METRIC FIXED POINT RESULTS CONCERNING MEASURES OF NONCOMPACTNES

T. DOMÍNGUEZ, M.A. JAPÓN AND G. LÓPEZ

Contents

1. Preface

Fixed Point Theory has two main branches: on the one hand we can consider the results that are deduced from topological properties and on the other hand those which can be obtained using metric properties.

Among the former the most important result is Brouwer’s theorem, which assures that the euclidean unit ball \overline{B}_{n}(0, 1) in ℝ^{n} has the fixed point property for continuous functions, that is, if f : \overline{B}_{n}(0, 1) → \overline{B}_{n}(0, 1) is a continuous mapping, there exists a point x_{0} ∈ \overline{B}_{n}(0, 1) such that f(x_{0}) = x_{0}.

Schauder [?] generalized the Brouwer fixed point theorem to infinite dimensional Banach spaces for the case of compact operators. Compactness plays an essential role in the proof of the Schauder fixed point theorem. However, there are some important problems where the operators are not compact.

The first step to extend the Schauder theorem to the setting of noncompact operators was given by G. Darbo [?]. The main idea is to define a new class of operators which map any bounded set in a “more compact” set. In order to state the property: A set is mapped into a “more compact” set, we need to define some “measure of noncompactness”. The first such measure was defined by Kuratowski [?] in connection with certain problems of General Topology.

Probably the best known and important metric fixed point theorem is the Banach contraction principle, which assures that every contraction from a complete metric space into itself has a unique fixed point. A mapping T from a metric space (X, d) into itself is said to be a contraction if there exists k ∈ [0, 1) such that d(T x, T y) ≤ kd(x, y) for every x, y ∈ X. Simple examples show that the Banach theorem does not hold letting k to be equal to 1. Such mappings are called nonexpansive. However in the sixties Browder [?], Göhde [?] and Kirk [?] proved that under suitable geometric conditions nonexpansive mappings have the fixed point property.

Although historically the two branches of the fixed point theory have had a separated development, in recent years measures of noncompactness have also been used to define new geometrical properties of Banach spaces which are interesting for fixed point theory for nonexpansive mappings (see for instance [?], [?], [?], [?], [?] and references therein).

Our goal in this chapter is to study the use of measures of noncompactness in metric fixed point theory.

2. Kuratowski and Hausdorff measures of noncompactness

In this section we define the Kuratowski and Hausdorff measures of noncompactness (MNCs) and study their basic properties. As we know, if B is a bounded non precompact set, there is a number ε > 0 such that B cannot be covered by finitely many sets with diameter ≤ ε. Hence, we can give the following definition:

Definition 2.1. Let (X, d) be a complete metric space and B the family of bounded subsets of X. For every B ∈ B we define the Kuratowski measure of noncompactness α(B) of the set B as the infimum of the numbers d such that B admits a finite covering by sets of diameter smaller than d.

Remarks:

1) As usual, the diameter of a set B is the number sup{d(x, y) : x ∈ B, y ∈ B} denoted by diam(B), with diam(∅) = 0. It is clear that 0 ≤ α(B) ≤ diam(B) < +∞ for each nonempty bounded subset B of X and that diam(B) = 0 if and only if B is an empty set or consists of exactly one point.

2) The following properties are satisfied in any complete metric space and are a direct consequence of the definition:

(a) α(B) = 0 if and only if B is precompact.

(b) α(B̄) = α(B) for all B ∈ B.

(c) α(B_{1} ∪ B_{2}) = max{α(B_{1}), α(B_{2})}, ∀B_{1} ∈ B, ∀B_{2} ∈ B.

(d) B_{1} ⊂ B_{2} implies α(B_{1}) ≤ α(B_{2}), ∀B_{1} ∈ B, ∀B_{2} ∈ B.

(e) α(B_{1} ∩ B_{2}) ≤ min{α(B_{1}), α(B_{2})}, ∀B_{1} ∈ B, ∀B_{2} ∈ B.

(f) If B is a finite set, then α(B) = 0.

The α-measure of noncompactness was introduced by Kuratowski in order to generalize the Cantor intersection theorem

Theorem 2.2. Let (X, d) be a complete metric space and {B_{n}} be a decreasing sequence of nonempty, closed and bounded subsets of X and lim_{n→∞} α(B_{n}) = 0. Then the intersection B_{∞} of all B_{n} is nonempty and compact.

Proof. Let {x_{n}} be a sequence such that x_{n} ∈ B_{n} for all n ∈ ℕ and consider the decreasing sequence of sets {C_{n}} given by C_{n} = {x_{i} : i ≥ n}. Obviously C_{n} ⊂ B_{n} and α(C_{1}) = α(C_{n}) ≤ α(B_{n}) for every n ∈ ℕ.

Since lim_{n→∞} α(B_{n}) = 0, it follows that α(C_{1}) = 0 and so \{x_{n}\} is a precompact set. Let \overline{x} be the limit of a subsequence of \{x_{n}\}. Obviously \overline{x} ∈ B_{n} for all n ∈ ℕ and hence B_{∞} ≠ ∅.

Moreover, as α(B_{∞}) ≤ α(B_{n}) for every n ∈ ℕ and lim_{n→∞} α(B_{n}) = 0, we obtain α(B_{∞}) = 0 and so B_{∞} is compact since it is a closed set.

Horvath [?] has proved the following generalization of the Kuratowski theorem.

Theorem 2.3. Let (X, d) be a complete metric space and {B_{i}}_{i∈I} be a family of nonempty of closed and bounded subsets of X having the finite intersection property. If inf_{i∈I} α(B_{i}) = 0 then the intersection B_{∞} of all B_{i} is nonempty and compact.

Proof. Given n > 1 there exists i(n) ∈ I such that α(B_{i(n)}) < 1n. Let A_{k} = ∩_{n=1}^{k}B_{i(n)}, then {A_{k}} is a decreasing sequence of nonempty, closed and bounded subsets of X and lim_{n→∞} α(A_{k}) = 0. It follows by the Kuratowski theorem that K = ∩_{k≥1}A_{k} is a nonempty compact subset of X.

Set B̂_{i} = B_{i} ∩ K for each i ∈ I. It will be enough to prove that the family {B̂_{i}}_{i∈I} has the finite intersection property. In this situation, by compactness, ∩_{i∈I} B̂_{i} is nonempty, and since ∩_{i∈I} B̂_{i} ⊂ ∩_{i∈I}B_{i} the proof would be finished.

Let J be any non-empty finite subset of I and define

B_{J,1} = ∩_{i∈J}B_{i}, B_{J,k} = ∩_{i∈J}(B_{i} ∩ A_{k−1}) if k ≥ 2.

By the Kuratowski theorem ∩_{k≥1}B_{J,k} is nonempty and compact. But ∩_{k≥1}B_{J,k} = ∩_{i∈J}B_{i} ∩_{k≥1} A_{k} = ∩_{i∈J}B_{i} ∩ K. So the family {B̂_{i}}_{i∈I} has the finite intersection property and the proof is concluded.

Corollary 2.4. Let (X, d) be a complete metric space and f : X → ℝ a lower semicontinuous function such that inf_{x∈X} α{y ∈ X : f(y) ≤ f(x)} = 0. Then f is bounded from below and there exists x_{0} ∈ X such that f(x_{0}) = inf_{x∈X} f(x).

Proof. Let B_{x} = {y ∈ X : f(y) ≤ f(x)}, the family {B_{x}}_{x∈X} is in the hypothesis of the previous theorem. If x_{0} ∈ ∩_{x∈X}B_{x}, then f(x_{0}) ≤ f(x) for all x ∈ X.

Corollary 2.5. Let (X, d) be a complete metric space and g : X → X a function such that: 1) x → d(x, g(x)) is lower semicontinuous.

2) inf_{x∈X} d(x, g(x)) = 0.

3) inf_{x∈X} α{y ∈ X : d(y, g(y)) ≤ d(x, g(x))} = 0.

Then there exists x_{0} ∈ X such that g(x_{0}) = x_{0}.

In the framework of linear spaces additional properties can be deduced for the Kuratowski measure of noncompactness. We omit their well known proofs (see for instance [?]).

Proposition 2.6. Let X be a Banach space, then we have:

(a) α(tB) = |t|α(B) for any real number t and B ∈ B.

(b) α(B_{1} + B_{2}) ≤ α(B_{1}) + α(B_{2}), ∀B_{1} ∈ B, ∀B_{2} ∈ B.

(c) α(x_{0} + B) = α(B) for any x_{0} ∈ X and B ∈ B.

(d) α(B) = α( co(B)).

Associated to the notion of measure of noncompactness we can consider the class of condensing operators which generalize the notion of compact operator.

Definition 2.7. Let D be a nonempty subset of a metric spaces X. A mapping T : D ⊂ X → X is an α-condensing mapping if T is continuous and satisfies that α(T (A)) < α(A) for every bounded subset A of D with α(A) > 0.

For the class of compact operators we have the following generalization of Schauder theorem, usually called Darbo and Sadovskiĭ Theorem [?] and [?].

Theorem 2.8. Suppose M is a nonempty bounded closed and convex subset of a Banach space and suppose T : M → M is condensing. Then T has a fixed point.

Proof. Fix m ∈ M and denote by Σ the family of all closed and convex subsets K of M such that m ∈ K and T (K) ⊂ K. Now set

B = ⋂_{K∈Σ} K

and let

C = \overline{co}(T (B) ∪ \{m\}).

Obviously Σ ≠ ∅, m ∈ B and T : B → B.

Moreover B = C. Indeed, since m ∈ B and T (B) ⊂ B, it follows that C ⊂ B. This implies T (C) ⊂ T (B) ⊂ C and so C ∈ Σ, and hence B ⊂ C. Using now the properties of α we obtain α(B) = α(T (B)). Since T is α-condensing, it follows that B is compact. Therefore T is a continuous mapping of the compact convex set B into itself. By Schauder theorem T must have a fixed point.

In the definition of the Kuratowski measure we can consider balls instead of arbitrary sets. In this way we have the definition of the Hausdorff measure.

Definition 2.9. The Hausdorff measure of noncompactness χ(B) of the set B is the infimum of the numbers r such that B admits a finite covering by balls of radius smaller than r.

Remarks:

3) Recall that in a Banach space X a set S ⊂ X is called an r-net of B if B ⊂ S + r\overline{B}(0, 1) = \{s + rb : s ∈ S, b ∈ \overline{B}(0, 1)\}. So, the definition of the χ-measure in Banach spaces is equivalent to the following:

χ(B) = inf{r > 0 : B has a finite r − net}.

4) The Hausdorff MNC enjoys the same properties as the Kuratowski MNC, appearing in Remarks 2, Proposition 2.6 and Theorems 2.2, 2.3 and 2.8. However there is not a direct relationship between the Kuratowski and the Hausdorff MNCs as we will see in the following results.

Theorem 2.10. Let B(0, 1) be the unit ball in a Banach space X. Then α(B(0, 1)) = χ(B(0, 1)) = 0 if X is finite dimensional, and α(B(0, 1)) = 2, χ(B(0, 1)) = 1 otherwise.

Proof. If X is a finite dimensional Banach space, the result follows from the compactness of the unit ball.

It remains to consider the infinite dimensional case. We first prove the result for χ. Obviously χ(B(0, 1)) ≤ 1. Suppose χ(B(0, 1)) = r < 1. Let us choose ε > 0 such that r + ε < 1. Then there exist x_{1}, x_{2},..., x_{m} in X such that

B(0, 1) ⊂ ⋃_{k=1}^{m} B (x_{k}, (r + ε)) = ⋃_{k=1}^{m} (x_{k} + (r + ε)B(0, 1)).

¿From the properties of the MNC χ, it follows that

r = χ(B(0, 1)) ≤ (r + ε)χ(B(0, 1)) = r(r + ε)

and this implies r = 0. So χ(B(0, 1)) = 0 and hence B(0, 1) is precompact. This contradicts the infinite dimensionality of the space X. Therefore, χ(B(0, 1)) = 1.

To prove the result for α we make use of the Borsuk-Lyusternik-Shnirel’man theorem on antipodes (see [?]):

“If S_{n}(0, 1) is the unit sphere in an n-dimensional normed space and A_{k} (k = 1, ..., n) is a cover of S_{n}(0, 1) by closed subsets of that space, then at least one of the sets A_{k} contains a pair of diametrically opposite points, that is, diam(A_{k}) ≥ diam(S_{n}(0, 1))”.

Since diam(B(0, 1)) = 2, it is obvious that α(B(0, 1)) ≤ 2. Suppose α(B(0, 1)) < 2. Then we can find a finite number of closed subsets {B_{1}, B_{2}, ..., B_{n}} of X with diam(B_{k}) < 2 for all k = 1, ..., n such that B(0, 1) ⊂ ⋃_{k=1}^{n} B_{k}. Now, taking the section of B(0, 1) with an arbitrary ndimensional subspace X_{n} and setting A_{k} = B_{k} ∩ X_{n}, we obtain a contradiction with the theorem on antipodes.

Since α and χ are invariant under passage to the convex hull, we obtain the following corollary:

Theorem 2.11. Let S(0, 1) be the unit sphere in a Banach space X. Then α(S(0, 1)) = χ(S(0, 1)) = 0 if X is finite dimensional, and α(S(0, 1)) = 2, χ(S(0, 1)) = 1 otherwise.

Theorem 2.12. The Kuratowski and Hausdorff MNCs are related by the inequalities

χ(B) ≤ α(B) ≤ 2χ(B).

In the class of all infinite dimensional Banach spaces these inequalities are the best possible.

Proof. The inequalities are obvious from the definitions of α and χ. The sharpness of the second inequality follows from Theorem 2.10. The following example shows that the first inequality is also sharp. Let B = {e_{k} : k ≥ 1} the set of standard basis vectors in c_{0}. Since for all i ≠ j, ∥e_{i} − e_{j}∥ = 1, we have α(B) = 1. On the other hand, χ(B) = 1 because the distance from any infinite subset of B to any element of c_{0} cannot be smaller than 1.

Remark 5 ) Even though in general α and χ are different MNCs, in some Banach spaces we can find a close relation between them. This is, for instance, the case of hyperconvex spaces (see chapter by Khamsi-Espinola).

3. ϕ-minimal sets and the separation measure of noncompactness

The notion of a ϕ-minimal set, where ϕ is the α or χ MNCs, was introduced in [?] in order to study the relationships between condensing mappings for Kuratowski and Hausdorff’s measures of noncompactness. The notion has become very useful to study some properties of the separation measure of noncompactness, to obtain simpler forms for some moduli of noncompact convexity and to simplify several proofs.

Definition 3.1. Let X be a metric space and B the family of all bounded subsets of X. An infinite subset A ∈ B is said to be minimal for the measure ϕ (or, in short, ϕ-minimal) if ϕ(A) = ϕ(B) for every infinite subset B of A.

Examples:

(1) Every infinite precompact set is obviously ϕ-minimal.

(2) In particular, every Cauchy sequence with infinite range is ϕ-minimal.

(3) Every infinite subset of a ϕ-minimal set is a ϕ-minimal set.

(4) The standard bases in ℓ^{p} or c_{0} are minimal sets.

(5) A nonprecompact convex subset of a Banach space fails to be ϕ-minimal.

We are now going to prove the existence of ϕ-minimal sets in bounded sets.

Theorem 3.2. Let X be a bounded metric space. Then:

(a) There is a subset A of X such that A is ϕ-minimal.

(b) If X is not a precompact set, A can be chosen such that ϕ(A) > 0.

Proof. Let A_{0} = X; recursively let

ϕ_{n+1} = inf{ϕ(A) : A ⊂ A_{n}, A infinite}

and let A_{n+1} be chosen to be an infinite subset of A_{n} with

ϕ(A_{n+1}) < ϕ_{n+1} + 1n + 1.

Since A_{n} is an infinite set, for every n we can choose a_{n} ∈ A_{n} such that a_{n} ≠ a_{k} for k = 1, 2, ..., n − 1. Let A be the infinite set {a_{n} : n ∈ ℕ}. Then A\A_{n} is finite for each n ∈ ℕ. Let us see that A is ϕ-minimal. Indeed, let A′ be an infinite subset of A. Since A′\A_{n−1} is a finite set for each n > 1, we have

ϕ(A) ≤ ϕ(A_{n}) < ϕ_{n} + 1n ≤ ϕ(A′ ∩ A_{n−1}) + 1n = ϕ(A′) + 1n.

Hence ϕ(A′) = ϕ(A) and A is ϕ-minimal. This argument concludes part (a).

We now assume that ϕ(A) = 0 for every ϕ-minimal subset A of X. Let {x_{n}} be a sequence in X and we assume that the set {x_{n} : n ∈ ℕ} is infinite. Then there exists a ϕ-minimal subsequence {y_{n}} of {x_{n}}. Since ϕ({y_{n} : n ∈ ℕ}) = 0, {y_{n} : n ∈ ℕ} is precompact and so {y_{n}} has a Cauchy subsequence. Thus every sequence in X has a Cauchy subsequence. Hence X is a precompact set and the proof is complete.

Theorem 3.3. Let X be a separable metric space and B a bounded subset of X. Then there exists a χ-minimal subset A of B such that χ(A) = χ(B).

Proof. Without loss of generality, assume that χ(B) > 0. Let {p_{k}} be a dense sequence in X and let {B_{m}} be the sequence formed by all balls centered at each p_{k} with radius a rational positive number less than χ(B). Then for every positive integer n there are infinitely many points which are not in M_{n} = ∪_{m=1}^{n}B_{m}.

Choose x_{n} ∈ B \ M_{n} and denote by A the set {x_{n} : n ∈ ℕ}. For every n there exists k(n) > n such that x_{j} ∈ M_{k(n)} for j = 1, ..., n. Hence x_{k(n)} ≠ x_{j}, j = 1, ...., n. Therefore A is an infinite set.

Let C be an infinite subset of A. For each n ∈ ℕ, C is not contained in M_{n} because ∩M_{n} is a finite set. Now it is easy to prove that χ(C) ≥ χ(B) ≥ χ(A). Thus A is χ-minimal and χ(B) = χ(A)

Remark 6) This result holds for a wide class of spaces. We recall that a Banach space X is said to be weakly compactly generated if it contains a weakly compact set which is linearly dense in X. Obviously, reflexive and separable Banach spaces satisfy this condition. S. Prus ( see [?]) has proved that the previous theorem is also true for weakly compactly generated Banach spaces. However, this result does not hold for ℓ_{∞} (se [?]).

The situation for the Kuratowski MNC is different as we are going to show. We need some technical results

Lemma 3.4. Let A be an α-minimal subset of a metric space (X, d). Then, for every positive number ε, there exists an infinite subset B of A such that

α(A) − ε < d(x, y) < α(A) + ε

for every x ∈ B, y ∈ B, x ≠ y.

Proof. Without loss of generality we can assume that card(A)= χ_{0} and d(x, y) < α(A) + ε for every x, y ∈ A. By Ramsey’s theorem (see, for example, [?], pag. 392) there exists an infinite subset B of A such that d(x, y) ≤ α(A) − ε for every x, y ∈ B or d(x, y) ≥ α(A) − ε for every x, y ∈ B. Since the first possibility contradicts the α-minimality of A and the proof is concluded.

Remark 7) For a proof of the above lemma without using Ramsey’s theorem see ( [?], Lemma 3.1.3).

The following theorem which can be deduced from Lemma 3.4 is known ( see [?], pag 235)

Theorem 3.5. Let {x_{n}} be a bounded sequence in a metric space (X, d). Then there is a subsequence {y_{n}} of {x_{n}} such that lim_{n,m;n≠m} d(y_{n}, y_{m}) exists.

Let {x_{n}} be a bounded sequence in a Hilbert space H. Since H is reflexive, there is a subsequence {y_{n}} of {x_{n}} and a vector v ∈ H such that {y_{n}} is weakly convergent to v. Taking a subsequence if necessary, we can also assume that lim_{n→∞} ∥y_{n} − v∥ exists.

Consider the mapping Φ : H → ℝ : Φ(z) = lim sup_{n→∞} ∥y_{n} − z∥. This map takes its unique absolute minimum in v. This result is a consequence of the following lemma due to Opial [?]:

Lemma 3.6. If in a Hilbert space H the sequence {x_{n}} is weakly convergent to x_{0}, then for any x ≠ x_{0} we have

lim sup_{n→∞} ∥x_{n} − x∥ > lim sup_{n→∞} ∥x_{n} − x_{0}∥.  (1)

Proof. Since every weakly convergent sequence is necessarily bounded, both limits in (1) are finite. Thus, to prove this inequality, it suffices to observe that in the equality

∥x_{n} − x∥^{2} = ∥x_{n} − x_{0}∥^{2} + ∥x_{0} − x∥^{2} + 2(x_{n} − x_{0}) · (x_{0} − x)

the last term tends to zero as n tends to infinity.

Now it is easy to deduce that χ({y_{n} : n ∈ ℕ}) = Φ(v). Indeed, since Φ(v) = lim_{n→∞} ∥y_{n} − v∥ it follows that for every ε > 0 there exists n_{0} ∈ ℕ such that y_{n} ∈ B(v, Φ(v) + ε) for all n ≥ n_{0}, and hence χ({y_{n} : n ∈ ℕ}) ≤ Φ(v).

Conversely, let us suppose that {y_{n} : n ∈ ℕ} can be covered by finitely many balls with radius r < Φ(v). Then there is a ball B(u, r) containing infinitely many elements of this sequence. We still denote the subsequence contained in this ball by {y_{n} : n ∈ ℕ}. This sequence is weakly convergent to v, Φ(v) is still given by lim_{n→∞} ∥y_{n} − v∥ and the function Φ′ : H → ℝ : Φ′(z) = lim sup_{n→∞} ∥y_{n} − z∥ takes its unique absolute minimum in v. Therefore we obtain ∥y_{n} − u∥ ≤ r < Φ(v) for all n ∈ ℕ and thus Φ′(u) = lim sup_{n→∞} ∥y_{n} − u∥ ≤ r < Φ(v) = Φ′(v) contradicting the fact that Φ′ has an absolute minimum at v.

Lemma 3.7. Let {x_{n}} be an α-minimal sequence in a Hilbert space H. Assume that x_{n} ≠ x_{m} if n ≠ m. If {y_{n}}, v and Φ are constructed as above, then

α(\{x_{n} : n ∈ ℕ\}) = α(\{y_{n} : n ∈ ℕ\}) = Φ(v)\sqrt{2} = χ(\{y_{n} : n ∈ ℕ\})\sqrt{2}.

Proof. We only have to prove the equality α(\{y_{n} : n ∈ ℕ\}) = Φ(v)\sqrt{2}.

Let ε be an arbitrary positive number and define η = α({y_{n} : n ∈ ℕ}). By Lemma 3.4 there is a subsequence {z_{n}} of {y_{n}} such that

η − ε ≤ ∥z_{n} − z_{m}∥ ≤ η + ε, ∀n ≠ m.  (2)

Since lim_{n→∞} ∥y_{n} − v∥ = Φ(v), we can choose a positive integer k such that

|∥z_{n} − v∥ − Φ(v)| < ε, n ≥ k.  (3)

Fixing k ∈ ℕ, let n > k be large enough so that |(z_{n} − v) · (z_{k} − v)| < ε. From the identity

∥z_{k} − z_{n}∥^{2} = ∥z_{k} − v∥^{2} + ∥z_{n} − v∥^{2} − (z_{k} − v) · (z_{n} − v) − (z_{n} − v) · (z_{k} − v)

we obtain by (2) and (3)

2(Φ(v) + ε)^{2} + 2ε ≥ (η − ε)^{2} and (η + ε)^{2} ≥ 2(Φ(v) − ε)^{2} − 2ε.

Hence, letting ε → 0 we obtain η = \sqrt{2}Φ(v).

Theorem 3.8. Let H be an infinite dimensional Hilbert space and A an α-minimal subset of the unit sphere S(0, 1). Then

α(A) ≤ \sqrt{2} < 2 = α(S(0, 1)).

Proof. Let {x_{n}} be a sequence in A with x_{n} ≠ x_{m} for all n ≠ m. By Lemma 3.7 we have α(A)\sqrt{2} = inf\{Φ(z) : z ∈ H\}, where Φ(z) = lim sup_{n→∞} ∥y_{n} − z∥ for a subsequence \{y_{n}\} of \{x_{n}\}. Then α(A)\sqrt{2} ≤ Φ(0) = lim sup_{n→∞} ∥y_{n}∥ = 1. On the other hand we know from Theorem 2.11 that α(S(0, 1)) = 2.

The notion of minimal set is strongly related to the separation measure of noncompactness which was first introduced by Istrăţescu [?]. The set B is said to be r-separated if d(x, y) ≥ r for all x, y ∈ B, x ≠ y. The set B will be call an r-separation.

Definition 3.9. The measure of noncompactness β(B) of the subset B of the metric space X is the infimum of the numbers r > 0 for which B does not have an infinite r-separation or, equivalently, the supremun of those r > 0 for which B has an infinite r-separation.

Remarks:

8) It is easy to prove that

β(B) = sup{α(A) : A ⊂ B and A is α-minimal }.

9) The properties of the separation measure of noncompactness and related fixed point theorems has been studied by many authors (see [?]). The permanence under passage to the convex hull was proved independently by J. Arias [?] and Erzakova [?]

4. Moduli of noncompact convexity

A generalization of the moduli of k-uniform convexity, (see chapter by S. Prus), considering infinite dimensional subspaces, was given by Goebel and Sȩkowski in 1984 [?]. The corresponding concept to k-uniformly convex space is, in this case, that of nearly uniformly convex space. Goebel and Sȩkowski use the Kuratowski MNCs as a measure of the volume of the sets. The use of the measure of noncompactness is relevant since it implies a strong connection between topological methods and metric methods.

We are going to start this section with the definition of the moduli of noncompact convexity. Throughout this section ϕ will denote the α, χ or β measure of noncompactness.

Definition 4.1. Let X be a Banach space. We define the modulus of noncompact convexity associated to ϕ in the following way

Δ_{X,ϕ}(ε) = inf\{1 − d(0, A) : A ⊂ \overline{B}_{n}(0, 1) is convex , ϕ(A) > ε\}
= inf\{1 − d(0, co(A)) : A ⊂ \overline{B}_{n}(0, 1), ϕ(A) > ε\}.

We define the NUC-characteristic of X associated to the measure of noncompactness ϕ to be

ε_{ϕ}(X) = sup{ε ≥ 0 : Δ_{X,ϕ}(ε) = 0}.

The function Δ_{X,α}(ε) has been considered by Goebel and Sȩkowski [?], Δ_{X,χ}(ε) by Banaś [?] and Δ_{X,β}(ε) by Domínguez and López [?].

Remarks:

10) A Banach space is NUC if and only if ε_{ϕ}(X) = 0.

11) The following relationships among the different moduli are easy to obtain

δ_{X}(ε) ≤ Δ_{X,α}(ε) ≤ Δ_{X,β}(ε) ≤ Δ_{X,χ}(ε)

and consequently

ε_{0}(X) ≥ ε_{α}(X) ≥ ε_{β}(X) ≥ ε_{χ}(X).

We are going to introduce some equivalent definitions for the moduli Δ_{X,χ}(ε) and Δ_{X,β}(ε) in the case of reflexive spaces. The following technical lemma can be easily proved.

Lemma 4.2. Let X be a Banach space and {x_{n}} a sequence in X weakly convergent to w. Let A_{n} = \overline{co}(\{x_{k}\}_{k≥n}). Then

⋂_{n=1}^{∞} A_{n} = {w}.

Theorem 4.3. Let X be a reflexive Banach. Then

Δ_{X,β}(ε) = inf \{1 − ∥w∥ : \{x_{n}\} ⊂ \overline{B}_{n}(0, 1), x_{n} ⇀ w and Sep(\{x_{n}\}) > ε\}

where

Sep({x_{n}}) = inf{ε : ∥x_{n} − x_{m}∥ ≥ ε : n ≠ m}

and

Δ_{X,χ}(ε) = inf \{1 − ∥w∥ : \{x_{n}\} ⊂ \overline{B}_{n}(0, 1), x_{n} ⇀ w and χ(\{x_{n}\}) > ε\} .

Proof. Both equalities have a similar proof. So we are only going to prove the first one.

First we note that if A ⊂ \overline{B}_{n}(0, 1) with β(A) > ε, there exists a sequence \{x_{n}\} such that Sep({x_{n}}) > ε, {x_{n}} ⊂ A and {x_{n}} ⇀ w. Thus

Δ_{X,β}(ε) = inf\{1 − d( \overline{co}(\{x_{n}\}), 0) : x_{n} ⇀ w, \{x_{n}\} ⊂ \overline{B}_{n}(0, 1) and Sep(\{x_{n}\}) > ε\}

Now using the previous lemma it is easy to deduce the first inequality:

Δ_{X,β}(ε) ≥ inf \{1 − ∥w∥ : x_{n} ⇀ w, \{x_{n}\} ⊂ \overline{B}_{n}(0, 1) and Sep(\{x_{n}\}) > ε\} .

Conversely, let x_{n} ⇀ w with \{x_{n}\} ⊂ \overline{B}_{n}(0, 1) and Sep(\{x_{n}\}) > ε. As

1 − Δ_{X,β}(ε) = sup \{ d(0, \overline{co}(\{y_{n}\}) : \{y_{n}\} ⊂ \overline{B}_{n}(0, 1) Sep(\{y_{n}\}) > ε\} ,

if we take {y_{n}} = {x_{n}}_{n≥k}, we have

1 − Δ_{X,β}(ε) ≥ d(0, \overline{co} (\{x_{n}\}_{n≥k})).

Let η > 0. For all k there exists

z_{k} ∈ \overline{co}(\{x_{n}\}_{n≥k}) = A_{k}

such that

∥z_{k}∥ ≤ 1 − Δ_{X,β}(ε) + η.

Let z_{n_{k}} be a weakly convergent subsequence of \{z_{n}\}, and B_{n} = \overline{co}\{x_{n_{k}}\}_{k≥n}. Then B_{n} ⊂ A_{n}. So if z_{n_{k}} ⇀ z we obtain

z ∈ ⋂_{n=1}^{∞} B_{n} ⊂ ⋂_{n=1}^{∞} A_{n} = {w}.

Thus z_{n} ⇀ w and so

∥w∥ ≤ lim inf_{n→∞} ∥z_{n}∥ ≤ 1 − Δ_{X,β}(ε) + η.

Since the last inequality is true for every η, we obtain

Δ_{X,β}(ε) ≤ 1 − ∥w∥

and again this inequality is true for every x_{n} ⇀ w satisfying \{x_{n}\} ⊂ \overline{B}_{n}(0, 1) and Sep(\{x_{n}\}) > ε. So

Δ_{X,β}(ε) ≤ inf\{1 − ∥w∥ : x_{n} ⇀ w, (x_{n}) ⊂ \overline{B}_{n}(0, 1) Sep(\{x_{n}\}) > ε\}

and the proof is now complete.

Remarks:

12) The function

P_{X}(ε) = inf \{1 − ∥w∥ : x_{n} ⇀ w, \{x_{n}\} ⊂ \overline{B}_{n}(0, 1) and Sep(\{x_{n}\}) > ε\}

has been considered by Partington [?] as a modulus for the UKK property. Theorem 4.3 shows that for reflexive spaces Partington’s modulus is identical to the modulus of noncompact convexity associated to β.

13) Let X be a Banach space and τ a topology on X. Bearing in mind Theorem 4.3 it is natural to consider the following moduli (for more details see [?])

Δ_{X,β,τ} (ε) = inf \{1 − ∥w∥ : τ − lim\{x_{n}\} = w, \{x_{n}\} ⊂ \overline{B}_{n}(0, 1), and Sep(\{x_{n}\}) > ε\}

and

Δ_{X,χ,τ} (ε) = inf \{1 − ∥w∥ : τ − lim\{x_{n}\} = w, \{x_{n}\} ⊂ \overline{B}_{n}(0, 1), and χ(\{x_{n}\}) > ε\} ,

where τ − lim{x_{n}} denotes the limit of a sequence {x_{n}} with respect to the topology τ

Notice that a Banach X has the UKK(τ) property (see definition 8 in Chapter Prus) if and only if Δ_{X,β,τ} (ε) > 0 for all ε > 0.

Example 6. Let (Ω, Σ, μ) be a positive σ-finite measure space, X = (L_{p}(μ), ∥.∥_{p}) and τ the convergence locally in measure topology (clm) ( see Chapter by S. Prus for more details). We are going to prove that:

Δ_{X,β,clm} (ε) = 1 − (1 − (ε)^{p})^{1p} .

for 1 ≤ p < +∞.

We use the following result which can be derived from [?]:

If {f_{n}} is a clm-null sequence in L_{p}(μ), 1 ≤ p < +∞, and f is a function in L_{p}(μ), then

lim sup_{n} ∥f_{n} − f∥_{p}^{p} = ∥f∥_{p}^{p} + lim sup_{n} ∥f_{n}∥_{p}^{p}.  (4)

Let {f_{n}} be a sequence which is clm-convergent to f ∈ L_{p}(μ). Suppose that ∥f_{n}∥_{p} ≤ 1 and Sep({f_{n}}) > ε. Take a subsequence {g_{n}} ⊂ {f_{n}} such that lim_{n} ∥g_{n} − f∥_{p} and lim_{n} ∥g_{n}∥_{p} exist. If we denote h_{n} = g_{n} − f and apply (4) to the sequence {h_{n}} and −f, we obtain:

∥f∥_{p}^{p} = lim_{n} ∥h_{n} + f∥_{p}^{p} − lim_{n} ∥h_{n}∥_{p}^{p} = lim_{n} ∥g_{n}∥_{p}^{p} − lim_{n} ∥g_{n} − f∥_{p}^{p}
≤ 1 − lim_{n} ∥g_{n} − f∥_{p}^{p}.

Denote l = lim_{n} ∥g_{n} − f∥_{p} and let m ∈ ℕ. We have

ε^{p} ≤ lim sup_{n} ∥g_{n} − g_{m}∥_{p}^{p} = lim sup_{n} ∥g_{n} − f − (g_{m} − f)∥_{p}^{p}.

Applying again (4) we obtain

ε^{p} ≤ ∥g_{m} − f∥_{p}^{p} + lim_{n} ∥g_{n} − f∥_{p}^{p}.

Taking the limit when m tends to infinity, we deduce l ≥ ε/2^{1p}. Hence ∥f∥_{p}^{p} ≤ (1 − ε^{p}/2)^{1p} and

Δ_{X,β,clm} (ε) ≥ 1 − (1 − (ε)^{p})^{1p} .

Let us now study the converse inequality.

Suppose Ω = ∪_{n=1}^{∞}Ω_{n} with μ(Ω_{n}) < ∞. Given η > ε consider the function f = aχ_{Ω_{1}} where a^{p} = (1 − η^{P} /2)/μ(Ω_{1}) and the sequence f_{n} = f + b_{n}χ_{Ω_{n}} where b_{n}^{p} = η^{p}/(2μ(Ω_{n})) for n ≥ 2. It is clear that clm − lim_{n} f_{n} = f, ∥f∥_{p} = 1, ∥f_{n} − f_{m}∥_{p} = η and ∥f∥_{p} = (1 − η^{P} /2)^{1p}. Since η is any real number greater that ε, we finally obtain the desired equality.

Using similar arguments ( see [?]) we can also compute the modulus for the case of the measure χ

Δ_{X,χ,clm} (ε) = 1 − (1 − (ε)^{p})^{1p} .

Remarks:

14) Suppose that Ω = ℕ and μ is the counting measure. Then we obtain the space ℓ^{p} and clm-convergence of a sequence is equivalent to weak convergence for bounded sequences if p > 1. So

Δ_{ℓ^{p},β} (ε) = 1 − (1 − (ε)^{p})^{1p} .

and

Δ_{ℓ^{p},χ} (ε) = 1 − (1 − (ε)^{p})^{1p} ,

for 1 < p < +∞.

If p = 1 clm-convergence is equivalent to weak* convergence for bounded sequences if ℓ^{1} is seen as the dual space of c_{0}.

15) Goebel and Sȩkowski [?] computed the value of Δ_{ℓ^{p},α} obtaining

Δ_{ℓ^{p},α}(ε) = 1 − (1 − (ε2)^{p})^{1p}

which, for p ≥ 2, coincides with the value of the Clarkson modulus of convexity (see Chapter by S. Prus ).

5. Fixed point theorems derived from normal structure

In this section we are going to study fixed point results for nonexpansive mappings. Recall that if (C, d) is a metric space, a mapping T : C → C is said to be nonexpansive if d(T x, T y) ≤ d(x, y) for every x, y ∈ C. The framework where we are going to work is the following: Let X be a Banach space and τ an arbitrary topology considered on X. We shall say that X has the fixed point property with respect to τ (τ-FPP), if every nonexpansive mapping defined from a τ-sequentially compact bounded convex subset of X into itself has a fixed point. Notice that according to the Eberlein-Smulian Theorem, which proves the equivalence between weak compactness and weak sequential compactness, the previous definition is equivalent to the usual definition of the weak fixed point property whenever τ is the weak topology. Our main goal in this section is to obtain fixed point results derived from normal structure. We will finish using the moduli of noncompact convexity to deduce fixed point results for nonexpansive mappings.

We shall say that a Banach space X has normal structure with respect to a topology τ (τ-NS), if every τ-sequentially compact bounded convex set C ⊂ X with diam(C) > 0 contains a nondiametral point, i.e., there is x_{0} ∈ C such that sup{∥x − x_{0}∥ : x ∈ C} < diam(C). This notion has been defined previously in Chapter by S. Prus.

Before starting the main theorem of this section we need to prove the following Lemma which can be found essentially in [?], although the proof that we give is from [?]. Recall that the Chebyschev radius of a set C is defined as

r(C) = inf_{x∈C}{sup ∥x − y∥ : y ∈ C}

Lemma 5.1. Let X be a Banach space and C a bounded subset. Let S be a class of subsets of C which is stable under arbitrary intersections and it contains B ∩ C for every closed ball B of X. If T : C → C is a nonexpansive mapping, for every ϵ > 0 there exists C(ϵ) ∈ S such that T : C(ϵ) → C(ϵ) and diam(C(ϵ)) ≤ r(C) + ϵ.

Proof. If diam(C) = 0, take C(ϵ) = C. Otherwise, let ρ = r(C) + ϵ. We are going to construct C(ϵ).

Firstly, by definition of ρ, the set M = {z ∈ C : C ⊂ B(z, ρ)} is a nonempty set. Let

F = {D ∈ S : M ⊂ D, T : D → D}.

Notice that F ≠ ∅ since C ∈ F. Define

L = ⋂_{D∈F}D

Then L ∈ S, M ⊂ L and T : L → L. Let A = M ∪ T (L) . Let us prove that

L = ⋂{D ∈ S : A ⊂ D}.

Since A ⊂ L it is clear that ∩{D ∈ S : A ⊂ D} ⊂ L. This implies that T (∩{D ∈ S : A ⊂ D}) ⊂ T (L) ⊂ A ⊂ ∩{D ∈ S : A ⊂ D}. Also M ⊂ A ⊂ ∩{D ∈ S : A ⊂ D}. Thus the last set belongs to F and contains the set L.

Finally we define

C(ϵ) = {x ∈ L : L ⊂ B(x, ρ)} = { ⋂_{u∈L} B(u, ρ)} ∩ L

Notice that C(ϵ) ≠ ∅ since M ⊂ C(ϵ). On the other hand, C(ϵ) ∈ S because it is an arbitrary intersection of sets in S. It is also clear that diam(C(ϵ)) ≤ ρ. We need to prove that C(ϵ) is T -invariant.

Let x ∈ C(ϵ). Obviously T (x) ∈ L so we have to prove that L ⊂ B(T x, ρ). Firstly, we shall see that A ⊂ B(T x, ρ). If u ∈ M we have u ∈ B(T x, ρ) because C ⊂ B(u, ρ). Otherwise, if u ∈ T (L) there exists y ∈ L such that u = T (y). Hence d(u, T x) ≤ d(y, x) ≤ ρ because x ∈ C(ϵ). Consequently, A ⊂ B(T x, ρ) ∩ L and by the equivalent definition of L we finally deduce that L ⊂ B(T x, L) and C(ϵ) is T -invariant.

Recall that if X is a Banach space and τ is a topology on X, it is said that the norm is τ-sequentially lower semicontinuous (τ-slsc) if ∥x∥ ≤ lim inf_{n} ∥x_{n}∥ whenever τ − lim_{n} x_{n} = x.

Theorem 5.2. Let X be a Banach space and τ a topology on X such that the norm ∥ · ∥ is a τ-slsc function. If X has τ-NS, then X satisfies the τ-FPP.

Proof. Let C be a nonempty τ-sequentially compact bounded convex subset of X and let T : C → C be a nonexpansive mapping.

Denote A the following family of subsets of C:

A = {K ⊂ C : K ≠ ∅ τ-sequentially closed, convex and T : K → K} .

Notice that A ≠ ∅ since C ∈ A. For every K ∈ A we define

δ(K) = inf{diam(F) : F ∈ A, F ⊂ K}

Set K_{1} = C, and with K_{1}, ..., K_{n} given, select K_{n+1} ∈ A such that K_{n+1} ⊂ K_{n} and

diam(K_{n+1}) ≤ δ(K_{n}) + 1n

for every n ∈ ℕ.

Let K_{0} = ∩_{n=1}^{∞}K_{n}. Firstly, we are going to check that K_{0} ≠ ∅: Take x_{n} ∈ K_{n} for every n ∈ ℕ and consider the sequence {x_{n}}. Since C is a τ-sequentially compact set, there exists a subsequence {x_{n_{k}}} ⊂ {x_{n}} which is τ-convergent to a point x. Since the sets K_{n} are τsequentially closed and K_{n+1} ⊂ K_{n}, we know that x ∈ K_{n} for every n ∈ ℕ and so, K_{0} ≠ ∅. On the other hand, K_{0} is convex, τ-sequentially compact and T -invariant.

Define

S = {D ⊂ K_{0} : D is convex and τ-sequentially closed}

Clearly, S ≠ ∅ since K_{0} ∈ S, S is stable under arbitrary intersections and B ∩ C ∈ S for every closed ball in X due to the τ-sequential lower semi-continuity of the norm.

The above lemma implies that for each ϵ > 0 there exists K_{0}(ϵ) ∈ S T -invariant and such that

diam(K_{0}) − 1n ≤ diam(K_{n+1}) − 1n ≤ δ(K_{n}) ≤ diam(K_{0}(ϵ)) ≤ r(K_{0}) + ϵ

Letting n → ∞

diam(K_{0}) ≤ r(K_{0}) + ϵ

Since this is true for each ϵ > 0 we deduce that diam(K_{0}) = r(K_{0}) which implies that K_{0} = {x} since X has τ-NS. Therefore T (x) = x.

Remark 16) Notice that Theorem 5.2 generalizes Kirk’s Theorem [?] for general topologies because the norm is always a w-slsc function.

On the other hand, the existence of fixed points for nonexpansive mappings defined on domains which are compact with respect to certain topology, has been studied in several papers [?], [?], [?], [?], [?]. All of them require, as additional assumptions, the τ-sequential compactness of the domain. From Theorem 5.2 we check that the τ-compactness is not necessary to assure fixed point result in the above hypothesis

Next we are going to define a geometric property in a Banach space which implies normal structure with respect to a topology τ in very general conditions.

Definition 5.3. Let X be a Banach space and τ a topology on X. We shall say that X has the τ-GGLD property if:

lim_{n} ∥x_{n}∥ < lim_{n,m;n≠m} ∥x_{n} − x_{m}∥

for every bounded τ-null sequence {x_{n}} such that both limits exist and lim_{n} ∥x_{n}∥ ≠ 0.

Theorem 5.4. Let X be a separable Banach space. If τ is a linear topology and X has the τ-GGLD property then X has τ-NS. In case τ ≡ w the separability assumption is not necessary.

Proof. If X has not τ-NS there is a bounded convex τ-sequentially compact subset B which satisfies r(B) < diam(B) where r(B) is the Chebyshev radius of B. As in Corollary 52 in Chapter (Prus) we can obtain a sequence {x_{n}} ⊂ B which converges to x with respect to τ and

lim_{n} ∥x_{n} − x∥ = lim_{n} ∥x_{n} − x_{m}∥

for every m ∈ ℕ. Applying Theorem 3.5 there exists a subsequence {y_{n}} ⊂ {x_{n}} such that lim_{n,m;n≠m} ∥y_{n} − y_{m}∥ exists. Finally, if we set z_{n} = y_{n} − x it is clear that the sequence {z_{n}} is bounded, τ-null convergent since τ is a linear topology and also

lim_{n} ∥z_{n}∥ = lim_{n,m;n≠m} ∥z_{n} − z_{m}∥

which contradicts the τ-GGLD property.

Notice that if τ is the weak topology the separability is not necessary in the same way as in Corollary 52 in Chapter (Prus).

If X is a Banach space and τ a topology on X the following coefficient was defined in [?]

τCS(X) = inf { lim_{n,m;n≠m} ∥x_{n} − x_{m}∥ lim_{n} ∥x_{n}∥ }

where the infimum is taken over all norm bounded sequences which converge to 0 in τ, both limits exist and lim ∥x_{n}∥ ≠ 0. Property τ-GGLD is related to the coefficient τCS(X) in the following way: X has the τ-GGLD property if τCS(X) > 1. Consequently, we can state the following result which relates the τ-FPP and the coefficient τCS(X).

Corollary 5.5. Let X be a separable Banach space and τ a linear topology on X such that the norm is a τ-slsc function. If τCS(X) > 1 then X has the τ-FPP. If τ ≡ w the separability assumption is not necessary.

Remark 17)

If X = (L_{p}(μ), ∥.∥_{p} ) and τ is the clm-topology then τCS(X) = 2^{1/p} for 1 ≤ p < ∞ ( see chapter by S. Prus). So as a consequence of Corollary 5.5 we can deduce that L_{1}(μ) has the clm-FPP although fails the weak-FPP (see [?]). This result had been proved previously by Lennard [?].

To finish this section, we shall see how the coefficient τCS(X) is related to the moduli of noncompact convexity defined in Remark 13. As a consequence the moduli of noncompact convexity are also useful to deduce fixed point results for nonexpansive mappings.

Lemma 5.6. Let X be a Banach space and τ a linear topology on X. Then:

τCS(X) ≥ lim_{ε→1^{−}}11 − Δ_{X,β,τ} (ε) = 11 − Δ_{X,β,τ} (1^{−})

Proof. Let ϵ < 1 be an arbitrary positive number and {x_{n}} a sequence τ-convergent to zero, such that lim_{n} ∥x_{n}∥ exists and lim_{n,m;n≠m} ∥x_{n} − x_{m}∥ = (ϵ + 1)/2. If we take ϵ < η < (ϵ + 1)/2, we can find k_{0} ∈ ℕ such that η ≤ ∥x_{n} − x_{m}∥ ≤ 1 if m, n ≥ k_{0}.

Consider m ≥ k_{0} and the sequence {y_{n}} = {x_{n} − x_{m}}_{n≥k_{0}} which is τ-convergent to the point −x_{m}. In addition, the sequence {y_{n}} is contained in the closed unit ball of X and sep({y_{n}}) ≥ η > ϵ. From the definition of the modulus Δ_{X,β,τ} (·), we deduce that ∥x_{m}∥ ≤ 1 − Δ_{X,β,τ} (ϵ). Letting m → +∞ we obtain:

lim_{n,m;n≠m} ∥x_{n} − x_{m}∥ lim_{n} ∥x_{n}∥ ≥ ϵ + 121 − Δ_{X,β,τ} (ε)

Finally, taking the limit as ϵ → 1^{−} we deduce the desired inequality.

Remark 18)

If X is UKK(τ), from Lemma 5.6, τCS(X) > 1. Therefore, if X is a separable Banach space, τ a linear topology such that the norm is a τ-slcs function and X is UKK(τ), then has the τ-FPP [?]. Notice that from Lemma 5.6 we may deduce the same conclusion considering Δ_{X,β,τ} (1^{−}) > 0.

Recall that a Banach space X is said to have the nonstrict Opial property with respect to a topology τ is

lim inf_{n} ∥x_{n}∥ ≤ lim inf_{n} ∥x + x_{n}∥

for every τ-null bounded sequence {x_{n}} in X and x ∈ X. Before Lemma 3.7 it was proved for Hilbert spaces that χ({y_{n}}) = lim ∥y_{n} − v∥ if {y_{n}} is w-convergent to v and lim ∥y_{n} − v∥ exists. Following the same argument it is easy to deduce the following fact: If τ is a linear topology on X and X satisfies the nonstrict Opial property with respect to τ then χ({y_{n}}) = lim ∥y_{n} − v∥ if {x_{n}} is a sequence in X τ-convergent to v and lim ∥y_{n} − v∥ exists. This property will be used in the proof of the next result:

Lemma 5.7. Suppose that X verifies the nonstrict Opial property with respect to a linear topology τ defined on X. Then τCS(X) ≥ h_{0} where

h_{0} = sup {t ≥ 1 : 1t + Δ_{X,χ,τ} (1^{−}t) ≥ 1} .

Proof. We define the function

h(t) = 1t + Δ_{X,χ,τ} (1^{−}t) .

Since Δ_{X,χ,τ} (·) is a nondecreasing function, h is increasing.

To obtain a contradiction, suppose that τCS(X) < h_{0}. Take l ∈ (τCS(X), h_{0}) and a normalized sequence which converges to zero for the topology τ and such that lim_{n,m;n≠m} ∥x_{n} −x_{m}∥ ≤ l.

Let ε be an arbitrary positive number with l + 2ε < h_{0}. There exists n_{0} ∈ ℕ such that ∥x_{n} − x_{m}∥ ≤ l + ε if n, m ≥ n_{0}. Fix k ≥ n_{0} and consider the sequence y_{n} = (x_{k} − x_{n})/(l + ε) with n ≥ n_{0}. Then, {y_{n}} is contained in the closed unit ball of X, τ-converges to the point x_{k}/(l + ε) and

∥y_{n} − x_{k}l + ε∥ = ∥x_{n}l + ε∥ = 1l + ε for every n ∈ ℕ.

¿From the non-strict Opial condition with respect to τ we deduce χ({y_{n}}) = 1/(l+ε) > 1/(l+2ε). Therefore, if we put l′ = l + 2ε, we obtain:

1l′ < 1l + ε = ∥x_{k}l + ε∥ ≤ 1 − Δ_{X,χ,τ} (1l + 2ε) = 1 − Δ_{X,χ,τ} (1l′) .

Hence h(l′) < 1, which contradicts the definition of h_{0} since l′ < h_{0}.

Remark 19) The lower bounds for τCS(X) given in the above lemmas are the best possible in general. Notice that for the space X = (L_{p}(μ), ∥ · ∥_{p}) endowed with the clm topology we have:

11 − Δ_{X,β,clm}(1^{−}) = 2^{1/p}, h_{0} = 2^{1/p}.

This corresponds to the actual value of (clm)CS(L_{p}(μ)). However, the inequalities can be strict in some Banach spaces (see Chapter VI in [?]).

6. Fixed points in NUS spaces

In Section 5 we studied the w-FPP as a property which is implied by either the GGLD property or normal structure. However, there are Banach spaces without these properties which have the FPP. For instance, ℓ^{p,∞} does not have normal structure (see [?], Example VI.2), but this space has the FPP. This fact can be proved, for instance, checking that the Banach-Mazur distance between ℓ^{p,∞} and ℓ^{p} is 2^{1/p} and applying a stability result in [?].

On the other hand, NUS spaces (see definition in Prus chapter ) can also fail to have normal structure. The space ℓ^{p,∞} is again an example of this assertion. However, we will prove that NUS spaces have the FPP.

We are going to prove a basic result in metric fixed point theory: Goebel-Karlovitz’s lemma. We need some previous definitions.

Definition 6.1. Let (X, d) be a metric space and T a mapping from X into X. A sequence {x_{n}} is called an approximated fixed point sequence for T if d(x_{n}, T x_{n}) → 0 as n → ∞.

If we assume that C is a convex bounded closed subset of a Banach space X and T : C → C a nonexpansive mapping, it is easy to prove that an approximated fixed point sequence exists in C.

Definition 6.2. The asymptotic radius and centre of a sequence {x_{n}} in a Banach space X will be defined by:

r_{a}({x_{n}}, B) = inf{lim sup_{n→∞} ∥x_{n} − y∥ : y ∈ B},

Z_{a}({x_{n}}, B) = {y ∈ B : lim sup_{n→∞} ∥x_{n} − y∥ = r_{a}({x_{n}}, B)},

for a subset B of X. When B = \overline{co}(\{x_{n}\}) we will denote r_{a}(\{x_{n}\}, co(\{x_{n}\})) and Z_{a}(\{x_{n}\}, co(\{x_{n}\})) respectively by r_{a}({x_{n}}) and Z_{a}({x_{n}}).

Proposition 6.3. Let K be a weakly compact convex subset of a Banach space X, and T : K → K be a nonexpansive mapping. Assume that K is minimal for T, that is, no closed convex bounded proper subset of K is invariant for T. If {x_{n}} is an approximated fixed point sequence in K, then

Z_{a}({x_{n}}, K) = K.

Proof. Let

Z_{a,ε}({x_{n}}, K) = {y ∈ K : lim sup_{n→∞} ∥x_{n} − y∥ ≤ r_{a}({x_{n}}, K) + ε}.

It is easy to check that Z_{a,ε}({x_{n}}, K) is nonempty, closed, convex and invariant for T. Thus Z_{a,ε}({x_{n}}, K) = K and Z_{a}({x_{n}}, K) = ⋂_{ε>0} Z_{a,ε}({x_{n}}, K) = K.

Lemma 6.4. (Goebel-Karlovitz) Let K be a weakly compact convex subset of a Banach space X, and T : K → K a nonexpansive mapping. Assume that K is minimal for T and {x_{n}} is an approximated fixed point sequence for T. Then

lim_{n→∞} ∥y − x_{n}∥ = diam(K)

for every y ∈ K.

Proof. Since T (K) ⊂ K we have \overline{co}(T (K)) ⊂ K. Thus \overline{co}(T (K)) is a weakly compact convex subset of K which is also invariant under T. The minimality of K implies K = \overline{co}(T (K)). Since K is a weakly compact and convex set we know that Z(K) is a nonempty set. Let x ∈ Z(K). For every y ∈ K we have ∥T y − T x∥ ≤ ∥y − x∥ ≤ r(K). Thus T (K) is contained in the closed ball \overline{B}(T x, r(K)) which implies that \overline{co}(T (K)) = K ⊂ \overline{B}(T x, r(K)). Hence sup\{∥y − T x∥ : y ∈ K} ≤ r(K) which means T x ∈ Z(K). Thus Z(K) is a convex weakly compact subset of K and is invariant under T. Again the minimality of K implies Z(K) = K and so K is a diametral set.

We claim now that lim sup_{n→∞} ∥y −x_{n}∥ = diam(K) for every y ∈ K. Indeed, we assume that a vector y ∈ K exists such that lim sup_{n→∞} ∥y−x_{n}∥ < diam(K). We let r = lim sup_{n→∞} ∥y−x_{n}∥ and d = diam(K) and consider the family \{\overline{B}(z, (r + d)/2) ∩ K : z ∈ K\}. Choose any positive ε < (d−r)/2. From Proposition 6.3 we know that lim sup_{n→∞} ∥x_{n} −z∥ = r for any z ∈ K. Thus, for any finite subset {z_{1}, ..., z_{k}} of K an integer N exists such that ∥x_{N} − z_{i}∥ ≤ r + ε = (r + d)/2 for i = 1, ..., k. Hence x_{N} belongs to ⋂_{i=1}^{k} \overline{B}(z_{i}, (r + d)/2). The weak compactness of K implies the existence of x_{0} ∈ ⋂_{z∈K} \overline{B}(z, (r + d)/2) ∩ K and this point is not diametral because

sup_{z∈K} ∥z − x_{0}∥ < r + d2 < d = diam(K).

This contradiction proves the claim. If lim inf_{n→∞} ∥y − x_{n}∥ < diam(K) for some y ∈ K there is a subsequence {y_{n}} of {x_{n}} such that lim sup_{n→∞} ∥y_{n} − y∥ = lim inf_{n→∞} ∥x_{n} − y∥ < diam(K), contradicting the claim, because every subsequence of an approximated fixed point sequence is also an approximated fixed point sequence.

Lemma 6.5. Let K be a weakly compact convex subset of a Banach space X, and T : K → K be a nonexpansive mapping. Assume that K is minimal for T, diam(K) = 1 and {x_{n}} is an approximated fixed point sequence for T which is weakly null. Then, for every ε > 0, there exists a sequence {z_{n}} in K such that:

(i) {z_{n}} is weakly convergent to a point z ∈ K.

(ii) ∥z_{n}∥ > 1 − ε for every n ∈ ℕ.

(iii) ∥z_{n} − z_{m}∥ ≤ 1/2 for every n, m ∈ ℕ.

(iv) lim sup_{n} ∥z_{n} − x_{n}∥ ≤ 1/2.

Proof. Let 0 ∈ K. If {w_{n}} is an approximated fixed point sequence in K, we know from Lemma 6.4 that lim_{n} ∥w_{n}∥ = 1. Therefore, for every ε > 0 there exists δ(ε) > 0 such that ∥x∥ > 1 − ε if x ∈ K and ∥T x − x∥ < δ(ε). Indeed, otherwise, there exists ε_{0} > 0 such that we can find w_{n} ∈ K with ∥T w_{n} − w_{n}∥ < 1/n and ∥w_{n}∥ ≤ 1 − ε_{0} for every n ∈ ℕ. Thus, the sequence {w_{n}} is approximated fixed point sequence in K which satisfies lim sup_{n} ∥w_{n}∥ ≤ 1 − ε_{0}.

Let ε > 0. We choose γ < min{1, δ(ε)} and for every n ∈ N we define the contractive mapping S_{n} : K → K by

S_{n}(x) = (1 − γ)T (x) + γ/2x_{n}.

Banach’s Contraction Principle assures the existence of a (unique) fixed point z_{n} for S_{n}. Since K is weakly compact, we can assume, taking a subsequence if necessary that {z_{n}} satisfies (i). Since

∥z_{n} − T z_{n}∥ ≤ γ∥T z_{n} − 12x_{n} ≤ γδ(ε)∥

we know that {z_{n}} satisfies (ii).

Condition (iii) easily follows and (iv) is a consequence of the inequalities

∥z_{n} − x_{n}∥ ≤ ∥(1 − γ)T z_{n} + γx_{n}2 − x_{n}∥
≤ (1 − γ)∥T z_{n} − T x_{n}∥ + (1 − γ)∥T x_{n} − x_{n}∥ + γ∥x_{n}∥2.

Thus,

∥z_{n} − x_{n}∥ ≤ 12 + 1 − γγ∥T x_{n} − x_{n}∥.

Taking limit sup as n tends to infinity we obtain (iv).

The following definition was given in [?]:

Definition 6.6. Let X be a Banach space. We define the coefficient

R(X) = sup{lim inf_{n→∞} ∥x_{n} + x∥}

where the supremum is taken over all x ∈ X with ∥x∥ ≤ 1 and over all weakly null sequences in \overline{B}(0, 1).

Theorem 6.7. Let X be a Banach space and assume that R(X) < 2. Then X has the weak fixed point property.

Proof. If we assume that X fails to have the w-FPP, we can find a weakly compact and convex subset K of X such that diam(K) = 1 and K is minimal invariant for a nonexpansive mapping T and an approximated fixed point sequence {x_{n}} in K. We can assume that {x_{n}} is weakly convergent, and by translation that {x_{n}} is weakly null. Consider a sequence {z_{n}} satisfying (i)- (iv) in Lemma 6.5. Taking again a subsequence, if necessary, we can assume that lim_{n} ∥z_{n} − z∥ exists. Furthermore, lim_{n} ∥z_{n} − z∥ ≤ lim sup_{n} lim sup_{m} ∥z_{n} − z_{m}∥ ≤ 1/2.

A positive η can be chosen such that ηR(X) < 1 − R(X)/2. For a large enough n, we have ∥z_{n} − z∥ ≤ 1/2 + η. Furthermore ∥z∥ ≤ lim inf_{n→∞} ∥z_{n} − x_{n}∥ ≤ 1/2. Hence

∥z_{n}1/2 + η∥ = ∥z_{n} − z1/2 + η + z1/2 + η∥ ≤ R(X).

Thus lim sup_{n→∞} ∥z_{n}∥ ≤ R(X)(1/2 + η) < 1 which is a contradiction because 0 ∈ K.

Remark 20) Assume X = c_{0}. It is easy to check that R(c_{0}) = 2 and so c_{0} has the w-FPP although c_{0} does not satisfy the GGLD property.

Theorem 6.8. Let X be an NUS Banach space. Then R(X) < 2.

Proof. Since X is NUS we know that for any ε > 0 there exists δ > 0 such that for any weakly null sequence in B_{X} and any x ∈ B_{X} a positive integer k exists such that ∥x + tx_{k}∥ ≤ 1 + εt if 0 < t < δ. Hence,

∥x + x_{k}∥ ≤ ∥x + tx_{k}∥ + (1 − t)∥x_{k}∥ ≤ 2 + t(1 − ε).

Furthermore k can be chosen arbitrarily large because the same argument can be applied to the sequence {y_{n}} where y_{1} = x and y_{n} = x_{k+n} for n > 1. Hence R(X) < 2.

7. Asymptotically regular mappings

We shall study in this section the existence of fixed points for a different class of mappings, called asymptotically regular mappings. The concept of asymptotically regular mappings is due to Browder and Petryshyn [?]. The fixed point theorems which we shall study are based upon coefficients concerning the modulus of NUC or Opial modulus.

Definition 7.1. Let (X, d) be a metric space. A mapping T : M → M is called asymptotically regular if

lim_{n→∞} d(T^{n}x, T^{n+1}x) = 0

for all x ∈ M.

Example 7: Let T : [0, 1] → [0, 1] be an arbitrary nonexpansive mapping. It is easy to check that S = (I + T)/2 is also nonexpansive. Thus

|S^{n+1}x − S^{n}x| ≤ ... ≤ |S^{2}x − Sx| ≤ |Sx − x|.

Furthermore S is a nondecreasing function. Indeed, if x ≤ y and Sx > Sy we have (x+T x)/2 > (y + T y)/2 which implies |T x − T y| ≥ T x − T y > y − x = |x − y|. Thus

1 ≥ |S^{n+1}x − x| = ∑_{k=1}^{n} |S^{k+1}x − S^{k}x| ≥ n|S^{n+1}x − S^{n}x|

which implies |S^{n+1}x − S^{n}x| ≤ 1/n. Then S is asymptotically regular. In fact, it can be proved (see [?], Theorem 9.4, page 98) that if T is a nonexpansive mapping from a bounded convex subset C of a Banach space into C, then T_{λ} = λI + (1 − λ)T is asymptotically regular for all 0 < λ < 1.

We will show the possibility of using a modulus of noncompact convexity and Opial modulus to obtain fixed point theorems for asymptotically regular mappings. We need some notations: If S is a mapping from a set C into itself, then we use the symbol |S| to denote the Lipschitz constant of S, that is,

|S| = sup {∥Sx − Sy∥∥x − y∥ : x, y ∈ C, x ≠ y} .

Now if T is a mapping on C, we set

s(T) = lim inf_{n→∞} |T^{n}|.

Theorem 7.2. Let X be a Banach space, C a nonempty closed τ-sequentially compact subset of X, and T : C → C an asymptotically regular mapping. Assume that one of the following conditions is satisfied:

(a) s(T) < 1 + r_{X,τ} (1)

(b) s(T) < \sqrt{τCS(X)}

(c) s(T) < 1/(1 − Δ_{X,β,τ} (1^{−}))

Then T has a fixed point.

Proof. Choose a subsequence {n_{k}} of positive integers such that s(T) = lim_{k} |T^{n_{k}}|. For any x ∈ C we can construct a subsequence {n_{k}(x)} ⊂ {n_{k}} such that {T^{n_{k}(x)}x}_{k} is τ-convergent, say to l(x), and lim_{l,k;l≠k} ∥T^{n_{k}(x)}x − T^{n_{l}(x)}x∥ and lim_{k} ∥T^{n_{k}(x)}x − l(x)∥ exist. We construct a sequence {x_{m}} ⊂ C and sequences of positive integers {n_{k}(x_{m})} in the following way:

x_{0} ∈ C arbitrary

x_{m} = l(x_{m−1}) = τ − lim_{k} T^{n_{k}(x_{m−1})}x_{m−1},

{n_{k}(x_{m})}_{k} ⊂ {n_{k}(x_{m−1})}_{k} for all k ≥ 1.

By using a diagonal argument we consider the sequence of positive integers n_{i} = n_{i}(x_{i}) for all i ∈ ℕ. In this way, we have that the sequence {T^{n_{i}}x_{m}} is weakly convergent to x_{m+1}, lim_{i,j;i≠j} ∥T^{n_{i}}x_{m} − T^{n_{j}}x_{m}∥ and lim_{i} ∥T^{n_{i}}x_{m} − x_{m+1}∥ exist for all m ∈ ℕ and s(T) = lim_{i} |T^{n_{i}}|.

We are going to prove that {x_{m}} is norm convergent to a vector in C.

Denote B_{m} = lim_{i} ∥T^{n_{i}}x_{m} − x_{m+1}∥. Our proof is based upon the following claim:

Claim. Assume that s(T) satisfies either (a) or (b) or (c). Then, there exists α < 1 such that B_{m} < αB_{m−1} for any m ∈ ℕ.

Using this claim we can go on the proof. Firstly, we know that lim_{m} B_{m} = 0. By using that the norm on X is τ-sequentially lower semicontinuous we deduce

∥x_{m} − x_{m+1}∥ ≤ lim sup_{i} ∥x_{m} − T^{n_{i}}x_{m}∥ + lim sup_{i} ∥T^{n_{i}}x_{m} − x_{m+1}∥ ≤

lim sup_{i} (lim inf_{j} ∥T^{n_{j}}x_{m−1} − T^{n_{i}}x_{m}∥) + B_{m} ≤ s(T)B_{m−1} + B_{m}.

which implies that {x_{m}} is a Cauchy sequence. Let z = lim_{m} x_{m}. Since C is closed, z ∈ C. Furthermore

∥z − T^{n_{i}}z∥ ≤ ∥z − x_{m+1}∥ + ∥x_{m+1} − T^{n_{i}}x_{m}∥ + ∥T^{n_{i}}x_{m} − T^{n_{i}}z∥ ≤
∥z − x_{m+1}∥ + ∥x_{m+1} − T^{n_{i}}x_{m}∥ + L_{i}∥x_{m} − z∥.

Taking the limit as i → ∞ yields

lim sup_{i} ∥z − T^{n_{i}}z∥ ≤ ∥z − x_{m+1}∥ + B_{m} + s(T)∥x_{m} − z∥ → 0

as m → ∞. Hence T^{n_{i}}z → z. From the assumption on s(T) there exits a positive integer k such that |T^{k}| < +∞. If we choose a positive number ϵ, the asymptotic regularity of T implies that there exists i_{0} ∈ ℕ such that ∥T^{n_{i}+k}z − T^{n_{i}}z∥ < ϵ if i ≥ i_{0}. We also assume ∥T^{n_{i}}z − z∥ ≤ ϵ.

Hence, for i ≥ i_{0} we obtain

∥T^{k}z − z∥ ≤ ∥T^{k}z − T^{n_{i}+k}z∥ + ∥T^{n_{i}+k}z − T^{n_{i}}z∥ + ∥T^{n_{i}}z − z∥ ≤
(|T^{k}| + 1 + 1)ϵ → 0 as ϵ → 0.

yielding T^{k}z = z. It is easily verified by induction that T^{ks}z = z for s = 1, 2, · · ·. Therefore

∥T z − z∥ = ∥T^{ks+1} − T^{ks}z∥ → 0

as s → ∞. So z is a fixed point for T in C.

It remains to prove the claim. From the definition of τCS(X) we obtain

B_{m} ≤ lim_{i,j;i≠j} ∥T^{n_{i}}x_{m} − T^{n_{j}}x_{m}∥τCS(X)

The asymptotic regularity of T on C ensures that lim_{j} ∥T^{n_{j}−l}x_{m} − x_{m}∥ = lim_{j} ∥T^{n_{j}}x_{m} − x_{m}∥ for any l ∈ ℕ. Thus

lim_{i,j;i≠j} ∥T^{n_{i}}x_{m} − T^{n_{j}}x_{m}∥ = lim sup_{i} (lim sup_{j} ∥T^{n_{i}}x_{m} − T^{n_{j}}x_{m}∥) ≤