This is a reprint of the paper

On the stability of the fixed point property in l_{p} spaces

by Mohamed A. Khamsi

published in Rev. Colombiana Mat. 28 (1994), pp. 1–6

ON THE STABILITY OF THE FIXED

POINT PROPERTY IN l_{p} SPACES

Mohamed A. Khamsi

The University of Texas at El Paso

Abstract. We prove in this work that for any p > 1 there exists an improved constant c_{p}, such that if d(X, l_{p}) < c_{p} then X has the fixed point property. Key words and phrases. Fixed point property, nonexpansive mapping, normal structure 1991 Mathematics Subject Classification. Primary 47H10, Secondary 46B20.

1. Introduction

Let X be a Banach space and let K be a nonempty weakly compact convex subset of X. We will say that K has the fixed point property (f.p.p. for short) if every nonexpansive T : K → K (i.e., ∥T x − T y∥ ≤ ∥x − y∥ for every x, y ∈ K) has a fixed point, i.e., there exists x ∈ K such that T (x) = x. We will say that X has the f.p.p. if every weakly compact convex subset of X has the f.p.p.

The fixed point property, as stated above, originated in four papers which appeared in 1965. They mainly assert that the presence of a geometric property, called “normal structure”, implies the f.p.p., [10]. A number of abstract results were brought to light, along with important discoveries related both to the structure of the fixed point sets and to techniques for approximating fixed points. The first negative result to the existence part of the theory goes to Alspach, cf. [2], who gave an example of a weakly compact convex subset K of L^{1} and an isometry T : K → K which fails to have a fixed point. This example showed that further assumptions are needed in addition to weak compactness,

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and at the same time it set the stage for Maurey’s surprising discovery, [12] (see also [8], [11]).

For more on f.p.p., one can consult [1], [5].

2. Notations, definitions and basic facts.

Let K be a nonempty weakly compact convex subset of a Banach space X. Suppose that T : K → K is nonexpansive. By Zorn’s lemma, K contains a closed nonempty convex subset K_{0} which is minimal for T. This means that T (K_{0}) ⊂ K_{0} and that no strictly smaller closed nonempty convex subset of K_{0} is invariant under T. A classical argument shows that any closed nonempty convex subset of K which is invariant under T contains an approximate fixed point sequence (a.f.p.s.) (x_{n}), i.e., lim_{n→∞} ∥x_{n} − T (x_{n})∥_{X} = 0.

The following lemma, cf. [4], [7], proved to be fundamental for the study of the f.p.p.

Lemma 1. Suppose K_{0} is a minimal weakly compact convex set for T and (x_{n}) is an a.f.p.s. for T. Then for all x ∈ K_{0} we have

lim_{n→n∞} ∥x_{n} − x∥ = diam(K_{0}).

Since we will be using Maurey’s technique in proving our main result, we recall some basic definitions and facts.

Definition 1. Let X be a Banach space and let U be a free ultrafilter over N. The ultraproduct X̃ of X is the quotient space of

l_{∞}(X) = {(x_{n}) | x_{n} ∈ X for all n ∈ N and ∥(x_{n})∥_{∞} = sup_{n} ∥x_{n}∥ < ∞},

by

N = {(x_{n}) ∈ l_{∞}(X) | lim_{n→U} ∥x_{n}∥ = 0}.

For (x_{n}) ∈ l_{∞}(X), we will denote (x_{n}) + N by (x_{n})_{U} ∈ X̃. Clearly we have

∥(x_{n})_{U}∥X̃ = lim_{n→U} ∥x_{n}∥.

It is also clear that X is isometric to a subspace of X̃ by the mapping x → (x, x, …). Hence, we will see X as a subspace of X̃ and we will write x̃, ỹ, z̃ for the general elements of X̃ and x, y, z for those of X.

Let K and T be as described above. Define K̃ and T̃ by

K̃ = {x̃ ∈ X̃ | there exists a representative (x_{n}) of x̃ with x_{n} ∈ K for n ≥ 1}

ON THE STABILITY OF THE FIXED POINT PROPERTY IN l_{p} SPACES 3

and T̃(x̃) = (T (x_{n}))_{U} for any x̃ ∈ K̃. Then K̃ is a bounded closed convex subset of X̃ and T̃(K̃) ⊂ K̃. We remark that K̃ is not minimal for T. Indeed, if (x_{n}) ⊂ K is an a.f.p.s. then T̃(x̃) = x̃, where x̃ = (x_{n})_{U}. On the other hand, if x̃ = (x_{n})_{U} with x_{n} ∈ K and T̃(x̃) = x̃, then there exists a subsequence (x′_{n}) of (x_{n}) which is an a.f.p.s. for T. Finally we recall that if K_{0} is a minimal set for T and x̃ is a fixed point for T in K̃_{0}, then for any x ∈ K_{0} we have, from Lemma 1, that

∥x̃ − x∥X̃ = diam(K_{0}).

The next lemma was proved by Maurey in [12].

Lemma 2. Suppose x̃ and ỹ are two fixed points of T̃ in K̃. Then for every r ∈ (0, 1), there exists a fixed point z̃ of T̃ so that

∥x̃ − z̃∥ = r∥x̃ − ỹ∥ and ∥ỹ − z̃∥ = (1 − r)∥x̃ − ỹ∥.

3. Main result

Let p ∈ (1, ∞) and consider the function defined on [0,1] by

φ_{p}(x) = \frac{1 + (1 − x)^{p}}{x^{p} + (1 − x)^{p}} .

Then sup_{x∈[0,1]} φ_{p}(x) = φ_{p}(x_{p}) where x_{p} is the only root of

(1 − x^{p−1})(1 − x)^{p−1} − x^{p−1} = 0

in [0,1]. It can be easily proved that

lim_{p→1} x_{p} = 2, lim_{p→∞} x_{p} = 2 and φ_{p}(x_{p}) = \frac{1}{x_{p}^{p−1}} .

Also one can check that x_{p} < _{\frac{1}{\frac{1}{2^{p−1}}}} .

Now recall that the Banach–Mazur distance between two isomorphic Banach spaces X and Y, denoted d(X, Y), is infimum of ∥U∥ ∥U^{−1}∥ taken over all bicontinuous linear operators U from X onto Y.

We now state and prove the main result of this work.

Main Theorem. Let X be a Banach space such that

d(X, l_{p}) < c_{p} = φ_{p}(x_{p})^{\frac{1}{p}}

for some p > 1. Then X has the f.p.p.

Proof. It is enough to prove that X = (l_{p}, | · |) has the f.p.p., where | · | is an equivalent norm to ∥ · ∥_{p} satisfying

∥ · ∥_{p} ≤ | · | ≤ d ∥ · ∥_{p}

with d < c_{p}.

Assume on the contrary that X fails to have the f.p.p. Then there exist K, a nonempty weakly compact convex subset, and T : K → K, a nonexpansive map, with no fixed point. Without any loss of generality, we can assume that K is minimal for T and that diam(K) = 1. Classical arguments imply that K contains an a.f.p.s. (x_{n}) which can be assumed to be weakly convergent to 0. Passing to subsequences, if needed, we may suppose that there exist coordinate projections P_{F_{n}} on X (with respect to the canonical Schauder basis of l_{p}) such that

(1) F_{n} ∩ F_{m} = ∅ for n ≠ m,

(2) lim_{n→∞} |x_{n} − P_{F_{n}}(x_{n})| = 0,

(3) lim_{n→∞} |x_{n} − x_{n+1}| = 1.

The subsets (F_{n}) can be chosen to be successive intervals, and (3) follows from Lemma 1. Put u_{n} = P_{F_{N}} (x_{n}) for all n ∈ N. Then for z ∈ l_{p} we have

∥z∥_{p}^{p} + ∥z − u_{n} − u_{n+1}∥_{p}^{p} = ∥z − u_{n}∥_{p}^{p} + ∥z − u_{n+1}∥_{p}^{p}.  (∗)

Let X̃ be an ultraproduct of X and K̃, T̃ be as defined in the previous section. Set x̃ = (x_{n})_{U} and ỹ = (x_{n+1})_{U}. Then

x̃ = (u_{n})_{U} and ỹ = (u_{n+1})_{U}.

Relation (∗) translates into X̃ as

∥z̃∥_{p}^{p} + ∥z̃ − x̃ − ỹ∥_{p}^{p} = ∥z̃ − x̃∥_{p}^{p} + ∥z̃ − ỹ∥_{p}^{p}  (∗∗)

for every z̃ ∈ X̃. Let r ∈ (0, 1), and let z̃ be a fixed point of T̃, as given by Lemma 2, such that

|z̃ − x̃| = r and |z̃ − ỹ| = 1 − r.

Then,

∥z̃ − x̃ − ỹ∥_{p}^{p} ≥ \frac{1}{d^{p}} |z̃ − x̃ − ỹ|^{p} ≥ \frac{1}{d^{p}} (1 − |z̃ − x̃|)^{p} ≥ \frac{1}{d^{p}} (1 − r)^{p}.

Hence,

\frac{1}{d^{p}} |z̃|^{p} + \frac{1}{d^{p}} (1 − r)^{p} ≤ ∥z̃∥_{p}^{p} + ∥z̃ − x̃ − ỹ∥_{p}^{p} ≤ |z̃ − x̃|^{p} + |z̃ − ỹ|^{p} = r^{p} + (1 − r)^{p}.

Then

|z̃|^{p} ≤ d^{p}(r^{p} + (1 − r)^{p}) − (1 − r)^{p},

and since |z̃| = 1, we get φ_{p}(r) ≤ d^{p}. Since r was arbitrary in (0,1), we deduce that

sup_{r∈(0,1)} φ_{p}(r) = φ_{p}(x_{p}) ≤ d^{p},

which contradicts our assumption on d. The proof of the main theorem is therefore complete. □

Remarks

(1) It is known [3], that if d(X, l_{p}) < 2^{\frac{1}{p}} then X has the normal structure property and therefore, via a Theorem of Kirk [10], has the f.p.p. If d(X, l_{p}) = 2^{\frac{1}{p}} then Bynum [3] has proved that X has the f.p.p. He also gave an example of a situation where X fails to have normal structure. For

p > \frac{ln(2)}{ln (\frac{\sqrt{33}−3}{2})},

one has to use the result of Lin [11], to prove that if d(X, l_{p}) < \frac{\sqrt{33}−3}{2} then X has the f.p.p. It is worth mentioning that

c_{p} ≥ c_{2} = \frac{1}{\sqrt{x_{2}}} = (\frac{3 + \sqrt{5}}{2})^{\frac{1}{2}}

for every p > 1 and c_{2} > \frac{\sqrt{33}−3}{2}. Therefore we get through the main theorem an improvement to all the well known results.

(2) It is a surprising fact that the constants (c_{p}) do no decrease as p goes to ∞. On the contrary, for p ≥ 2 the constants c_{p} increase to 2, which by itself project new light on the stability of the fixed point property (for the l_{p} spaces).

(3) For p = 2 the main theorem reduces to the main result of [6].

References

1. A.G. Aksoy and M.A. Khamsi, Nonstandard Methods in Fixed point theory, Springer– Verlag, Heidelberg, New York, 1990.

2. D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423–424.

3. W.L. Bynum, Normal structure coefficients for Banach spaces, Pac. J. Math. 86 (1980), 427–436.

4. K. Goebel, On the structure of minimal invariant sets for nonexpansive mappings, Ann. Univ. Mariae Curie–Sklodowska 29 (1975), 73–77.

5. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory, to appear in Cambidge University Press.

6. A.M. Jimenez and E.F. Llorens, On the stability of the fixed point property for nonexpansive mappings, preprint.

7. L.A. Karlovitz, Existence of a fixed point for a nonexpansive map in a space without normal structure, Pac. J. Math. 66 (1966), 153–159.

8. M.A. Khamsi, La proprite du point fixe dans les espaces de Banach avec base inconditionelle, Math. Annalen 277 (1987), 727–734.

9. M.A. Khamsi, Normal structure for Banach spaces with Schauder decomposition, Canad. Math. Bull. 32(3) (1989), 344–351.

10. W.A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly 72 (1965), 1004–1006.

11. P.K. Lin, Unconditional bases and fixed points of nonexpansive mapping, Pac. J. Math. 116 (1985), 69–76.

12. B. Maurey, Point fixes des contractions sur un convex fermé de L^{1}, École Polytechnique, 1980.

(Recibido en julio de 1993)

Mohamed A. Khamsi

Department of Mathematical Sciences The University of Texas at El Paso El Paso, Texas 79968-0514, USA e–mail: khamsi@math-1.sci.kuniv.edu.kw mohamed@banach.math.ep.utexas.edu.co