OPIAL MODULUS AND STABILITY OF THE FIXED POINT PROPERTY

A. Jiménez-Melado

Dept. Análisis Matemático Universidad de Málaga Facultad de Ciencias

29071 Málaga. Spain

E. Llorens-Fuster

Dept. Análisis Matemático Facultad de Matemáticas

46100 Burjassot. Valencia. Spain

OPIAL MODULUS AND STABILITY OF THE FIXED POINT PROPERTY

A. Jiménez-Melado ^{1} E. Llorens-Fuster ^{2}

Abstract Our main result shows that a Banach space Y has the fixed point property whenever the Banach-Mazur distance from Y to a Hilbert space is less than \sqrt{2 + \sqrt{2}}.

1 Introduction and preliminary results

Let C be a nonempty subset of a Banach space X, with norm ∥ · ∥. A mapping T: C → X is called nonexpansive if ∥T (x) − T (y)∥ ≤ ∥x − y∥ for all x, y ∈ C. We say that X has the weak fixed point property (WFPP) if every nonexpansive mapping T: C → C defined on a nonempty convex and weakly compact subset K of X has a fixed point.

Perhaps, the most classical result in this area is that of Kirk [K] which asserts that every Banach space with normal structure has the WFPP. Since then, many efforts has been directed into finding geometrical conditions on a Banach space X which guarantee the WFPP for X.

Although Alspach [A] showed that L^{1}[0, 1] lacks the WFPP, an earlier result of Day-James-Swaminathan ([DJS]) assures that there exists a renorming of L^{1}[0, 1] with the WFPP. Hence the WFPP is not invariant under topological isomorphisms. Nevertheless, many stability results has beenestablished for the WFPP in terms of the Banach-Mazur distance and othercoefficients of Banach spaces.

Recently, J. García-Falset [Ga] showed that a Banach space Y has the WFPP if there exists a Banach space X such that d(X, Y) R(X) < 2, where d(X, Y) is the Banach-Mazur distance from X to Y and R(X) is the supremum of the numbers lim inf ∥x + x_{n}∥, the supremum being taken over all weakly null sequences (x_{n}) in the unit ball of X and all points x in the unit ball of X. In the proof of this result, García-Falset uses a nice trick combined with ”nonstandard” techniques, a method introduced in Fixed Point Theory by Maurey [Ma] and afterwards developped by Khamsi, Lin and Borwein and Sims, among others. (See [AK]).

1The first author has been partially supported by D.G.I.C.Y.T. PB94-1496 and by a grant from La Junta de Andalucía.

2The second author has been partially supported by D.G.I.C.Y.T., PB93-117-CO2-02.

Inspired on the proof of [Ga], Dominguez-Benavides [Do] has improved the stability constant 2R(X)^{−1}. In order to do this, he defined, for a Banach space X and a nonnegative real number a, the coefficient

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}) = lim sup_{n} (lim sup_{m} ∥x_{n} − x_{m}∥) ≤ 1 .

It is shown in [Do] 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 X, M(X) = \sqrt{3}. Hence, the result of [Do] improves all previously known stability results in Hilbert spaces and l_{p}, 1 < p < ∞ (see [BeS, By, P2, Kh, JL1]).

In this paper we improve the stability constant M(X) for a certain class of Banach spaces. Since we follow the proof of [Do], we need to state some basic results.

Suppose that C is a nonempty, convex and weakly compact subset of the Banach space X and that T: C → C is nonexpansive. Standard arguments show that C contains a subset K which is minimal for the properties of being nonempty, convex, weakly compact and T -invariant, and that K contains an approximate fixed point sequence (afps in short) for T (i.e., a sequence (x_{n}) in K such that lim ∥x_{n} − T (x_{n})∥ = 0). The well known Goebel-Karlovitz lemma (see [GK]) ensures that if K is minimal for T and (x_{n}) is an afps for T in K, then (x_{n}) is diametral, i.e., lim ∥x_{n} −x∥ = diam(K) for every x ∈ K.

We denote by [X] the quotient space \frac{l_{∞}(X)}{C_{0}(X)} endowed with the norm ∥[z_{n}]∥ = lim sup ∥z_{n}∥, where [z_{n}] denotes the class of (z_{n}) ∈ l_{∞}(X). Any x ∈ X will be considered as an element of [X] by identifying x with the class which contains the constant sequence (x, x, …). If C is a bounded subset of X and T: C → C is a mapping, we denote by [C] the set

[C] = {[c_{n}]: (c_{n}) is a sequence in C}

and by [T] the mapping [T]: [C] → [C] defined by [T]([c_{n}]) = [T (c_{n})]. With these notations we can state the following version of the Goebel-Karlovitz lemma (see [Ga, Do]):

lemma 1 . Let K be a nonempty, convex and weakly compact subset of X which is minimal for the nonexpansive mapping T: K → K. If [W] is any nonempty, closed and convex subset of [K] which is invariant under [T], then

sup {∥[w_{n}] − x∥: [w_{n}] ∈ W } = diam(K)

for every x ∈ K.

It is shown in [Ga] that if (x_{n}) ∈ l_{∞}(X) and (y_{n}) ∈ C_{0}(X) then D(x_{n}) = D(x_{n} + y_{n}) and so we can define D[x_{n}] = D(x_{n}). The next lemma is also a result from [Ga]:

lemma 2 Let X be a Banach space and 0 ≤ t ≤ 1. Then

[W] = {[z_{n}] ∈ [X]: D[z_{n}] ≤ t}

is a closed subset of [W].

2 The results

To state our main result we need some definitions. Recall that a Banach space X has the Uniform Opial Property [P3] 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 modulus of Opial [LTX] 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}.

Since the well known inequality (see [X])

c − 1 ≤ r_{X}(c) ≤ c

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

theorem 1 Let X be a Banach space with norm ∥ · ∥ and denote by r_{X} its modulus of Opial. Let | · | 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 inequation

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

then Y has the fixed point property.

Proof

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 there exists a nonempty, convex and weakly compact set K ⊂ X and a fixed point free | · |-nonexpansive mapping T: K → K such that K is minimal for T. Since T has no fixed point in K, then d = diam(K) > 0 and we can suppose that d = 1. Let (x_{n}) be an afps for T in K. Since every subsequence of (x_{n}) is again an afps for T, we can suppose that (x_{n}) is an afps which is weakly convergent and then, by translation of K, that K is minimal for T, 0 ∈ K and that (x_{n}) is weakly null.

Consider the subset [W] of [X] defined by

[W] = {[z_{n}] ∈ [K]: |[z_{n}] − [x_{n}]| ≤ 1 − t and D[z_{n}] ≤ t}

where t = \frac{1}{1+a}.

It is easy to see that [W] is a nonempty, closed, convex and [T]-invariant set. Since 0 ∈ K, we have, by Lemma 1, that

sup {|[w_{n}]|: [w_{n}] ∈ [W]} = 1.

Let [z_{n}] be any element of [W]. Since K is weakly compact, there exists a subsequence (z_{n_{k}}) of (z_{n}) such that

|[z_{n}]| = lim sup |z_{n}| = lim |z_{n_{k}}|

and such that (z_{n_{k}}) converges weakly to y ∈ K. By passing to subsequences, we can assume that the following limits exist:

lim ∥z_{n_{k}} − y∥ lim ∥z_{n_{k}} − y − x_{n_{k}}∥ lim ∥x_{n_{k}} − z_{n_{k}}∥ lim ∥z_{n_{k}}∥ .

Fix k ∈ N. Since the sequence (z_{n_{k}} − z_{n_{r}}) converges weakly to z_{n_{k}} − y, then

|z_{n_{k}} − y| ≤ lim inf_{r} |z_{n_{k}} − z_{n_{r}}|

so that

lim sup_{k} |z_{n_{k}} − y| ≤ D(z_{n_{k}}) ≤ D(z_{n}) ≤ t .

Hence, we have that

lim ∥z_{n_{k}} − y∥ ≤ lim sup_{k} |z_{n_{k}} − y| ≤ t .

Since (x_{n_{k}}) is an afps for T, we have that lim |x_{n_{k}}| = 1, by the Goebel-Karlovitz lemma, and then

1 = lim |x_{n_{k}}| ≤ lim sup |x_{n_{k}} − z_{n_{k}} + y| + lim sup |z_{n_{k}} − y| ≤
≤ lim sup |x_{n_{k}} − z_{n_{k}} + y| + t .

¿From this we get that

lim ∥x_{n_{k}} − z_{n_{k}} + y∥ ≥ \frac{1}{B} lim sup |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 ∥u_{k}∥ ≥ 1. By the definition of r_{X} we have that

lim inf ∥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 that

lim inf ∥u_{k} + \frac{B}{1 − t}y∥ = \frac{B}{1 − t} lim ∥z_{n_{k}} −x_{n_{k}}∥ ≤ \frac{B}{1 − t} lim sup |z_{n_{k}} −x_{n_{k}}| ≤ B .

Hence, we have that B − 1 ≥ r_{X} (\frac{B}{1−t}∥y∥) and then ∥y∥ ≤ \frac{1−t}{B}C(B). Pick any ε > 0. Since lim ∥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 ∥\frac{y_{k}}{t + ε}∥ = lim ∥\frac{y_{k} − y}{t + ε} + \frac{y}{t + ε}∥ ≤ R (a\frac{C(B)}{B}, X) .

and, since ε > 0 is arbitrary,

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

Finally, we get that

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

which contradicts (1).

corollary 1 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 that r_{X}(c) = \sqrt{1 + c^{2}} − 1 for all c ≥ 0 and then C(B) = \sqrt{B^{2} − 1}. On the other hand,

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

for all a ≥ 0 and then

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\} ,

i.e. 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} = \frac{\sqrt{2}}{2}. Then a solution of the above inequation 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 a of a 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 that 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.

Moreover, for p > 2

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

Therefore, 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. By arguments from elementary Calculus, it is straightforward to see that a_{p} is the unique maximum of F_{p}. One can search also an interval for this maximum in the case 1 < p < 2, but we do not repeat this for shake of brevity.

corollary 2 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 (X, | · |) has the WFPP.

Proof

Since (see [X]) r_{ℓ_{p}}(c) = (1 + c^{p})^{\frac{1}{p}} − 1 for all c ≥ 0, then C_{ℓ_{p}}(B) = C(B) = (B^{p} − 1)^{\frac{1}{p}}. On the other hand, (see [Do]), 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

i.e.

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

which 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 takes its maximum value at some a_{p} > \frac{1}{2}. Then a solution of the above inequation is every B > 1 such that

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

and this happens if B < c_{p}.

Remark 1 .

The effective computation of the constant c_{p} in the above corollary seems to be hard, spcecially if p is not a natural number. Nevertheles, 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 fournishes bounds of stability in ℓ_{p} which are greater than M(ℓ_{p}), at least for p < 6.

Remark 2 .

Let X be a Banach space with Opial modulus r_{X}. In [LTX] 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}}

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

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

Therefore, for the 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 our theorem is M(X).

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

As a counterpart of the above remark we can state the following result.

corollary 3 Let X be a Banach space. If r_{X}(1) > 0 then X has the weak fixed point property.

Proof

Since r_{X} is a continuous function, there exists c_{0} < 1 such that r_{X}(c_{0}) > 0. By definition we obtain

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

Now, we can apply the theorem for B = 1. Since for every a > 0 one has R(aC_{X}(1), X) ≤ 1 + aC_{X}(1), we have

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

≥ sup \{\frac{1 + a}{1 + aC_{X}(1)} : a ≥ 0\} ≥ sup \{\frac{1 + a}{1 + ac_{0}} : a ≥ 0\} = \frac{1}{c_{0}} > 1.

Notice that in the Theorem 3.3. of [Do] has been proved that r_{X^{∗}}(1) > 0 implies M(X) > 1, for reflexive Banach space, in some sense, a dual version of last Corollary.

Acknowledgement. The authors would like to thank Professor T. Domínguez Benavides for his comments on the first version of this paper, mainly for the result of Corollary 3.

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