ r or sep((y_{n})) ≥ r.

Proof. We have

P_{2}(C) = A ∪ B

where

A := {{x, y} ∈ P_{2}(C) : d(x, y) ≤ r}

and

B := {{x, y} ∈ P_{2}(C) : d(x, y) ≥ r} .

Using Ramsey’s theorem we obtain an infinite subset D of C such that P_{2}(D) ⊂ A or P_{2}(D) ⊂ B. We can define in D an infinite sequence (y_{n}). If P_{2}(D) ⊂ A then diam((y_{n})) ≤ r. If P_{2}(D) ⊂ B then sep((y_{n}) ≥ r. ■

Because of the indefiniteness of y in the expression which defines W CS(X) it is convenient to get easier equivalent formulae for this coefficient. Among others, we have the following.

Lemma 2.7 Let (X, ∥ · ∥) be a non-Schur Banach space. The following constants are equal.

a) W CS(X).

b) A(X):= inf \{\frac{diam_{a}((x_{n}))}{r_{a}((x_{n}))} : (x_{n}) is a weakly (non strongly) convergent seq. in X\} .

c) β(X) := inf{D[(x_{n})] : x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1}.

d) B(X) := sup{M > 0 : ∀(x_{n}), with x_{n} ⇀^{w} 0, M lim sup ∥x_{n}∥ ≤ diam_{a}((x_{n}))}.

e) C(X) := inf{diam_{a}((x_{n})) : x_{n} ⇀^{w} 0, ∥x_{n}∥ → 1}.

Proof. Let

M_{0} ∈ \{M : ∀(x_{n}) ∈ WCS(X) ∃y ∈ \overline{co}\{x_{n}\} s.t. M lim sup_{n} ∥x_{n} − y∥ ≤ diam_{a} ((x_{n}))\}

where WCS(X) is the set of all the weakly convergent sequences in X. Let WCS^{∗}(X) be the set of all the weakly convergent but not norm convergent sequences in X.

In particular, for each (x_{n}) ∈ WCS^{∗}(X) there exists y ∈ \overline{co}\{x_{n}\} such that

M_{0} ≤ \frac{diam_{a} ((x_{n}))}{lim sup_{n} ∥x_{n} − y∥} ≤ \frac{diam_{a} ((x_{n}))}{r_{a}((x_{n}))}.

Thus, taking the infimum on the right hand side of the above inequality,

M_{0} ≤ A(X) := inf \{\frac{diam_{a} ((x_{n}))}{r_{a}((x_{n}))} : (x_{n}) ∈ WCS^{∗}(X)\} .

Then it follows that,

W CS(X) ≤ A(X).

On the other hand, fix λ > 1. Then for all (x_{n}) ∈ WCS^{∗}, r_{a}((x_{n}))λ > r_{a}((x_{n})) and hence there exists y ∈ \overline{co}(\{x_{n}\}) such that

λr_{a}((x_{n})) > lim sup_{n} ∥x_{n} − y∥ ≥ r_{a}((x_{n})).

This gives

\frac{1}{λr_{a}((x_{n}))} < \frac{1}{lim sup_{n} ∥x_{n} − y∥}.

Hence

\frac{A(X)}{λ} ≤ diam_{a} ((x_{n}))λr_{a}((x_{n})) < \frac{diam_{a} ((x_{n}))}{lim sup_{n} ∥x_{n} − y∥},

which yields

\frac{A(X)}{λ} lim sup_{n} ∥x_{n} − y∥ ≤ diam_{a} ((x_{n})) .

In summary, for each (x_{n}) ∈ WCS^{∗}(X) there exists y ∈ \overline{co}((x_{n})) for which the above inequality holds. Of course, if (x_{n}) is a convergent sequence, then there exists y = lim_{n} x_{n} for which this inequality also holds. By the definition of W CS(X) this gives

\frac{A(X)}{λ} ≤ W CS(X).

Letting λ → 1^{+} we have A(X) ≤ W CS(X). We thus get A(X) = W CS(X). Let M_{0} be such that for all weakly null sequence (x_{n}),

M_{0} lim sup ∥x_{n}∥ ≤ diam_{a}((x_{n})).

In particular, if (x_{n}) is weakly null and ∥x_{n}∥ → 1, M_{0} ≤ diam_{a}((x_{n})). Therefore M_{0} ≤ C(X), and hence B(X) ≤ C(X).

On the other hand, let (x_{n}) be a weakly null sequence. If ρ := lim sup ∥x_{n}∥ > 0 then for a subsequence (x_{n_{k}}) of (x_{n})

\frac{1}{ρ}x_{n_{k}} ⇀^{w} 0 and ∥\frac{1}{ρ}x_{n_{k}}∥ → 1.

Then

C(X) ≤ diam_{a} ((\frac{1}{ρ}x_{n_{k}})) = \frac{1}{ρ} diam_{a}((x_{n_{k}})) ≤ \frac{1}{ρ} diam_{a}(x_{n}).

That is,

C(X)ρ = C(X) lim sup_{n} ∥x_{n}∥ ≤ diam_{a}((x_{n}))

which holds trivially if ρ = 0. This yields C(X) ≤ B(X) and thus we have shown that C(X) = B(X).

Since for each bounded sequence (x_{n}) in X, D[(x_{n})] ≤ diam_{a}((x_{n})), it is immediate that β(X) ≤ C(X).

Suppose, for a contradiction, that β(X) < C(X). Then there exists a sequence (x_{n}) with x_{n} ⇀^{w} 0 and ∥x_{n}∥ → 1, such that D[(x_{n})] < C(X) ≤ diam_{a}((x_{n})) ≤ diam((x_{n})).

Let r be such that D[(x_{n})] < r < C(X). As the set {x_{n} : n ∈ ℕ} is infinite (since x_{n} ⇀^{w} 0 and ∥x_{n}∥ → 1), then we can apply Lemma 2.6 to this set . Hence there exists a subsequence (y_{n}) of (x_{n}) such that either diam((y_{n})) ≤ r or sep((y_{n})) ≥ r. But y_{n} ⇀^{w} 0 and ∥y_{n}∥ → 1. Hence r < C(X) ≤ diam_{a}((y_{n})) ≤ diam((y_{n})), and then diam((y_{n})) ≤ r which is absurd. Thus we have

sep((y_{n})) ≥ r > D[(x_{n})] ≥ D[(y_{n})],

which is absurd again. Hence we have that β(X) = C(X).

Finally we will show that W CS(X) = B(X). Let M be a constant such that for all weakly null sequences (x_{n}), M lim sup ∥x_{n}∥ ≤ diam_{a}((x_{n})). Let (y_{n}) be a sequence which converges weakly to y ∈ X. Thus (y_{n} − y) is weakly null and hence

M lim sup ∥y_{n} − y∥ ≤ diam_{a}((y_{n} − y)) = diam_{a}((y_{n})).

As y ∈ \overline{co}(\{y_{n} : n ∈ ℕ\}, M ≤ W CS(X). Thus, B(X) ≤ W CS(X).

On the other hand, let

M_{0} ∈ \{M : ∀(x_{n}) ∈ WCS(X) ∃y ∈ \overline{co}\{x_{n}\} s.t. M lim sup_{n} ∥x_{n} − y∥ ≤ diam_{a} ((x_{n}))\} .

Let (y_{n}) be a weakly null sequence in X, and define

A_{k} := \overline{co}\{y_{n} : n ≥ k\}.

An easy application of Mazur’s theorem (see [A-D-L 97]) gives

⋂_{k=1}^{∞} A_{k} = {0}.

The function Φ : X → ℝ defined by Φ(z) := lim sup_{n} ∥z − y_{n}∥ is norm-continuous and convex. Hence it is weak sequentially lower semicontinuous. Since each A_{k} is weakly compact, for each positive integer k there exists z_{k} ∈ A_{k} such that

Φ(z_{k}) = inf{Φ(z) : z ∈ A_{k}}.

As \{z_{k} : k ∈ ℕ\} ⊂ \overline{co}\{y_{n} : n ∈ ℕ\} and \overline{co}\{y_{n} : n ∈ ℕ\} is weakly compact, (z_{k}) admits a subsequence (z_{n_{p}}) such that z_{n_{p}} ⇀^{w} z ∈ \overline{co}\{y_{n} : n ∈ ℕ\}. Bearing in mind that the sequence of sets (A_{k}) is decreasing, as well as Lemma 1.9. of [A-D-L 97],

\{z\} = ⋂_{r≥1} \overline{co}\{z_{k_{p}} : p ≥ r\} ⊂ ⋂_{r≥1} A_{k_{r}} = ⋂_{k∈ℕ}A_{k} = \{0\}.

As 0 ∈ A_{k} for each positive integer k, and the sequence (Φ(z_{k})) is nondecreasing because A_{k} ⊃ A_{k+1}, there exists

lim_{k} Φ(z_{k}) ≤ Φ(0) ≤ lim_{k} Φ(z_{k}).

For each positive integer k let us consider the sequence (y_{n+k})_{n}. By definition of M_{0}, there exists y ∈ \overline{co}\{y_{n+k} : n ≥ 1\} such that

M_{0} lim sup_{n} ∥y − y_{n+k}∥ ≤ diam_{a}((y_{n+k})_{n}).

Thus,

M_{0}Φ(z_{k}) ≤ M_{0} lim sup_{n} ∥y − y_{n+k}∥ ≤ diam_{a}((y_{n+k})_{n}) = diam_{a}((y_{n})).

Letting k → ∞,

M_{0}Φ(0) ≤ diam_{a}((y_{n})).

That is

M_{0} lim sup_{n} ∥y_{n}∥ ≤ diam_{a}((y_{n})).

It follows that

W CS(X) lim sup_{n} ∥y_{n}∥ ≤ diam_{a}((y_{n})),

which yields W CS(X) ≤ B(X), and the proof is complete. ■

The coefficient β(X) was defined by Jiménez-Melado in [J 92] twelve years after Bynum’s paper was published. In addition, he defined a sufficient condition for weak normal structure as follows.

Definition 2.8 A Banach space (X, ∥ · ∥) has the generalized Gossez-Lami Dozo property (GGLD) whenever D[(x_{n})] > 1 for every weakly null sequence (x_{n}) such that lim ∥x_{n}∥ = 1.

ΛGeneralized Gossez-Lami Dozo property Of course, β(X) > 1 ⇒ X has the GGLD property.

One of the most important results about stability of weak normal structure is the following one.

Theorem 2.9 [By 80] For X, Y isomorphic Banach spaces,

W CS(X) ≤ d(X, Y) W CS(Y)

or equivalently

β(X) ≤ d(X, Y) β(Y).

Proof. Let U : Y → X be a bicontinuous isomorphism. Given any weakly null sequence (y_{n}) in Y with ∥y_{n}∥ → 1, there exists a subsequence (y_{n_{k}}) of (y_{n}) such that

λ := lim sup_{n} ∥U(y_{n})∥ = lim_{k} ∥U(y_{n_{k}})∥ ≤ ∥U∥.

As

∥y_{n}∥ = ∥U^{−1}U(y_{n})∥ ≤ ∥U^{−1}∥∥U(y_{n})∥,

letting n → ∞ we have

1 ≤ ∥U^{−1}∥λ.

In particular λ > 0.

The sequence (\frac{1}{λ}U(y_{n_{k}}))_{k≥0} is weakly null in X and ∥\frac{1}{λ}U(y_{n_{k}})∥ → 1. Thus, by definition of β(X), we have:

β(X) ≤ D [(\frac{1}{λ}U(y_{n_{k}}))_{k≥0}]

= lim sup_{p} [lim sup_{q} ∥\frac{1}{λ}U(y_{n_{p}}) − \frac{1}{λ}U(y_{n_{q}})∥]

≤ lim sup_{p} [lim sup_{n} ∥\frac{1}{λ}U(y_{n_{p}}) − \frac{1}{λ}U(y_{n})∥]

≤ lim sup_{m} [lim sup_{n} ∥\frac{1}{λ}U(y_{m}) − \frac{1}{λ}U(y_{n})∥]

≤ \frac{∥U∥}{λ} lim sup_{m} [lim sup_{n} ∥y_{m} − y_{n}∥]

= \frac{∥U∥}{λ}D[(y_{n})]

≤ ∥U∥∥U^{−1}∥D[(y_{n})].

As the inequality β(X) ≤ ∥U∥∥U^{−1}∥D[(y_{n})] holds for each weakly null sequence (y_{n}) in Y with ∥y_{n}∥ → 1, it is immediate that

β(X) ≤ ∥U∥∥U^{−1}∥β(Y).

Finally, as the above inequality holds for each bicontinuous isomorphism U : Y → X,

β(X) ≤ d(X, Y)β(Y)

which completes the proof. ■

Corollary 2.10 [By 80], [J 92] If W CS(X) > 1 and d(X, Y) < W CS(X) then Y has the GGLD property, and hence WNS and the WFPP.

Proof. By the above theorem we have that

\frac{W CS(X)}{d(X, Y)} ≤ W CS(Y) .

Thus, W CS(Y) > 1 and Y has weak normal structure. ■

We present now a well known computation of weakly convergent sequence coefficient for X = ℓ_{p} that yields stability results.

Corollary 2.11 Let 1 < p < ∞. If Y is a Banach space such that d(ℓ_{p}, Y) < 2^{\frac{1}{p}} then Y has the GGLD property and hence NS and the FPP.

Proof. Let (x_{n}) be a weakly null sequence in ℓ_{p}. Given x ∈ ℓ_{p} it is well know that

lim sup ∥x_{n} − x∥_{p}^{p} = ∥x∥_{p}^{p} + lim sup ∥x_{n}∥_{p}^{p}.

Thus, for each positive integer m,

lim sup_{n} ∥x_{n} − x_{m}∥_{p}^{p} = ∥x_{m}∥_{p}^{p} + lim sup_{n} ∥x_{n}∥_{p}^{p}.

If, in particular, ∥x_{n}∥_{p} → 1, then

lim sup_{m} (lim sup_{n} ∥x_{n} − x_{m}∥_{p}^{p})) = 2,

that is, D[(x_{n})] = 2^{\frac{1}{p}}. Hence W CS(ℓ_{p}) = β(ℓ_{p}) = 2^{\frac{1}{p}}. ■

The coefficient W CS(X) has been computed for many other Banach spaces. The reader is referred to the book [A-D-L 97] for a comprehensive study. We list some of these computations.

1 For x ∈ ℓ_{p}, 1 ≤ p < ∞, we denote by x^{+} and x^{−} the vectors whose i component are respectively

x^{+}(i) := max\{x(i), 0\} = \frac{x(i) + |x(i)|}{2}, x^{−}(i) := max\{−x(i), 0\} = \frac{−x(i) + |x(i)|}{2}.

For any q ∈ [1, ∞) and for x ∈ ℓ_{p} we denote

∥x∥_{p,q} := (∥x^{+}∥_{p}^{q} + ∥x^{−}∥_{p}^{q})^{\frac{1}{q}}
∥x∥_{p,∞} := max{∥x^{+}∥_{p}, ∥x^{−}∥_{p}}.

It is easy to check that all these norms are equivalent to the usual norm in ℓ_{p}. The Banach spaces ℓ_{p,q} := (ℓ_{p}, ∥.∥_{p,q}) were introduced by Bynum. In [D-L-X 96] the authors showed that for p ∈ (1, ∞) and q ∈ [1, ∞),

W CS(ℓ_{p,q}) = min\{2^{\frac{1}{p}} , 2^{\frac{1}{q}} \}.

2 For β > 1 the Banach space E_{β}, introduced by James, is ℓ_{2} renormed accordingly to |x|_{β} = max{∥x∥_{2}, β∥x∥_{∞}}. It was shown in [A-X 93] that

W CS(E_{β}) = \{ \begin{matrix}\frac{\sqrt{2}}{β} & 1 ≤ β\sqrt{2} \\ 1 & \sqrt{2} < β < ∞. \end{matrix}

3 Let J be the real James space which consists of all real sequences x = (x_{n}) for which lim x_{n} = 0 and ∥x∥ < ∞, where

∥x∥ := sup \{[(x_{p_{1}} − x_{p_{2}})^{2} + … + (x_{p_{m−1}} − x_{p_{m}})^{2} + (x_{p_{m}} − x_{p_{1}})^{2}]^{\frac{1}{2}} \}

and the supremum is taken over all choices of m and p_{1} < p_{2} < … < p_{m}.

Other equivalent norms on J are

∥x∥_{1} := sup_{p_{1}<...<p_{m}^{m}} [∑_{j=2}^{m}(x_{p_{j}−1} − x_{p_{j}})^{2}]^{\frac{1}{2}}

and

∥x∥_{2} := sup_{p_{1}<...<p_{2^{m}}^{m}} [∑_{j=2}^{m}(x_{p_{2^{j−1}}} − x_{p_{2^{j}}})^{2}]^{\frac{1}{2}} .

In [J 92] Jiménez-Melado showed that W CS(J) = \sqrt{\frac{3}{2}}. Later on, Domínguez

and Xu ([D-X 95]) showed that W CS((J, ∥.∥_{2})) = W CS((J, ∥.∥_{1})) = \sqrt{2}. We will write J_{i} for the space (J, ∥·∥_{i}) (i = 1, 2).

4 Let (Ω, Σ, μ) be a σ-finite measure space, 1 ≤ p < ∞, and assume that L^{p}(Ω) is infinite dimensional. Then

N(L^{p}(Ω)) = min \{2^{1− \frac{1}{p}} , 2^{\frac{1}{p}} \} .

Moreover W CS(L^{p}(Ω)) = N(L^{p}(Ω)) if either p ≥ 2 or μ is not purely atomic. (See [A-D-L 97]).

As a direct consequence of the computations of W CS(X) we have the following stability result.

Corollary 2.12 a) Let 1 < p < ∞. If Y is a Banach space such that

d (ℓ_{p,q}, Y) < min \{2^{\frac{1}{p}} , 2^{\frac{1}{q}} \}

then Y has the GGLD property and hence NS and the FPP.

b) Let 1 < β < \sqrt{2}. If Y is a Banach space such that d(E_{β}, Y) < \sqrt{2}/β then Y has the GGLD property and hence NS and the FPP.

c) If d(Y, J) < \sqrt{3/2} then Y has property GGLD and hence WNS and the WFPP.

d) If d(Y, L^{p}(Ω)) < min \{2^{1− \frac{1}{p}} , 2^{\frac{1}{p}} \} then Y has property NS and hence the FPP.

2.2. Further remarks about the stability of WNS

Next we list some stability theorems for some other conditions which are sufficient for normal structure.

Stability of condition ε_{1}(X) < 1 [S 86], [G-K 90] . ΛStability, ε_{1}(X) < 1

Theorem 2.13 Let X be a Banach space and let X_{1} := (X, ∥.∥_{1}) and X_{2} := (X, ∥.|_{2}) where ∥.∥_{1} and ∥.∥_{2} are equivalent norms on X where there exists α, β > 0 such that,

α∥x∥_{1} ≤ ∥x∥_{2} ≤ β∥x∥_{1}

for all x ∈ X. If k := \frac{β}{α} then

Δ_{X_{2}}(ε) ≥ 1 − k (1 − Δ_{X_{1}} (\frac{ε}{k} )) .

Consequently, if ε_{1}(X_{1}) < 1 and

k (1 − Δ_{X_{1}} (\frac{1}{k} )) < 1

then ε_{1}(X_{2}) < 1.

Stability of condition ε_{χ}(X) < \frac{1}{2} [Ba 87] . ΛStability, ε_{χ}(X) < \frac{1}{2}

Theorem 2.14 Let X be a Banach space with ε_{χ}(X) < \frac{1}{2}. Let B > 1 satisfy

1 − \frac{1}{B} = Δ_{χ} (\frac{1}{2B} ) .

(which exists in view of continuity of the function Δ_{χ}) If Y is another Banach space with d(X, Y) < B then Δ_{χ}(Y) < 1/2.

Bernal-Sullivan convexity property. ΛBernal-Sullivan convexity property

Theorem 2.15 Let (H, ∥.∥) be a Hilbert space, and let | · | be a norm on H such that, for all x ∈ H,

\frac{1}{β} |x| ≤ ∥x∥ ≤ |x|

for some β with 1 ≤ β < \sqrt{2}. Given ε > 0, there exists δ > 0 and M, a positive integer, such that for m ≥ M, if x_{1}, …, x_{m} ∈ B_{(H,|.|)} and

|\frac{1}{m}(x_{1} + … + x_{n})|^{2} > 1 − δ,

then D(| · |, x_{1}, …, x_{m}) < ε. Here

D(| · |, x_{1}, …, x_{k+1}) = |x_{k} − x_{k−1}|. dist (x_{k−1}, [x_{k}, x_{k+1}]). . . . . dist (x_{1}, [x_{2}, …, x_{k+1}]).

where [x_{i}, …, x_{j}] is the affine span of the vectors {x_{i}, …, x_{j}}.

Corollary 2.16 Let (H, ∥.∥) be a Hilbert space, and let | · | be a norm on H such that, for all x ∈ H,

\frac{1}{β} |x| ≤ ∥x∥ ≤ |x|

for some β with 1 ≤ β < \sqrt{2}. Then (H, | · |) has normal structure.

3. Stability for weakly orthogonal Banach lattices

ΛStability, weakly orthogonal Banach lattices In 1980-81 Maurey [M 80] showed that the classical Banach space c_{0} enjoys the WFPP. In order to generalize Maurey’s ideas to a larger class of Banach lattices, Borwein and Sims [B-S 84] introduced the notion of a weakly orthogonal Banach lattice as well as the concept of Riesz angle.

Definition 3.1 A subset C of a Banach lattice X will be called weakly orthogonal if

lim inf_{n} lim inf_{m} ∥|x_{n} − x_{0}| ∧ |x_{m} − x_{0}|∥ = 0

whenever (x_{n}) is a sequence in C, weakly convergent to x_{0} ∈ C. A Banach lattice X is said to be weakly orthogonal whenever all its weakly compact subsets are weakly orthogonal.

ΛWeakly orthogonal It is not hard to show that c_{0}(S), c(S), ℓ_{p}(S) (1 ≤ p < ∞) are weakly orthogonal while l_{∞}(ℕ) and non-atomic spaces are not.

Definition 3.2 The Riesz angle of a Banach lattice X is defined by

a(X) := sup{∥|x| ∨ |y|∥ : x, y ∈ B_{X}}.

ΛRiesz angle The fundamental inequality involving the Riesz angle is:

Proposition 3.3 Let (X, ∥ · ∥) a Banach lattice and consider x, y, z ∈ X. Then

∥z∥ ≤ a(X)(∥x − z∥ ∨ ∥z − y∥) + ∥|x| ∧ |y|∥.

Proof. By the definition of Riesz angle it is clear that

a(X) ≥ ∥|\frac{x − z}{∥x − z∥ ∨ ∥y − z∥}| ∨ |\frac{y − z}{∥x − z∥ ∨ ∥y − z∥}|∥ = \frac{∥|x − z| ∨ |y − z|∥}{∥x − z∥ ∨ ∥y − z∥}.

Therefore,

∥|x − z| ∨ |y − z|∥ ≤ a(X)(∥x − z∥ ∨ ∥y − z∥).  (3.1)

On the other hand, since

|z| ≤ (|x − z| ∨ |y − z| + |x|) ∧ (|x − z| ∨ |y − z| + |y|)

we have

|z| ≤ ||x − z| ∨ |y − z| + |x| ∧ |y|| .  (3.2)

Since X is Banach lattice from (3.2) we obtain that

∥z∥ ≤ ∥|x − z| ∨ |y − z|∥ + ∥|x| ∧ |y|∥.  (3.3)

Finally by (3.1) and (3.2) we conclude

∥z∥ ≤ a(X)(∥x − z∥ ∨ ∥z − y∥) + ∥|x| ∧ |y|∥.
■

By using these notions they obtained the following stability result.

Theorem 3.4 [B-S 84] If X is a weakly orthogonal Banach lattice such that

d(Y, X)a(X) < 2

then Y has the WFPP.

Proof. By standard procedures we can suppose, to get a contradiction, that T is a nonexpansive mapping on a minimal invariant weakly compact subset C of X such that diam(C) = 1, and we can select a weakly null approximately fixed point sequence {x_{n}}.

Since d(Y, X)a(X) < 2 we can pick an isomorphism U of Y onto X with

∥U∥U^{−1}∥a(X) < 2.

Since X is weakly orthogonal, we can find subsequences {a_{n}} and {y_{n}} of {x_{n}} such that

lim_{n→∞} ∥|Ua_{n}| ∧ |Uy_{n}|∥ = 0.

Since any approximate fixed point sequence is diameterizing we can also assume that

lim_{n→∞} ∥a_{n} − y_{n}∥ = 1.

On the other hand, it is well known that given {a_{n}} and {y_{n}} we can consider another approximate fixed point sequence {z_{n}} satisfying

lim_{n→∞} ∥a_{n} − z_{n}∥ = lim_{n→∞} ∥z_{n} − y_{n}∥ = \frac{1}{2}.

Now by using the above proposition, we have

lim sup_{n} ∥Uz_{n}∥ ≤ a(X) lim sup_{n}(∥Ua_{n} − Uz_{n}∥ ∨ ∥Uy_{n} − Uz_{n}∥) + lim sup_{n}(∥|Ua_{n}| ∧ |Uy_{n}|∥).

Therefore,

1 = lim sup_{n} ∥U^{−1}Uz_{n}∥ ≤ ∥U^{−1}∥ lim sup_{n} ∥Uz_{n}∥ ≤ ∥U^{−1}∥∥U∥a(X)\frac{1}{2} < 1,

which is a contradiction. ■

As a consequence of the above theorem it is clear that every weakly orthogonal Banach lattice (X, ∥ · ∥) with Riesz angle a(X) < 2 has the WFPP. However, this result does not work when the Riesz angle is equal to 2. To avoid this difficulty in 1988 Sims [Si 88] introduced a strong version of weakly orthogonal Banach lattices as follows.

Definition 3.5 Let (X, ∥ · ∥) a Banach lattice. We say that X is strong weakly orthogonal if

lim_{n→∞} ∥|x_{n}| ∧ |x|∥ = 0

for all x ∈ X, whenever {x_{n}} is a weakly null sequence.

ΛStrong weakly orthogonal For this class of Banach lattices, it was shown in [Si 88] that

Theorem 3.6 a) If (X, ∥.∥) is a strongly weak orthogonal Banach lattice then (X, ∥·∥) has the WFPP.

b) If (X, ∥ · ∥) is a strongly weak orthogonal Banach lattice then (X, ∥ · ∥) has the non-strict Opial property.

Recall that a Banach space (X, ∥ · ∥) has the non-strict Opial property if for every weakly null sequence (x_{n}) in X the following inequality holds

lim inf ∥x_{n}∥ ≤ lim inf ∥x_{n} − x∥

for every x ∈ X.

It is not difficult to see that the Banach lattices c_{0}(S) and ℓ_{p}(S) (1 ≤ p < ∞) are strongly weak orthogonal. Since it is well known that c(S) does not enjoys the non-strict Opial property, then c(S) cannot be a strongly weak orthogonal Banach lattice.

Stability results for the strong weakly orthogonality have been found by Sims [Si 86], Khamsi and Turpin [Kh-T 89] (for other topologies besides the weak topology) and Dalby [Da 97]. The ‘radius of stability’ in each case was, respectively, \sqrt{5} − 1, 4/3, (\sqrt{33} − 3)/2. In this context the following observation due to Garcia-Falset [GF 94] is of interest.

Let (X, ∥ · ∥) a Banach space. In [GF 94] the author defines the coefficient

R(X) := sup{lim inf ∥x_{n} + x∥}

where the supremum is taken over all x ∈ B_{X} and all weakly null sequences in B_{X}. ΛR(X)

Proposition 3.7 If (X, ∥·∥) is a strong weakly orthogonal Banach lattice, then R(X) ≤ a(X).

Proof. Let {x_{n}} be a weakly null sequence in B_{X} and let x ∈ B_{X}. It is well known that

|x_{n}| + |x| = |x_{n}| ∨ |x| + |x_{n}| ∧ |x|.

Then

∥|x_{n}| + |x|∥ ≤ ∥|x_{n}| ∨ |x|∥ + ∥|x_{n}| ∧ |x|∥.

Therefore, by the definition of the Riesz angle

∥|x_{n}| + |x|∥ ≤ a(X) + ∥|x_{n}| ∧ |x|∥.

Since X is a strong weakly orthogonal Banach lattice, we have

lim inf_{n} ∥|x_{n}| + |x|∥ ≤ a(X),

and hence,

lim inf_{n} ∥x_{n} + x∥ ≤ lim inf_{n} ∥|x_{n} + x|∥ ≤ lim inf_{n} ∥|x_{n}| + |x|∥ ≤ a(X).

This implies that R(X) ≤ a(X). ■

Later, in [GF 97] the author showed that a Banach space (X, ∥ · ∥) has the WFPP whenever R(X) < 2 and an easy computation allows us to see that if (X, ∥ · ∥) and (Y, | . |) are isomorphic Banach spaces then R(Y) ≤ d(X, Y)R(X). Thus, in order to obtain stability results for weakly orthogonal Banach lattices (X, ∥ · ∥) it is clear from the above proposition that computing the coefficient R(X) is better than a computation of the Riesz angle. Moreover, the Banach space ℓ_{p,∞}, which is ℓ_{p} renormed by |x| := max{∥x^{+}∥_{p}, ∥x^{−}∥_{p}}, does not have asymptotically normal structure [By 80] and it is not a Banach lattice. However, we have

Proposition 3.8 [GF 94] A Banach space (X, ∥ · ∥) has the WFPP whenever there exists 1 < p < ∞ such that d(X, ℓ_{p,∞}) < 2^{1− \frac{1}{p}}.

Proof. It is enough to show that R(ℓ_{p,∞}) < 2^{\frac{1}{p}} .

Let (x_{n}) be a weakly null sequence in the unit ball of ℓ_{p,∞} and let x be an element in the unit ball. Then we have

|x_{n} + x| = max {∥(x_{n} + x)^{+}∥_{p}, ∥(x_{n} + x)^{−}∥_{p}} .

Since (x_{n}) is a weakly null sequence there exists a subsequence (x_{n_{k}}) of (x_{n}) and a sequence (P_{k}) of projections of the form P_{k} = P_{[a_{k},b_{k}]}, where

[a_{k}, b_{k}] := {s ∈ ℕ : a_{k} ≤ s ≤ b_{k}}

and lim_{k→∞} a_{k} = ∞, such that

lim_{k→∞} ∥x_{n_{k}} − P_{k}x_{n_{k}}∥_{p} = 0  (3.4)

and

lim_{k→∞} |x_{n_{k}} − P_{k}x_{n_{k}}| = 0.  (3.5)

Moreover, we can assume that x has finite support, that is x = P_{[a,b]}x. Therefore, there exists k_{0} ∈ ℕ such that a_{k} > b for all k ≥ k_{0}.

Consequently, for each k ≥ k_{0} we have

∥(P_{k}(x_{n_{k}} + x)^{+}∥_{p} = ∥P_{k}x_{n_{k}}^{+} + x^{+}∥_{p}  (3.6)

and

∥(P_{k}x_{n_{k}} + x)^{−}∥_{p} = ∥P_{k}x_{n_{k}}^{−} + x^{−}∥_{p} .  (3.7)

On the other hand, by (3.5) we have

lim inf_{n→∞} |x_{n} + x| ≤ lim inf_{k→∞} |x_{n_{k}} + x| ≤ lim inf_{k→∞} |P_{k}x_{n_{k}} + x|  (3.8)

and then by using (3.4), (3.6), (3.7) and (3.8) we obtain

lim inf_{n→∞} |x_{n} + x| ≤ lim inf_{k→∞} |P_{k}x_{n_{k}} + x| ≤ max{lim inf_{k→∞} ∥P_{k}x_{n_{k}}^{+} + x^{+}∥_{p} ,

lim inf_{k→∞} ∥P_{k}x_{n_{k}}^{−} + x^{−}∥_{p} ≤ 2^{\frac{1}{p}} .

Since (x_{n}) and x are arbitrary, we get that R(ℓ_{p,∞}) ≤ 2^{\frac{1}{p}} . ■

On the other hand, if (X, ∥ · ∥) is a weakly orthogonal Banach lattice which is not strong weakly orthogonal, the coefficient R(X) can be bigger than the Riesz angle of (X, ∥ · ∥) . This is the case for the classical space (c, ∥.∥_{∞}). Indeed,

Proposition 3.9 R(c, ∥.∥_{∞}) = 2 and a(c, ∥.∥_{∞}) = 1.

Proof. First, let us see that a(c) = 1. Indeed, given x, y ∈ c with ∥x∥_{∞} ≤ 1 and ∥y∥_{∞} ≤ 1. It is clear that there exists n ∈ ℕ such that

∥|x| ∨ |y|∥_{∞} = |x_{n}| ∨ |y_{n}|

but, we know that

|x_{n}| ∨ |y_{n}| ≤ ∥x∥_{∞} ∨ ∥y∥_{∞}.

Consequently a(c) = 1.

On the other hand, if we consider the sequence (e_{n}) where e_{n} = (δ_{i,n}) and the element e = (1, 1, 1, ....) it is easy to see that ∥e_{n} + e∥_{∞} = 2 and therefore R(c) = 2. ■

As a consequence of both Borwein-Sims’s result and the above comment we have the following stability corollaries.

Corollary 3.10 [B-S 84] If X is a Banach space such that

d(X, ℓ_{p})) < 2^{1− \frac{1}{p}}

for some 1 < p < ∞, then X has the WFPP.

Corollary 3.11 [B-S 84] If X is a Banach space such that

d(X, c(S))) < 2 or d(X, c_{0}(S))) < 2

then X has the WFPP.

4. Stability of the property M(X) > 1

ΛStability, M(X) Inspired by the proof of [GF 97], Domínguez-Benavides [D 96] in 1996 improved the stability constant 2R(X)^{−1}. In order to do this, he stated the following definitions.

Definition 4.1 [D 96] Let (X, ∥·∥) be a Banach space. We define the coefficient M(X) as

sup \{\frac{1 + a}{R(a, X)} : a ≥ 0\}

where for a nonnegative real number a,

R(a, X) := sup { lim inf ∥x + x_{n}∥} ,

where the supremum is taken over all x ∈ X with ∥x∥ ≤ a and all weakly null sequences (x_{n}) in the unit ball of X such that D[(x_{n})] ≤ 1 .

It is shown in [D 96] that a Banach space Y has the WFPP if there exists a Banach space X such that

d(X, Y) < M(X) = sup \{\frac{1 + a}{R(a, X)} : a ≥ 0\} .

It turns out that for 1 < p < ∞,

M(ℓ_{p}) = (1 + 2^{\frac{1}{p−1}} )^{\frac{p−1}{p}}

and that for a Hilbert space H, M(H) = \sqrt{3}. Hence, the result of [D 96] improves all previously known stability results in Hilbert spaces and ℓ_{p}, 1 < p < ∞. See [B-S 84], [By 80], [Pr 90], [J-L 92], [Kh 94], [J-L 93].

In this section we improve the stability constant M(X) for a certain class of Banach spaces

Recall that a Banach space X has the uniform Opial property [Pr 92] if for every c > 0 there exists r = r(c) > 0 such that 1 + r ≤ lim inf ∥x + x_{n}∥ for every x ∈ X with ∥x∥ ≥ c and every weakly null sequence (x_{n}) in X with lim inf ∥x_{n}∥ ≥ 1.

To any Banach space X we can associate its Opial’s modulus [L-T-X 95] by

r_{X}(c) = inf {lim inf ∥x + x_{n}∥ − 1}

where c ≥ 0 and the infimum is taken over all x ∈ X with ∥x∥ ≥ c and all weakly null sequences (x_{n}) in X with lim inf ∥x_{n}∥ ≥ 1.

In order to simplify the statement of the following theorem, for B ≥ 1 we define the number

C_{X}(B) := sup{c ≥ 0 : r_{X}(c) ≤ B − 1}.

The inequality (see [X 97])

c − 1 ≤ r_{X}(c) ≤ c

holds for all c ≥ 0, so C_{X}(B) ≥ B − 1.

Theorem 4.2 Let X be a Banach space with norm ∥ · ∥ and denote by r_{X} its Opial’s modulus. Let | · | be a norm on X equivalent to ∥ · ∥ and let B be a positive real number such that, for all x ∈ X,

∥x∥ ≤ |x| ≤ B∥x∥.

Let Y be the Banach space X with the new norm | · | and let C(B) = C_{X}(B). If B is a solution of the inequality

B < sup \{\frac{1 + a}{R (\frac{a}{B} C(B), X)} : a ≥ 0\}

then Y has the weak fixed point property.

To prove the above theorem we need some notation and a preliminary lemma. Let (X, ∥ · ∥) be a real Banach space and let l_{∞}(X) denote the space

{(y_{n}) : y_{n} ∈ X and {∥y_{n}∥} ∈ l_{∞}}

with the norm ∥(y_{n})∥_{l_{∞}(X)} := sup_{n} ∥y_{n}∥ and let c_{0}(X) be the closed subspace of l_{∞}(X)

{(y_{n}) : y_{n} ∈ X and {∥y_{n}∥} ∈ c_{0}}.

Set

X̃ = l_{∞}(X)/c_{0}(X).

Thus for any [y_{n}] ∈ X̃,

∥[y_{n}]∥X̃ = lim sup_{n→∞} ∥y_{n}∥.

Finally, given a nonexpansive mapping T : K → K where K is a nonempty, convex weakly compact subset of X we consider

K̃ = {[y_{n}] ∈ X̃ : y_{n} ∈ K}.

Clearly, K̃ is a closed convex subset of X̃. Let T̃ : K̃ → K̃ be the mapping defined by

T̃[y_{n}] = [T y_{n}].

Since T is nonexpansive on K, T̃ is a well-defined nonexpansive mapping on K̃. Moreover, if (x_{n}) is an approximate fixed point sequence (a.f.p.s.) in K, then [x_{n}] is a fixed point of T̃.

Let (x_{n}) be any fixed a.f.p.s. of T. By passing to a weakly convergent subsequence of (x_{n}), and then translating K, we may assume that (x_{n}) converges to 0 weakly. For convenience, we assume that diam(K) = 1 and we denote the fixed point [x_{n}] by x̃. For any 0 < t < 1 define, in X̃,

W_{t}^{1} = B_{(1−t)}[x_{n}]
W_{t}^{2} = {[w_{n}] ∈ K̃ : lim sup_{n} lim sup_{m} ∥w_{m} − w_{n}∥ ≤ t}
W_{t}^{3} = {[w_{n}] ∈ K̃ : ∃x ∈ K with ∥[w_{n}] − x∥X̃ ≤ t} .

Each set is closed, convex and T̃ invariant, so

W_{t} = W_{t}^{1} ∩ W_{t}^{2} ∩ W_{t}^{3}

is closed, convex and T̃ invariant and non-empty as tx̃ ∈ W_{t}.

Lemma 4.3 [L 99] For 0 < t < 1, let [w_{n}] be an element in W̃_{t}.

(a) lim_{n→∞} ∥x_{n} − w_{n}∥ = 1 − t.

(b) There is x ∈ K such that lim_{n→∞} ∥w_{n} − x∥ = t.

(c) For any x ∈ K, lim inf_{n→∞} ∥w_{n} − x∥ ≥ t.

(d) lim sup_{m→∞} lim sup_{n→∞} ∥w_{n} − w_{m}∥ ≤ t. Moreover, if (w_{n_{k}}) converges weakly to w ∈ K, then lim_{n→∞} ∥w_{n} − w∥ = t and lim sup_{k→∞} ∥x_{n_{k}} − w_{n_{k}} + w∥ ≥ 1 − t.

Proof of Theorem 4.2. First, let us observe that we can suppose that there exists a > 0 such that

\frac{1 + a}{R (\frac{a}{B} C(B), X)} > B .

Suppose, for a contradiction, that Y lacks the WFPP. Then let t = 1/(1 + a) and consider the corresponding set W̃_{t} as in Lemma 4.3. From Lin’s lemma [L 85], W̃_{t} contains elements of norm arbitrarily close to 1.

Let [z_{n}] be any element of W̃_{t}. Since K is weakly compact, there exists a subsequence (z_{n_{k}}) of (z_{n}) which converges weakly to y ∈ K. By Lemma 4.3, (d) we have that

lim_{n→∞} ∥z_{n} − y∥ ≤ t .

Again from Lemma 4.3 (d) and the relationship between the norms we get that

lim_{k→∞} ∥x_{n_{k}} − z_{n_{k}} + y∥ ≥ \frac{1}{B} lim sup_{k→∞} |x_{n_{k}} − z_{n_{k}} + y| ≥ \frac{1 − t}{B} .

Since t < 1, we can consider the sequence (u_{k}) defined by

u_{k} = \frac{B}{1 − t} (−x_{n_{k}} + z_{n_{k}} − y) .

Then (u_{k}) is weakly convergent to 0 and lim_{k→∞} ∥u_{k}∥ ≥ 1. By the definition of r_{X} we have that

lim inf_{k→∞} ∥u_{k} + \frac{B}{1 − t}y∥ ≥ 1 + r_{X} (\frac{B}{1 − t}∥y∥) .

On the other hand, since u_{k} + \frac{B}{1−t}y = (z_{n_{k}} − x_{n_{k}})\frac{B}{1−t}, we have

lim inf_{k→∞} ∥u_{k} + \frac{B}{1 − t}y∥ = \frac{B}{1 − t} lim_{k→∞} ∥z_{n_{k}} − x_{n_{k}}∥ ≤ \frac{B}{1 − t} lim sup_{k→∞} |z_{n_{k}} − x_{n_{k}}| ≤ B .

Hence, we have B − 1 ≥ r_{X}(\frac{B}{1−t}∥y∥) and then ∥y∥ ≤ \frac{1−t}{B}C(B).

Pick any ε > 0. Since lim_{k→∞} ∥z_{n_{k}} − y∥ ≤ t, there exists a subsequence (y_{k}) of (z_{n_{k}}) such that ∥y_{k} − y∥ < t + ε for every positive integer k.

On the other hand, we have that

D (\frac{y_{k} − y}{t + ε} ) = D (\frac{y_{k}}{t + ε}) ≤ D (\frac{z_{n}}{t + ε}) ≤ \frac{t}{t + ε} < 1

and also that

∥\frac{y}{t + ε}∥ ≤ \frac{1 − t}{t + ε} \frac{C(B)}{B} ≤ \frac{1 − t}{t} \frac{C(B)}{B} = a\frac{C(B)}{B} .

¿From the above facts we conclude that

lim_{k→∞} ∥\frac{y_{k}}{t + ε}∥ = lim_{k→∞} ∥\frac{y_{k} − y}{t + ε} + \frac{y}{t + ε}∥ ≤ R (a\frac{C(B)}{B}, X) .

and, since ε > 0 is arbitrary,

lim_{k→∞} ∥y_{k}∥ ≤ \frac{1}{1 + a}R (a\frac{C(B)}{B}, X) .

Finally, we get that

|[z_{n}]| = lim_{k→∞} |z_{n_{k}}| = lim_{k→∞} |y_{k}| ≤ \frac{B}{1 + a}R (a\frac{C(B)}{B}, X) < 1

which contradicts Lin’s lemma. ■

Corollary 4.4 Let X be a Hilbert space with norm ∥ · ∥. Suppose that | · | is a norm on X such that

∥x∥ ≤ |x| ≤ B∥x∥ for all x ∈ X.

If B < \sqrt{2 + \sqrt{2}}, then (X, | · |) has the WFPP.

Proof. Since ∥ · ∥ is an Euclidean norm, we have r_{X}(c) = \sqrt{1 + c^{2}} − 1 for all c ≥ 0 and C(B) = \sqrt{B^{2} − 1}. On the other hand,

R(a, X) = \sqrt{\frac{1}{2} + a^{2}}

for all a ≥ 0 and

R (a\frac{C(B)}{B}, X) = \sqrt{\frac{1}{2} + a^{2} \frac{B^{2} − 1}{B^{2}}}.

By the Theorem, we only have to show that

B < sup \{\frac{1 + a}{R (\frac{a}{B} C(B), X)}: a ≥ 0\} = sup \{\frac{1 + a}{\sqrt{\frac{1}{2} + a^{2} \frac{B^{2}−1}{B^{2}}}} : a ≥ 0\} .

That is, that there exists a ≥ 0 such that

\frac{1 + a}{\sqrt{\frac{1}{2} + a^{2} \frac{B^{2}−1}{B^{2}}}} > B

which is equivalent to

F (a) := 2\frac{(1 + a)^{2} + a^{2}}{1 + 2a^{2}} > B^{2}.

The left hand side F (a) in the above inequality takes its maximum value at a_{0} = \sqrt{2}/2. Then a solution of the above inequality is every B > 1 such that

F (\frac{\sqrt{2}}{2}) > B^{2}

and this happens if B < \sqrt{2 + \sqrt{2}}. ■

In order to obtain a similar result for ℓ_{p} when 1 < p < ∞ and p ≠ 2 we will need to find the maximum of the positive real function

F_{p}(a) := 2\frac{(1 + a)^{p} + a^{p}}{1 + 2a^{p}}

defined for a ≥ 0.

It is easy to see that

2 = F_{p}(0) = lim_{a→∞} F_{p}(a).

On the other hand, for nonnegative a

F′_{p}(a) = \frac{2p}{(2a^{p} + 1)^{2}} (a^{p−1} + (a + 1)^{p−1} − 2 (a(a + 1))^{p−1}) .

Since the term \frac{2p}{(2a^{p}+1)^{2}} is positive, the sign of the derivative F′(a) is the same as the sign of the continuous function

G_{p}(a) := a^{p−1} + (a + 1)^{p−1} − 2 (a(a + 1))^{p−1} .

We have

G_{p} ((\frac{1}{2})^{\frac{1}{p−1}} ) = \frac{1}{2} > 0.

We will give the argument only for p > 2. Since for p > 2 we have

G_{p} ((\sqrt{\frac{1}{2}})^{\frac{1}{p−1}} ) = \sqrt{\frac{1}{2}} + (1 − \sqrt{2}) ((\sqrt{\frac{1}{2}})^{\frac{1}{p−1}} + 1)^{p−1}

< \sqrt{\frac{1}{2}} + (1 − \sqrt{2}) (\sqrt{\frac{1}{2}} + 1)^{p−1}

< \sqrt{\frac{1}{2}} + (1 − \sqrt{2}) (\sqrt{\frac{1}{2}} + 1) = 0,

there exists a_{p} in the open interval

((\frac{1}{2})^{\frac{1}{p−1}} , (\sqrt{\frac{1}{2}})^{\frac{1}{p−1}} )

for which F′_{p}(a_{p}) = 0. Then, using arguments from elementary calculus we conclude that a_{p} is the unique maximum of F_{p}.

Corollary 4.5 Let p > 1 be a real number, and c_{p} := (F_{p}(a_{p}))^{\frac{1}{p}}. Suppose that | · | is a norm on ℓ_{p} such that, for all x ∈ ℓ_{p},

∥x∥ ≤ |x| ≤ B∥x∥

where ∥ · ∥ is the ordinary norm of ℓ_{p}. If B < c_{p}, then (ℓ_{p}, | · |) has the WFPP.

Proof. Since (see [X 97]) r_{ℓ_{p}}(c) = (1 + c^{p})^{\frac{1}{p}} − 1 for all c ≥ 0, we have

C_{ℓ_{p}}(B) = C(B) = (B^{p} − 1)^{\frac{1}{p}} .

On the other hand, (see [D 96]), for all a ≥ 0

R (a\frac{C(B)}{B}, ℓ_{p}) = (\frac{1}{2} + \frac{a^{p}(B^{p} − 1)}{B^{p}})^{\frac{1}{p}} .

By the Theorem, we only have to show that there exists a ≥ 0 such that

\frac{1 + a}{R(a\frac{C(B)}{B}, ℓ_{p})} = \frac{1 + a}{(\frac{1}{2} + \frac{a^{p}(B^{p}−1)}{B^{p}})^{\frac{1}{p}}} > B.

That is

(1 + a)^{p} > B^{p} (\frac{1}{2} + a^{p} \frac{B^{p} − 1}{B^{p}}) .

This is equivalent to

2\frac{(1 + a)^{p} + a^{p}}{1 + 2a^{p}} > B^{p}.

The left hand-side F_{p}(a) in the above inequality attains its maximum value at some a_{p} > (1/2)^{1/(p−1)}. Then a solution of the above inequality is every B > 1 such that

F_{p} (a_{p}) > B^{p}

and this happens if B < c_{p}. ■

Remarks

1 The effective computation of the constant c_{p} in the above corollary seems to be hard, specially if p is not a natural number. Nevertheless, it is clear that

c_{p} ≥ max \{(F_{p} ((\frac{1}{2})^{\frac{1}{p−1}} ))^{\frac{1}{p}} , (F_{p} ((\sqrt{\frac{1}{2}})^{\frac{1}{p−1}} ))^{\frac{1}{p}} \} .

The right hand-side in this inequality furnishes bounds of stability in ℓ_{p} that are greater than M(ℓ_{p}), at least for p < 6.

2. Let X be a Banach space with Opial modulus r_{X}, and let B > 1. Since r_{X}(c) ≥ c − 1 for every c ≥ 0, if r_{X}(c) ≤ c − 1 then c ≤ B. Therefore, by the definition of C_{X}(B) it is clear that C_{X}(B) ≤ B. Hence, for every a ≥ 0,

R (\frac{aC_{X}(B)}{B}, X) ≤ R(a, X).

Consequently

M(X) ≤ sup \{\frac{1 + a}{R (\frac{aC_{X}(B)}{B}, X)} : a ≥ 0\} .

Thus, we have that the above theorem is more general than the stability theorem due to Domínguez-Benavides [D 96]. Nevertheless, in [L-T-X 95] the authors proved that r_{X} is continuous on [0, ∞) and that, for 0 < c_{1} ≤ c_{2},

r_{X}(c_{2}) − r_{X}(c_{1}) ≤ (c_{2} − c_{1})\frac{r_{X}(c_{1}) + 1}{c_{1}}

If r_{X} satisfies the condition r_{X}(1) = 0, then for c ≥ 1, we have

c − 1 ≤ r_{X}(c) ≤ (c − 1)(r_{X}(1) + 1) = c − 1.

Therefore, for those Banach spaces X with r_{X}(1) = 0, C_{X}(B) = B (B ≥ 1).

It is obvious that, for such spaces, the stability bound given by the above theorem is M(X).

Spaces verifying r_{X}(1) = 0 are, for example, X = ℓ_{2} ⊕ ℝ (see [L-T-X 95], Remark 3.1.), and the Bynum spaces ℓ_{p,∞} (1 < p < ∞).

3. It is worthwhile to notice that for all Banach space X, R(0, X) = (W CS(X))^{−1} and therefore M(X) ≥ W CS(X). (See [A-D-L 97], and [D-J 98] for more information about the coefficient M(X)).

5. Stability for Hilbert spaces. Lin’s Theorem

ΛStability, Hilbert spaces ΛLin’s Theorem The stability constant for a Hilbert space was improved by several authors over a short period of time. We present here the last of these improvements, due to P.K. Lin ([L 99]). His result gives a lower bound for the stability constant that is slightly greater than 2.07, dispelling the conjecture that for Hilbert spaces the constant was 2.

Theorem 5.1 Let (H, ∥ · ∥) be a real Hilbert space. Let | · | be an equivalent renorming of H such that for all v ∈ H

∥v∥ ≤ |v| ≤ b∥v∥.

If b < \sqrt{\frac{5+\sqrt{13}}{2}} then (H, | · |) has the FPP.

Proof. Suppose, for a contradiction, that (H, | · |) does not have the FPP. Then we may suppose that there exists T, a nonexpansive mapping on a minimal invariant weakly compact subset C of H such that |·|−diam(C) = 1, and we can select a weakly null approximately fixed point sequence (x_{n}).

Let

D := inf {lim inf_{n} ∥y_{n} − y∥ : (y_{n}) is an a.f.p.s. in C, for T, y_{n} ⇀^{w} y} .

Notice that by the Goebel-Karlovitz’s Lemma, D ≥ \frac{1}{b}. As a first step, let us show that for all t ∈ (0, 1), if W̃_{t} is defined as in Lemma 4.3 we have

sup \{ lim_{n→U} ∥w_{n}∥ : { \atop (w_{n})} ∈ W̃_{t}\} ≥ D.

Indeed, suppose that there exists t_{0} ∈ (0, 1) such that

sup \{ lim_{n→U} ∥w_{n}∥ : { \atop (w_{n})} ∈ W̃_{t_{0}}\} < D.

Then there exists ε > 0 such that

sup \{ lim_{n→U} ∥w_{n}∥ : { \atop (w_{n})} ∈ W̃_{t_{0}}\} < D − ε.

Since the bounded closed convex set W̃_{t_{0}} is T̃ invariant, we can select an a.f.p.s. (w̃_{m}) for T̃, with

w̃_{m} := { \atop (w_{n}^{m})}_{n≥1}.

Fix a positive integer k. Then there exists m_{k} ∈ ℕ such that

lim_{n→U} ∥w_{n}^{m_{k}} − T (w_{n}^{m_{k}}) ∥ < \frac{1}{k} .

Thus the set A_{k} := \{n ∈ N : ∥w_{n}^{m_{k}} − T (w_{n}^{m_{k}})∥ < \frac{1}{k}\} ∈ U. Moreover, since

lim_{n→U} ∥w_{n}^{m_{k}}∥ < sup \{ lim_{n→U} ∥w_{n}∥ : { \atop (w_{n})} ∈ W̃_{t_{0}}\} < D − ε,

we have that the set

B_{k} := {n ∈ ℕ : ∥w_{n}^{m_{k}}∥ < D − ε} ∈ U.

As U is a free ultrafilter we know that A_{k} ∩ B_{k} is an infinite set, and therefore we can select a strictly increasing sequence (n_{k}) of positive integers such that for any k ∈ ℕ,

∥w_{n_{k}}^{m_{k}} − T (w_{n_{k}}^{m_{k}}) ∥ < \frac{1}{k}

and

∥w_{n_{k}}^{m_{k}} ∥ < D − ε.

Put z_{k} := w_{n_{k}}^{m_{k}}. It is clear now that the sequence (z_{k}) is an a.f.p.s. for T and moreover it satisfies ∥z_{k}∥ < D − ε for each k ∈ ℕ.

Since the set C is weakly compact, we may suppose without loss of generality that z_{n} ⇀^{w} z ∈ C. Then, as (H, ∥.∥) is a Hilbert space, we obtain

lim inf ∥z_{k} − z∥ ≤ lim inf ∥z_{k}∥ ≤ D − ε < D

which is a contradiction.

As a second step, we show that D^{2} ≥ \frac{1}{b^{2}−1}.

Given any t with \sqrt{1 − 1/b} < t < 1, by the definition of D, there exists an a.f.p.s. (y_{n}) for T in C with y_{n} ⇀^{w} y ∈ C, such that

D ≤ lim ∥y_{n} − y∥ ≤ D + (1 − t)^{2}.

After a translation, if necessary, we may suppose without loss of generality that y = 0.

Let W̃_{t} be defined as in Lemma 4.3, with x̃ = ˜(y_{n}), we can apply the first step to obtain

sup \{ lim_{n→U} ∥w_{n}∥ : { \atop (w_{n})} ∈ W̃_{t}\} ≥ D.

Hence, we can select w̃ := { \atop (w_{n})} ∈ W̃_{t} such that

lim_{n→U} ∥w_{n}∥ ≥ D − (1 − t)^{2}.

As (H, ∥.∥) is a Hilbert space, the ultrapower H̃_{U} is also a Hilbert space. Let P = x̃, Q_{t} = tx̃ and R_{t} = w̃. Consider the triangle △P Q_{t}R_{t}. First note that this is a proper triangle because w̃ does not lie on the line that contains x̃ and tx̃. Otherwise, since by Lemma 4.3 |w̃ − x̃| = 1 − t, we would have that w̃ = tx̃. But then, the above estimates on ∥w̃∥ and ∥x̃∥ give D − (1 − t)^{2} ≤ ∥w̃∥ = ∥tx̃∥ ≤ tD + t(1 − t)^{2} and so 1 − t^{2} > D ≥ \frac{1}{b} which contradicts the choice of t.

Denoting by γ the angle at P we obtain

2∥w̃ − x̃∥∥x̃∥ cos(γ) = ∥x̃∥^{2} − ∥w̃∥^{2} + ∥w̃ − x̃∥^{2}

≤ (D + (1 − t)^{2})^{2} − (D − (1 − t)^{2})^{2} + (\frac{1 − t}{b})^{2}

≤ (4D + 1/b^{2})(1 − t)^{2}

Thus we may choose t near enough to 1 so that γ is arbitrarily close to or larger than \frac{π}{2} . Let the perpendicular from P meet the line through R_{t} and Q_{t}, at F. Let a be the distance, in ∥.∥, between P and Q_{t} and let h be the distance, again in ∥.∥, from P to F. For convenience, let d be the distance between P and Q_{t}. Then

d = (1 − t)D,
a ≤ 1 − t

and

\frac{1 − t}{b} ≤ h.

To see this last inequality note that, if F = λtx̃ + (1 − λ)w̃,then for λ ∈ [0, 1] we have F ∈ W̃_{t} and, by Lemma 4.3 (a), |P − x̃| = 1 − t. For λ > 1 we have again, by Lemma 4.3 (a), that |P − x̃| ≥ |tx̃ − x̃| − (1 − λ)|w̃ − x̃| = 1 − t. For λ < 0 we can use similar reasoning to obtain the same conclusion. In any case, |P − x̃| ≥ 1 − t.

Now, the distance from R_{t} to F is \sqrt{a^{2} − h^{2}} and the distance from F to O_{t} is \sqrt{d^{2} − h^{2}}, thus given ϵ > 0, using the above estimates and the observation about γ we may choose t sufficiently close to 1 so that we have

(\sqrt{a^{2} − h^{2}} + \sqrt{d^{2} − h^{2}})^{2} ≥ a^{2} + d^{2} − ϵ.

This leads to

d^{2} ≥ h^{2} (1 + \frac{1}{(a/h)^{2} − 1} + O(ϵ)) .

Substituting the value for d and using the bounds on a and h, we obtain

(1 − t)^{2}D^{2} ≥ (\frac{1 − t}{b})^{2} (1 + \frac{1}{(_{\frac{1−t}{\frac{1−t}{b}}} )^{2} − 1} + O(ϵ))

and so

D^{2} ≥ \frac{1}{b^{2} − 1} + O(ϵ).

Since ϵ is arbitrary this establishes the inequality and completes the proof step two. For the third step, let us see that for any t ∈ [0, 1] the following inequality holds

sup \{ lim_{n→U} ∥w_{n}∥^{2} : { \atop (w_{n})} ∈ W̃_{t}\} ≤ \frac{t^{2}}{2} + (1 − t)^{2}(1 − \frac{1}{b^{2}} ) .

Indeed, consider { \atop (w_{n})} ∈ W̃_{t}. Since C is a weakly compact subset of H, we know that there exists w ∈ C such that w = (w − lim_{n→U})w_{n}. Then it is clear that

lim_{n→U} ∥x_{n} − w_{n}∥^{2} = ∥w∥^{2} + lim_{n→U} ∥x_{n} − w_{n} + w∥^{2}.

Now, since { \atop (w_{n})} ∈ W̃_{t} by Lemma 4.3 (a), we have

∥x_{n} − w_{n}∥^{2} ≤ |x_{n} − w_{n}|^{2} ≤ (1 − t)^{2}.

Moreover, by Lemma 4.3 (d), it is easy to see that

∥x_{n} − w_{n} + w∥^{2} ≥ \frac{1}{b^{2}} |x_{n} − w_{n} + w|^{2} ≥ (\frac{1 − t}{b})^{2} .

Therefore,

∥w∥^{2} ≤ (1 − t)^{2}(1 − \frac{1}{b^{2}} ).

Now, by using again the facts that (H, ∥.∥) is a Hilbert space and that w = (w − lim_{n→U})w_{n} it is not difficult to see that

lim_{m→U} lim_{n→U} ∥w_{n} − w_{m}∥^{2} = 2 lim_{n→U} ∥w_{n} − w∥^{2}.

Thus, we may apply Lemma 4.3 (d) in the above equality to obtain

lim_{n→U} ∥w_{n} − w∥^{2} = \frac{1}{2} lim_{m→U} lim_{n→U} ∥w_{n} − w_{m}∥^{2} ≤ \frac{t^{2}}{2} .

Consequently,

lim_{n→U} ∥w_{n}∥^{2} = ∥w∥^{2} + lim_{n→U} ∥w_{n} − w∥^{2} ≤ \frac{t^{2}}{2} + (1 − t)^{2} (1 − \frac{1}{b^{2}} ) .

Combining Lemma 4.3 together with the inequalities of first, second and third steps, we finally derive

\frac{1}{b^{2} − 1} ≤ sup \{ lim_{n→U} ∥w_{n}∥^{2} : { \atop (w_{n})} ∈ W̃_{t}\} ≤ \frac{t^{2}}{2} + (1 − t)^{2} (1 − \frac{1}{b^{2}} ) .

In particular, taking t = \frac{2b^{2}−2}{3b^{2}−2} this yields

b ≥ \sqrt{\frac{5 + \sqrt{13}}{2}}.

■

We end this section with a question raised in the introduction: it was noted there that every |·|-nonexpansive mapping in a Banach space X is ∥·∥-uniformly lipschitzian with respect to any norm ∥ · ∥ on X equivalent to | · |. Then, the question of whether any equivalent renorming of a Hilbert space X, with norm ∥ · ∥, has the FPP would have an affirmative answer if the same were true for the following second question: does any Hilbert space have the fixed point property for uniformly lipschitzian mappings? Some well known examples show that this last question has a negative answer. One of them, due to J.B. Baillon [B 78], is the mapping T : B → B defined on the closed unit ball of the Euclidean space ℓ_{2} by

T (x) = \{ \begin{matrix}cos (\frac{π}{2} ∥x∥_{2}) e + sin (\frac{π}{2} ∥x∥_{2}) \frac{S(x)}{∥x∥_{2}} & x ≠ 0 \\ e & x = 0 \end{matrix}

where (e_{n}) is the usual Schauder basis in ℓ_{2} and S is the right shift operator associated to (e_{n}).

It is known that T is a fixed point free uniformly \frac{π}{2} -lipschitzian mapping, and this suggest that an strategy for answering in the negative our first question could be to show that T is nonexpansive with respect to some equivalent norm on ℓ_{2}. Unfortunately, the following theorem [G-J-L 97] shows that this is not the case.

Theorem 5.2 The mapping T is not nonexpansive with respect to any norm on ℓ_{2} equivalent to the Euclidean norm.

Proof. Suppose that | · | is a norm on ℓ_{2} equivalent to the Euclidean norm, ∥ · ∥_{2}, such that T is | · |-nonexpansive.

Let a > 0 be a real number such that a∥x∥_{2} ≤ |x| for all x ∈ ℓ_{2}. Assume that a is optimal and choose y such that ∥y∥_{2} = 1 and a ≤ |y| < a\sqrt{2}. Then we have that

\sqrt{2} = ∥T (0) − T (y)∥_{2} ≤ \frac{1}{a}|T (0) − T (y)| ≤ \frac{1}{a}|y| ,

which is a contradiction. ■

6. Stability for the τ-FPP

ΛStability, τ-FPP As we said in the introduction, the fixed point theorems for uniformly lipschitzian mappings can be used to obtain stability results for nonexpansive mappings. Although in the case of the WFPP these results have not showed to be better. However, for a general topology τ it is possible to obtain stability results by using some fixed points results for asymptotically regular and uniformly lipschitzian mappings. We recall the following.

Let (X, ∥·∥) be a Banach space and let C be a closed convex subset of X, a mapping T : C → C is said to be k-uniformly lipschitzian whenever the inequality

∥T^{n}x − T^{n}y∥ ≤ k∥x − y∥

holds for every n ∈ ℕ and for every x, y ∈ C.

Recall that T is asymptotically regular on C if

lim_{n→∞} ∥T^{n+1}x − T^{n}x∥ = 0

for all x ∈ C.

It is well known (see [I 76],[G-K 90]) that if (X, ∥ · ∥) is a Banach space and C is a closed convex and bounded nonempty subset of X and S : C → C is nonexpansive then the mapping T := \frac{I+S}{2} : C → C is asymptotically regular on C.

Definition 6.1 Let M be a bounded convex subset of (X, ∥ · ∥) .

(a) A number b ≥ 0 is said to have the property (P_{τ}) with respect to M if there exists a > 1 such that for all x, y ∈ M and r > 0 with ∥x−y∥ ≥ r and each τ-convergent sequence (ζ_{n}) ⊂ M for which lim sup_{n} ∥ζ_{n} − x∥ ≤ ar and lim sup_{n} ∥ζ_{n} − y∥ ≤ br, there exists some z ∈ M such that lim inf_{n} ∥ζ_{n} − z∥ ≤ r.

(b) κ_{τ} (M) := {b > 0 : b ∈ (P_{τ})}.

(c) κ_{τ} (X) := inf{κ_{τ} (M)} where M is any subset of X as above.

Theorem 6.2 Let (X, ∥ · ∥) be a Banach space, τ an arbitrary topology on X. Assume that | · | is an equivalent norm on X such that

|x| ≤ ∥x∥ ≤ k|x|

for every x ∈ X and k < κ_{τ} (X). Then (X, | · |) has the τ-FPP.

Proof. Let C be a τ-sequentially compact norm-bounded convex subset of X, and S : C → C a | · |-nonexpansive mapping. By [I 76] we know that T := (I + S)/2 is a | · |-nonexpansive and asymptotically regular mapping on C, therefore it is clear that T is k-uniformly lipschitzian for the norm ∥ · ∥.

Let us define a function r on C as follows

r(x) = inf {r > 0 : ∃y ∈ C : lim inf_{n→∞} ∥x − T^{n}y∥ ≤ r} .

Since k < κ_{τ} (C), it is readily seen that there exists positive numbers α, μ ∈ (0, 1) such that for every x, y ∈ C with ∥x − y∥ ≥ (1 − μ)r and any τ-convergent sequence (ζ_{n}) in C for which lim sup_{n} ∥ζ_{n} − x∥ ≤ (1 − μ)r and lim sup_{n} ∥ζ_{n} − y∥ ≤ k(1 − μ)r, there exists some z ∈ C such that lim sup_{n} ∥ζ_{n} − z∥ ≤ αr. By the definition of r(x), there exist some integer m ≥ 1 such that ∥x − T^{n_{m}}∥ > r(1 − μ) and a y ∈ C such that

lim inf_{n→∞} ∥x − T^{n}y∥ ≤ (1 + μ)^{\frac{1}{2}} r(x).  (6.1)

Take a subsequence (T^{n_{k}}y) of (T^{n}y) such that

lim inf_{n→∞} ∥T^{n}y − x∥ = lim_{k→∞} ∥T^{n_{k}}y − x∥.

Since C is τ-sequentially compact, we may assume that (T^{n_{k}}y) is τ-convergent. Now using the asymptotically regularity of T on C, we obtain

lim sup_{k→∞} ∥T^{n_{k}}y − T^{n_{m}}x∥ ≤ k lim sup_{k→∞} ∥T^{n_{k}−n_{m}}y − x∥
= k lim sup_{k→∞} ∥T^{n_{k}}y − x∥ ≤ (1 + μ)kr(x).  (6.2)

It follows from (6.1) and (6.2) that there exists z = z(x) ∈ C such that

lim inf_{k→∞} ∥T^{n_{k}}y − z∥ ≤ αr(x).

Which implies that

r(x) ≤ αr(x).

Also, we have

∥z − x∥ ≤ lim inf_{k→∞} ∥z − T^{n_{k}}y∥ + lim inf_{k→∞} ∥x − T^{n_{k}}y∥ ≤ αr(x) + (1 + μ)r(x) = Ar(x),

where A = 1 + α + μ. Proceeding in this fashion, we get a sequence (z_{n}) in C (z_{0} = x and z_{k} = z(z_{k−1})) such that

r(z_{n+1}) ≤ αr(z_{n}) hboxand ∥z_{n+1} − z_{n}∥ ≤ Ar(z_{n}).  (6.3)

Therefore, from (6.3) it is not difficult to see that (z_{n}) is a norm-Cauchy sequence and thus strongly-convergent. Let z_{∞} = lim_{n→∞} z_{n}. Clearly r(z_{∞}) = 0, which implies that z_{∞} is a fixed point of T. Indeed, for any ϵ > 0, since r(z_{∞}) = 0, there exists y ∈ C such that

lim_{k→∞} ∥z_{∞} − T^{n_{k}}y∥ ≤ ϵ.  (6.4)

Then,

∥z_{∞} − T z_{∞}∥ ≤ lim sup_{k→∞}(∥z_{∞} − T^{n_{k}+1}y∥ + ∥T^{n_{k}+1}y − T z_{∞}∥)

≤ (1 + k) lim_{k→∞} ∥T^{n_{k}}y − z_{∞}∥ < (1 + k)ϵ.

■

The definition of κ_{τ} (X) makes its computation difficult . In some classes of Banach spaces we can use an easier definition. In this way, let us recall the following.

Definition 6.3 A Banach space (X, ∥ · ∥) is said to enjoy the property L(τ) if there exists a function δ : [0, +∞[×[0, +∞[→ [0, +∞[ satisfying

(1) δ is continuous

(2) δ(., s) is nondecreasing

(3) δ(r, .) is nondecreasing

(4) δ(lim sup_{n} ∥x_{n}∥, ∥y∥) = lim sup_{n} ∥y−x_{n}∥ for all y ∈ X and every τ-null sequence.

Lemma 6.4 Let (X, ∥ · ∥) a Banach space with the property L(τ) where τ is a linear topology on X. Then the norm ∥.∥ is τ-slsc.

Proof. We are going to show that the unit ball B_{X} is τ-sequentially closed which easily implies that the norm is τ-slsc. Let (x_{n}) be a sequence in the unit ball which is τ-convergent to x_{0}. Since (x_{n} − x_{0}) is τ-null, we obtain from the property L(τ)

1 ≥ lim sup_{n→∞} ∥x_{n}∥ = lim sup_{n→∞} ∥x_{n} − x_{0} + x_{0}∥ = lim sup_{n→∞} ∥x_{n} − 2x_{0}∥.

Hence,

∥2x_{0}∥ ≤ lim sup_{n→∞} ∥x_{n} − 2x_{0}∥ + lim sup_{n→∞} ∥x_{n}∥ ≤ 2,

and thus we may conclude that x_{0} ∈ B_{X}. ■

It is not difficult to show that the Banach spaces which satisfy the L(τ) property such that δ(1, .) is an increasing function are a subclass of Banach spaces with the τ-uniformly Opial condition. That is,

Definition 6.5 A Banach space is said to have the τ-uniformly Opial condition if for c > 0 there exists r > 0 such that lim inf_{n} ∥x + x_{n}∥ ≥ 1 + r for every x ∈ X such that ∥x∥ ≥ c and every τ-null sequence (x_{n}) in X with lim inf_{n} ∥x_{n}∥ ≥ 1.

If (X, ∥ · ∥) is a Banach space satisfying the τ-uniformly Opial condition and with the norm ∥·∥ τ-slsc, then

κ_{τ} (M) = sup{b > 0 : b ∈ (P_{τ})′}

where a number b ≥ 0 is said to have the property (P_{τ})′ with respect to M, b ∈ (P_{τ})′, if for all z, y ∈ M and r > 0 with ∥x − y∥ ≥ r and each sequence (ζ_{n}) ⊂ M τ-convergent to z ∈ M such that lim sup_{n} ∥ζ_{n} − y∥ ≤ br, we have that lim inf_{n} ∥ζ_{n} − z∥ ≤ r. (See [D-X 95]).

Proposition 6.6 Let (X, ∥ · ∥) be a Banach space and consider τ a linear topology on X. Assume that X satisfies both property L(τ) and the τ-uniformly Opial condition. Then κ_{τ} (X) ≥ δ(1, 1). If, in addition, there exists a τ-null sequence that is not normconvergent, then κ_{τ} (X) = δ(1, 1).

Proof. By above Lemma the norm of X is τ-slsc, so we may use the equivalent definition of κ_{τ} (M). Without loss of generality we can assume that r = 1 and z = 0 in such definition of κ_{τ} (M). Suppose that b < δ(1, 1). If (ζ_{n}) is τ-null sequence, ∥y∥ ≥ 1 and lim sup_{n} ∥ζ_{n} + y∥ ≤ b we have lim sup_{n} ∥ζ_{n}∥ ≤ 1, since otherwise we arrive to the contradiction

b ≥ lim sup_{n} ∥ζ_{n} − y∥ = δ(lim sup_{n} ∥ζ_{n}∥, ∥y∥) ≥ δ(1, 1).

Thus κ_{τ} (X) ≥ δ(1, 1).

To prove the second assertion, let (x_{n}) be a τ-null normalized sequence and M = co((x_{n})∪{0}). Consider y = x_{1} and ζ_{n} = x_{n}, n ≥ 2. Then ∥y∥ = 1, lim sup_{n} ∥ζ_{n} −y∥ = δ(1, 1) and lim inf_{n} ∥ζ_{n}∥ = 1. Thus κ_{τ} (X) ≤ δ(1, 1). ■

Example 6.7 Let (Ω, Σ, μ) be a positive σ-finite measure space. For every 1 ≤ p < ∞, consider the Banach space L_{p}(Ω) with the usual norm. Now, consider L_{p}(Ω) with the topology of the convergence locally in measure (clm). If (f_{n}) is a clm-null sequence in L_{p}(Ω), 1 ≤ p < ∞, and fis a function in L_{p}(Ω), then

lim sup_{n} ∥f_{n} − f∥_{p}^{p} = ∥f∥_{p}^{p} + lim sup_{n} ∥f_{n}∥_{p}^{p}.

Thus, L_{p}(Ω) endowed with the clm topology satisfies the property L(clm) with δ(r, s) = (r^{p} + s^{p})^{1/p}. In particular, if Ω = ℕ with the cardinal measure and p > 1 we obtain ℓ_{p} with the weak topology. For ℓ_{1}, the clm-topology coincides with the weak* topology on the norm-bounded subset of ℓ_{1} = c_{0}^{⋆}.

As a consequence of the above results we have:

Corollary 6.8 If Y is a Banach space isomorphic to L_{p}(Ω), 1 ≤ p < ∞, such that d(Y, L_{p}(Ω)) < 2^{1/p} then Y has the clm-FPP.

In particular, if Y is a Banach space isomorphic to ℓ_{1} and d(Y, ℓ_{1}) < 2, then Y has the weak*-FPP. On the other hand, it must be noted that 2 is the best possible stability bound for ℓ_{1} with respect to the weak*-topology and thus for L_{1}(Ω) with respect to the clm topology. Indeed,

Example 6.9 [Li 85] Consider the classical Banach space ℓ_{1} endowed with the norm

|x| = max{∥x^{+}∥_{1}, ∥x^{−}∥_{1}}.

Now we define the set,

K := {(x_{n}) ∈ ℓ_{1}^{+} : ∑_{n=1}^{∞} x_{n} ≤ 1} .

It is easy to see that K is convex and compact with respect to the weak*-topology of ℓ_{1}. On the other hand, if we define the mapping T : K → K as

T ((x_{n})) = (1 − ∑_{n=1}^{∞} x_{n}, x_{1}, x_{2}, ..., x_{n}, ...) .

Clearly T is |.|-nonexpansive and fixed point free on K.

Finally, an easy computation shows that d((ℓ_{1}, ∥.∥_{1}), (ℓ_{1}, |.|)) = 2.

7. Further remarks 7.1. Stability for uniformly convex spaces

ΛStability, uniformly convex spaces Bynum’s paper [By 80] also contains the following theorem of stability:

Theorem 7.1 If (X, ∥·∥) is a uniformly convex Banach space and if (Y, |.|) is a Banach space such that

d(X, Y) ≤ W CS(X),

then Y has the FPP.

Note that the condition d(X, Y) < W CS(X) yields weak normal structure and hence the WFPP for Y, without the assumption of uniform convexity for (X, ∥ · ∥) . On the other hand there are spaces Y with d(ℓ_{2}, Y) = W CS(ℓ_{2}) = \sqrt{2} and without normal structure. So it is clear that this theorem improves Theorem 2.9. It is also clear from Lin’s result that the above theorem is not sharp concerning the stability of the FPP, at least for Hilbert spaces.

7.2. Stability of asymptotic normal structure,[B-K-R 98]

ΛStability, asymptotic normal structure In order to get a quantitative description of asymptotic normal structure, we may mimic the definition of the Bynums’s coefficients to obtain the following.

Definition 7.2 Recall that the bounded sequence (x_{n}) in X is asymptotically regular (a.r.) whenever x_{n} − x_{n+1} → 0.

We define

AN(X) := sup {k :k · inf_{(x_{n_{i}})} r_{a} ((x_{n_{i}})) ≤ diam_{a}((x_{n}))

for each a.r. bounded sequence (x_{n})}.

ΛAsymptotically normal structure coefficient, AN(X) If in the above definition we assume the additional condition of weak compactness of the set \overline{co}(\{x_{n} : n ∈ ℕ\}), then we get the asymptotic normal structure coefficient with respect to the weak topology

w − AN(X) := sup {k :k. inf_{(x_{n_{i}})} r_{a} ((x_{n_{i}})) ≤ diam_{a}((x_{n}))

for each a.r. bounded sequence (x_{n})

with \overline{co}(\{x_{n} : n ∈ ℕ\}) weakly compact\}.

ΛAsymptotic normal structure coefficient with respect to the weak topology, w−AN(X)

Definition 7.3 If AN(X) > 1 we say that (X, ∥ · ∥) has uniform asymptotic normal structure (UAN).

Definition 7.4 If w−AN(X) > 1 we say that (X, ∥·∥) has uniform asymptotic normal structure with respect to the weak topology (w-UAN).

We list some well known properties of these coefficients

1 1 ≤ AN(X) ≤ w − AN(X).

2 1 ≤ W CS(X) ≤ w − AN(X).

3 If a Banach space (X, ∥ · ∥) has AN(X) > 1 then it is reflexive.

4 w − AN(X) > 1 ⇒ W F P P.

5 AN(ℓ_{p}) = 2^{\frac{1}{p}}.

6 Example 7.5 For β, p > 1 let us consider ℓ_{p} endowed with the norm

∥x∥_{β} := max \{∥x∥_{∞}, \frac{1}{β} ∥x∥_{p}\} .

We denote this space by X_{β}^{p}. If we take the space Z := X_{√\overline{2}}^{2}×ℓ_{1} equipped with the ℓ_{1}-norm then this space is nonreflexive. Thus, AN(Z) = 1. But w−AN(Z) = \sqrt{2}.

Theorem 7.6 Let (X, ∥ · ∥) and (Y, | · |) be isomorphic Banach spaces. Then we have

AN(X) ≤ d(X, Y)AN(Y),
w − AN(X) ≤ d(X, Y).w − AN(Y).

Corollary 7.7 Let (X, ∥ · ∥) and (Y, | · |) be isomorphic Banach spaces.

a) If d(X, Y) < AN(X) then AN(Y) > 1, and hence (Y, | · |) has UAN.

b) If d(X, Y) < w −AN(X) then w −AN(Y) > 1, and hence (Y, |·|) has the WFPP.

7.3. Stability of Semi-Opial property, [B-K-R 98]

ΛStability, Semi-Opial property Semi-Opial property was defined in [B-K-R 98]. Recall that a Banach space is said to have the semi-Opial (weak semi-Opial) property, (SO) ((w-SO)), if for each bounded nonconstant asymptotically regular sequence (x_{n}) in X (with weakly compact closed convex hull), there exists a subsequence (x_{n_{i}}) weakly convergent to x, such that

lim inf_{i} ∥x − x_{n_{i}}∥ < diam({x_{n}}).

Definition 7.8 The semi-Opial coefficient (with respect to the weak topology) is defined by

w − SOC(X) := sup {k :k. inf_{(x_{n_{i}}),x_{n_{i}} ⇀^{w} y} r_{a} (y, (x_{n_{i}})) ≤ diam_{a}((x_{n}))

for each a.r. boundeds sequence (x_{n})

with \overline{co}(\{x_{n} : n ∈ ℕ\}) weakly compact\}.

ΛSemi-Opial coefficient with to the weak topology, w − SOC(X)

Definition 7.9 If w − SOC(X) > 1 we say that (X, ∥ · ∥) has the uniform semi-Opial property, with respect to the weak topology (w-USO).

ΛUniform semi-Opial property The following are well known properties.

1 1 ≤ W CS(X) ≤ w − SOC(X) ≤ w − AN(X).

2 If a Banach space (X, ∥ · ∥) has N(X) > 1 then w − SOC(X) > 1.

3 If a Banach space (X, ∥·∥) has the nonstrict Opial property, then w −SOC(X) = w − AN(X).

4 The space L^{p}([0, 1]) 1 < p < ∞, p ≠ 2, has w − USO property but it does not satisfy the nonstrict Opial condition.

5 w − SOC(ℓ_{p}) = 2^{\frac{1}{p}}.

6

w − SOC(X_{β}^{p}) = max [1, min (2^{\frac{1}{p}} , \frac{4^{\frac{1}{p}}}{β} )] .

7 If we take the space Z := X_{√\overline{2}}^{2} × ℓ_{1} endowed with the ℓ_{1}-norm then this space is nonreflexive. Thus, AN(Z) = 1. But w − SOC(Z) = \sqrt{2}.

Theorem 7.10 Let (X, ∥ · ∥) and (Y, | · |) be isomorphic Banach spaces. Then we have

w − SOC(X) ≤ d(X, Y).w − SOC(Y).

Corollary 7.11 Let (X, ∥ · ∥) and (Y, | · |) be isomorphic Banach spaces. If d(X, Y) < w − SOC(X) then w − SOC(Y) > 1 and hence (Y, | · |) has the uniform semi-Opial property.

7.4. Stability of Orthogonal Convexity, [J 88]

ΛStability, orthogonal convexity If x, y ∈ X and λ is positive put

M_{λ}(x, y) = \{z ∈ X : max\{∥z − x∥, ∥z − y∥\} ≤ \frac{1}{2}(1 + λ)∥x − y∥\} .

If A is a bounded subset of X, |A| := sup{∥x∥ : x ∈ A}.

If (x_{n}) is a bounded sequence in X recall that

D[(x_{n})] := lim sup_{n}(lim sup_{m} ∥x_{n} − x_{m}∥).

Moreover we will use the notation

A_{λ}[(x_{n})] := lim sup_{n}(lim sup_{m} |M_{λ}(x_{n}, x_{m})|).

Definition 7.12 We say that a Banach space X is orthogonally convex (OC) if for every weakly null sequence (x_{n}) with D[(x_{n})] > 0 there exists λ = λ ((x_{n})) > 0 such that A_{λ}[(x_{n})] < D[(x_{n})].

Notice that |M_{λ}(x_{n}, x_{m})| is some measure of the size of the set of almost metric midpoints of the above segments [x_{m}, x_{n}], and we could say that this size is “small” when the unit ball B of X is rotund.

It is well known that in the unit ball of a uniformly convex space, if two points are far apart, then their midpoint must be within it. Similarly, property OC ensures that the “radial distances” |M_{λ}(x_{n}, x_{m})| are asymptotically strictly smaller than the length of the some linear segments [x_{m}, x_{n}]. This property was introduced in [J 88], as a sufficient condition for the WFPP. Every uniformly convex Banach space is OC. Other examples of OC spaces given in [J 88] are Banach spaces with the Schur property (hence ℓ_{1}), c_{0}, c and James’s space J. Moreover, Prus, Kutzarova and Sims have shown [Pr-K-S 93] that OC Banach spaces have the weak Banach-Saks property.

It turns that orthogonal convexity is reasonably well behaved under isomorphisms.

Definition 7.13 If (X, ∥.∥) is a Banach space, and Q = (Q_{n}), R = (R_{n}) are two bounded families of continuous linear operators from X to X, we shall refer to (Q, R) as a pair for OC if the following conditions hold

(a) For every weakly null sequence (x_{n}) in X,

lim_{n→∞} inf_{i} ∥R_{i}(x_{n})∥ = 0

and

lim_{n→∞} ∥Q_{i}(x_{n})∥ = 0 for all i ∈ Z^{+}.

(b) The coefficient

ρ_{(Q,R)}(X) := sup{∥x∥ : ∥Q_{i}(x)∥ ≤ 1, ∥R_{i}(x)∥ ≤ 1 for some i ∈ Z^{+}}

Stability of the fixed point propertyfor nonexpansive mappings 33

is finite. For simplicity we also write ρ(X) instead of ρ_{(Q,R)}(X). In particular if (X, ∥.∥) has a Schauder basis (e_{n}), a standard pair for OC is the sequence of natural projections

Q = (Q_{n}) = (P_{n}), R = (R_{n}) = (I − P_{n}).

Theorem 7.14 [J-L 93] If X admits a pair for OC and

d(X, Y) < 2(ρB)^{−1}

then Y has OC and hence the WFPP. Here

B := sup [{∥Q_{i}∥ : i = 1, 2, … } ∪ {∥R_{i}∥ : i = 1, 2, … }].

Corollary 7.15 [J 88]

a) If X is a Banach space such that

d(X, c_{0})) < 2

then X has OC and hence the WFPP. Moreover the bound 2 is sharp for OC.

b) If X is a Banach space such that

d(X, H) < \sqrt{2}

where H is a Hilbert space, then X has OC and hence the WFPP. Moreover the bound \sqrt{2} is sharp for OC.

Corollary 7.16 [J-L 93]

a) If X is a Banach space such that

d(X, ℓ_{p})) < 2^{1− \frac{1}{p}}

for 1 < p < ∞, then X has OC and hence the WFPP. This bound of stability for OC is sharp.

b) If X is a Banach space such that

d(X, ℓ_{p,∞})) < 2^{1− \frac{1}{p}}

for 1 < p < ∞, then X has OC and hence the WFPP.

c) Let V be the space ℓ_{2} endowed with the norm (defined by D. van Dulst)

|x| := max \{\frac{1}{3}∥x∥_{2}, sup |x_{1} + x_{n} + x_{n+1}|\}

where ∥x∥_{2} stands for the Euclidean norm of x ∈ ℓ_{2}. If X is a Banach space such that d(X, V)) < \sqrt{2} then X has OC and hence the WFPP.

8. Summary

For X =... inherits... whenever d(X, Y)

and the bound is sharp

References

H, a Hilbert space  NS  < \sqrt{2}  Yes  [By 80][M 84]
H, a Hilbert space  GGLD  < \sqrt{2}  Yes  [M 84]
L^{p}, 1 < p < ∞  NS  < min\{2^{\frac{1}{p}} , 2^{\frac{1}{p}} \}  No  [By 80]
ℓ_{p,q}, 1 < p < ∞, 1 ≤ q < ∞ GGLD  < min\{2^{\frac{1}{p}} , 2^{\frac{1}{q}} \}.  Yes  [By 80], [D-L-X 96]
E_{β}, 1 < β < \sqrt{2}  GGLD  < \frac{\sqrt{2}}{β}  Yes  [By 80], [A-X 93]
J  GGLD  < \sqrt{3/2}  ?  [J 92]
J_{i} (i = 1, 2)  GGLD  < \sqrt{2}  ?  [D-L-X 95]
c_{0}  WFPP  < 2  ?  [B-S 84]
c  WFPP  < 2  ?  [B-S 84]
ℓ_{p}, 1 < p < ∞  OC  < 2^{1− \frac{1}{p}}  Yes  [J 88]
ℓ_{p,∞}, 1 < p < ∞  OC  < 2^{1− \frac{1}{p}}  Yes  [J 88]
V  OC  < \sqrt{2}  Yes  [J 88]
c_{0}  OC  < 2  Yes  [J 88]
H, a Hilbert space  OC  < \sqrt{2}  Yes  [J 88]
Having property M  WFPP  < \frac{1+\sqrt{5}}{2}  No  [G-S 97]
ℓ_{p}, 1 < p < ∞  WUANS < 2^{\frac{1}{p}}  ?  [B-K-R 98]
ℓ_{p}, 1 < p < ∞  WSOC  < 2^{\frac{1}{p}}  ?  [B-K-R 98]
H a Hilbert space  FPP  < \sqrt{\frac{5+\sqrt{13}}{2}}  ?  [L 99]
ℓ_{p}, 1 < p < ∞  FPP  < (1 + 2^{\frac{1}{p−1}} )^{\frac{p−1}{p}} No  [D 96] [J-L 00] [L 99]

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