Measures of noncompactness in modular spaces and fixed point theorems for multivalued nonexpansive mappings

This paper is devoted to state some fixed point results for multivalued mappings in modular vector spaces. For this purpose, we study the uniform noncompact convexity, a geometric property for modular spaces which is similar to nearly uniform convexity in the Banach spaces setting. Using this property, we state several new fixed point theorems for multivalued nonexpansive mappings in modular spaces.


Introduction
The beginning of the Fixed Point Theory for nonexpansive mappings occurred in 1965 when Browder [5], Browder and Göhde [6,18] and Kirk [24] proved that every nonexpansive mapping defined on a convex closed bounded subset of, respectively, a Hilbert space, a uniformly convex Banach space or a reflexive Banach space with normal structure, has a fixed point. A natural problem is to extend these results to multivalued nonexpansive mappings (see Problem 8 in [32]). Using the uniqueness of the asymptotic center of a bounded sequence in a uniformly convex space, Browder-Göhde's Theorem was extended by Lim [27] (see also in [33] Corollary 3.5). Surprisingly, 55 years later, it is still an open problem the possibility of extending Kirk's Theorem. However, some partial extensions have been obtained assuming that the Banach space satisfies several different conditions which imply normal structure (see [9] and references therein). Another direction to research has been the development of the theory for single-valued nonexpansive mappings in modular function spaces (see, for instance [21,22]). Modular spaces were introduced by Nakano [29,30] and developed by Orlicz and Musielak [28]. A very relevant class of modular spaces are the variable exponent Lebesgue spaces, due to their applications to partial differential equations and variational integrals with non-standard growth conditions. This fact, especially after M. Ružička [34] discovered that they constitute a natural functional setting for the mathematical model of electrorheological fluids, has led to renewed attention on the modular function spaces.
Nearly uniform convexity is a geometric condition that has proved to be successful to obtain fixed points for multivalued nonexpansive mappings in Banach spaces [12,13]. Since this condition has a counterpart in modular spaces [21,Section 4.3], it is very natural to study the validity of these results in the setting of modular spaces. After several sections with preliminaries and technical results, in Sect. 6, we state some fixed point results for multivalued nonexpansive mappings in modular spaces. Our approach follows, in some parts, similar arguments to those in [12,13], but, in some other parts, we need some very different techniques. We avoid details in the first case and we will give complete proofs in the latter. We have included two examples of Orlicz sequential variable exponent spaces [31] where our results can be applied.
The formula defines a norm which is frequently called the Luxemburg norm. Thus, any convex modular space can be simultaneously studied as a normed space with the Luxemburg norm and any topic on these spaces can be split in two parts corresponding either to the modular space or the normed space. It should be noticed that in non-trivial cases, the Luxemburg norm is particularly difficult 40 to compute. Consequently, the results for the norm should be usually deduced from the properties of the modular space.
It is clear that the diameter of a set is preserved by translation, i.e., Given a subset A of X ρ , we denote co(A) its convex hull, i.e., the smallest convex subset of X ρ containing A. From the convexity of the modular ρ, we easily deduce that diam ρ (co(A)) = diam ρ (A).
Let us describe the relationship between the modular-convergence and norm-convergence in modular spaces. We can use the following proposition: Proposition 2.3. (Proposition 3.7 and 3.9 in [21]) Let ρ be a convex modular and let x ∈ X ρ . The following assertions are true: Remark 2.4. (1) From part (a) of Proposition 2.3 we conclude that a normconvergent sequence in X ρ is ρ-convergent. Therefore, every ρ-closed set in X ρ is norm-closed. (2) An easy consequence of part (b) of Proposition 2.3 is that any ρ-bounded subset of X ρ is also norm-bounded. (3) It is easy to check that the Fatou property implies that the ρ-balls are ρ-closed and that the ρ-diameter of a set is the same as the ρ-diameter of its ρ-closure. The Fatou property also implies that: if x ∈ X ρ and K is a nonempty ρ-compact subset of X ρ , then there exists y 0 ∈ K such that In the following, we always assume that the modular is ρ-complete.
Lemma 2.7. [11] Let ρ be a convex modular satisfying the Δ 2 -type condition. Then the growth function ω ρ has the following properties: Letting α go to x ρ and using the continuity of ω ρ (·), we obtain the wanted inequality. As a consequence any norm-bounded subset of X ρ is also ρ-bounded.
Furthermore, we also obtain that ρ-convergence is identical to normconvergence. Moreover, the modular is ρ-complete if and only if it is normcomplete and ρ-compact (ρ-closed) sets are the same as norm-compact (normclosed) sets. We will remove the prefix ρ in this case.
The following is a technical lemma which will be needed because of the lack of the triangular inequality. Lemma 2.9. [11] Assume that ρ is a convex modular satisfying the Δ 2 -type condition. Let {x n }, {y n } two sequences in X ρ . Then In the remainder of the paper, we will assume that X ρ satisfies the Fatou property even though no Δ 2 -type condition is assumed.
Some existence fixed point theorems for nonlinear mappings defined in modular spaces require a kind of uniform continuity of the modular. 40 Definition 2.10. A modular ρ is said to be uniformly continuous on bounded sets if for every bounded subset M of X ρ and for every ε > 0, there exists δ > 0 such that The following result is a particular case of Lemma 3.4 in [14] (see also [20]). The following property can be understood as the modular equivalence of the Banach space reflexivity. It will be a powerful tool to prove the fixed point property in modular spaces. Definition 2.12. [21] A modular space X ρ is said to satisfy property (R) if every nonincreasing sequence {C n } of nonempty, ρ-bounded, ρ-closed, convex subsets of X ρ has a nonempty intersection.
The method of asymptotic centers has played an important role in the fixed point theory for nonexpansive multivalued mappings in Banach spaces. Some definitions and results concerning asymptotic centers can be adapted to modular spaces in a straightforward way: Let C be a nonempty ρ-closed ρ-bounded subset of the space X ρ and {x n } be a bounded sequence in X ρ . We define The number r ρ (C, {x n }) and the (possible empty) set A ρ (C, {x n }) are called, respectively, the ρ-asymptotic radius and the ρ-asymptotic center of {x n } in C.
Obviously, A ρ (C, {x n }) is a convex set as C is. Furthermore, the set A ρ (C, {x n }) is nonempty and closed whenever the space satisfies the Δ 2type condition and satisfies property (R). Indeed, for any m ≥ 1 consider the set where r = r ρ (C, {x n }). Clearly A m is nonempty and convex. Also, A m is closed by Lemma 2.11. It follows from property (R) that The sequence {x n } is said to be regular relative to C if the asymptotic radii of all subsequences of {x n } (relative to C) are the same. If, in addition, Similarly as Goebel, Lim and Kirk [15,24,27] did for Banach spaces, the following lemma can be proved: Lemma 2.13. Let ρ be a convex modular satisfying the Δ 2 -type condition. Let C be a nonempty closed bounded separable subset of the space X ρ and {x n } be a bounded sequence in X ρ . Then {x n } contains a regular and asymptotically uniform subsequence relative to C.
If D is a ρ-bounded subset of X ρ , the ρ-Chebyshev radius of D relative to C is defined by Let C be a nonempty subset of the space X ρ . We shall denote by F ρ (C) the family of all nonempty ρ-closed subsets of C and by K ρ (C) (resp. KC ρ (C)) the family of all nonempty ρ-compact subsets of C (resp. ρ-compact convex). We can define the analogue to the Hausdorff distance for modular spaces by where for x ∈ X ρ and E ⊂ X ρ d ρ (x, E) := inf{ρ(x − y) : y ∈ E} is the ρ-distance from the point x to the subset E. This function will be called Hausdorff ρ-distance even though it is not a metric.
If C is a ρ-closed convex subset of X ρ , then a multivalued mapping Finally, we say that x ∈ C is a fixed point of T if and only if x ∈ T x. A counterpart of the Banach Contraction Principle for multivalued mappings in modular spaces has been proved in [7]. Theorem 2.14. Let ρ be a convex modular satisfying the Δ 2 -type condition, C a nonempty bounded closed subset of X ρ , and T : C → F ρ (C) a ρ−contraction mapping. Then T has a fixed point.

Measures of noncompactness in modular spaces
The definitions of both Kuratowski and Hausdorff measure of noncompactness were introduced in modular function spaces by Khamsi and Kozlowski [21]. Similarly, we can extend such concepts to modular abstract spaces in the following way. We will make the obvious convention that the infimum over an empty set is infinite.
We next summarize the basic properties of the above measures of noncompactness. These properties follow immediately from the definitions and some of them have been proved in [21]. Proposition 3.2. Let φ denote α or χ. Then the following properties hold: Let ρ a modular satisfying the Δ 2 -type condition. Assume that C is a subset of the modular space X ρ . We can consider the Hausdorff measure of noncompactness χ C defined for any ρ-bounded subset A of C by χ C (A) = inf{ε > 0 : A can be covered by finitely many ρ-balls centered at points in C with radius less than ε}. It must be noted that this measure depends on C and it is, in general, different from χ =: χ Xρ . However, if C is a convex closed set, it is easy to check that the arguments to prove the properties in Proposition 3.2 equally well apply to the measure χ C . We are going to prove that the Kuratowski and Hausdorff measure of noncompactness are invariant under passage to the convex hull.
Proof. We prove the result for χ. The proof for α is analogous.
We are going to follow an argument as in [2] to get a contradiction. Indeed, for every ε > 0 there exist Since Δ is compact, we can find σ (1) , · · · , σ (m) ∈ Δ such that for all σ ∈ Δ, we have we apply the convexity of the modular to obtain Hence, Therefore, This implies that Bearing in mind that ε was chosen arbitrary, we obtain χ(co(A) ≤ λχ(A) which is a contradiction.

Uniform noncompact convexity in modular spaces
Similarly as Goebel and Sekowski [17] did for Banach spaces, we can define a scaling of the convexity for modular spaces using the measures of noncompactness. T. D. Benavides, P. L. Ramírez.
Definition 4.1. [21] Let ρ be a convex modular. Let φ denote α or χ. The ρmodulus of noncompact convexity associated with φ is defined in the following way: for every r > 0, ε > 0. Let ρ be a convex modular. Let φ denote α or χ. The ρmodulus of noncompact convexity associated with φ is defined in the following way: for every r > 0, ε > 0.
Remark 4.6. In [1] (see also [5,21]), the authors introduced some interlinked notions of ρ-uniform convexity in a modular space. One of them leads to define the ρ-modulus of uniform convexity for every r > 0 and ε > 0 as follows: where the infimum is taken over all x, y ∈ X ρ such that ρ(x) ≤ r, ρ(y) ≤ r and ρ(x − y) ≥ εr.
Let us show the connection between this modulus and the ρ-modulus of noncompact convexity associated with φ = α or χ.
Consequently, the class of φ-uniformly ρ-noncompact convex spaces includes all ρ-(UUC1) spaces. The following example shows that the converse of this assertion is not true. Let {p n } be a sequence in [1, ∞) such that 1 < p =: lim inf n p n ≤ lim sup n p n < ∞. Consider the Orlicz space pn for the modular It is well known that ( pn , ρ) is a modular sequence space which satisfies the Δ 2 -type condition. Furthermore, ( pn , · ρ ) is a reflexive Banach space whenever · ρ is the corresponding Luxemburg norm [10, Theorem 18]. Note that ( pn , ρ) does not satisfy any uniform convexity condition (see [1, Definition 3.1]) whenever the sequence {p n } attains the value 1 more than once, because it contains R 2 with the 1-norm. It is clear that the modular is additive for disjointedly supported vectors. On the other hand, the functional Λ k : pn → R defined by Λ k (x) = x(k) is continuous and so every weakly null sequence in ( pn , · ρ ) converges to 0 coordinate-wise.
We are going to prove that Δ χ (ε, r) ≥ ε/2. To do that, let A be a convex subset of the closed ball B ρ (0, r) such that χ(A) > εr. We can find a sequence {x n } in A such that sep ρ (x n ) = inf{ρ(x n − x m ) : n, m ∈ N} ≥ εr. Taking a subsequence, we can assume that {x n } is weakly convergent, say to x ∈Ā, and lim n ρ(x n − x) =: lr does exist. After a further subsequence, we can assume that there exist two sequences {u n }, {v n } such that lim n ρ(x − u n ) = 0, lim n ρ(x n − x − v n ) = 0, supp u n < supp v n and supp v n < supp v n+1 for every n ∈ N, where supp(x) := {n ∈ N : x(n) = 0} and supp u < supp v means that there exists N ∈ N such that supp u ⊂ [1, N] and supp v ⊂ [N + 1, +∞). Choose an arbitrary η > 0. Using Lemma 2.11, we can choose n, m large enough such that Thus, Moreover, Next lemma provides an important tool to prove the relationship between modular uniform noncompact convexity and property (R) as well as one of the main fixed point theorems in our paper (Theorem 6.7 below). Lemma 4.8. Let ρ be a convex modular satisfying the Δ 2 -type condition. Assume that X ρ is φ-UNC. Let s ∈ (0, +∞) and ε > 0. Then there exists λ < 1 and η > 0 such that, for every r ∈ (s, s + η), we have 1 − Δ φ (r, ε) ≤ λ. (2) . Thus, Therefore, 1 − Δ φ (r, ε) ≤ λ.

Theorem 4.9. Let ρ be a convex modular satisfying the Δ 2 -type condition.
Assume that X ρ is φ-UNC. Then, for any closed bounded convex subset of X ρ and x ∈ X ρ , the set is nonempty compact and convex.
Proof. Since C is closed, we may assume without loss of generality that d := d ρ (x, C) > 0. Consider the sets C n = C ∩ B ρ (x, d + 1 n ) for any n ≥ 1. Clearly {C n } is a decreasing sequence of closed bounded convex subsets of X ρ and According to Lemma 4.8, there exists λ := λ(d, ε) < 1 and n 0 such that for every n ≥ n 0 contradicting the fact that d > 0. Hence lim n φ(C n ) = 0. By Proposition 3.2, we deduce that n≥1 C n is a nonempty ρ-compact convex subset of C and the proof is complete.

Theorem 4.10. Let ρ be a convex modular satisfying the Δ 2 -type condition.
Assume that X ρ is φ-UNC. Then X ρ has the property (R).
Proof. Let {C n } n≥1 be a decreasing sequence of closed bounded convex subsets of X ρ . Fix x ∈ C 1 . We have From Theorem 4.9, for every n ≥ 1 the set K n = C n ∩ B ρ (x, r) is nonempty. Clearly, {K n } is a decreasing sequence of closed bounded convex subsets of X ρ . Following the same argument as in the proof of Theorem 4.9, we can show that lim n φ(K n ) = 0. Indeed, if ε := inf for all n ≥ 1. Taking supremum and bearing in mind that Δ φ r, ε r > 0 we get a contradiction. So, n≥1 K n is a nonempty ρ-compact convex subset of X ρ . Therefore, n≥1 C n is nonempty.
Remark 4.11. In [21], Theorems 4.9 and 4.10 are stated without assuming the Δ 2 -type condition, but it is not clear for us their validity, because a positive lower bound of Δ φ (r + 1/n, ε), independent of n, is needed in both proofs. To obtain this lower bound, we have proved a kind of continuity of the modulus with respect to the first variable from the Δ 2 -type condition (Lemma 4.8). This bound can be also obtained if we, additionally, assume in the definition of uniform ρ-noncompact convexity that there is function η(·, ·) such that Δ φ (r, ε) > η(s, ε) > 0 for every r > s and ε > 0. (Compare with Theorem 4.1 and Theorem 4.2 in the same monograph when condition (UUC2) is assumed).

Asymptotic centers in UNC spaces
In this section, we shall give a connection between the asymptotic center of a sequence and the ρ-modulus of noncompact convexity. As in the Banach space setting it will play a crucial role to prove the existence of fixed point for multivalued ρ-nonexpansive mappings.
First, we state the following lemma.
Lemma 5.1. Let ρ be a convex modular satisfying the Δ 2 -type condition and C a closed convex subset of X ρ . Let {x n } be a bounded sequence in C and W the convex closed hull of its terms. Define Φ : W → [0, ∞) by Φ(x) = lim sup n ρ(x n − x). Let {z k } be a sequence in W such that z k ∈ A k =: co{x n : n ≥ k} (in particular, z k can be equal to x k ) and z ∈ ∩ ∞ k=1 co{z j : j ≥ k}. Then is a convex set. Since from Lemma 2.11 the function Φ is continuous, we have that Φ −1 ([0, a]) is also a closed set which contains {z j : j > k}. Thus, it contains the closed set co{z j : j ≥ k}. In particular, it contains z.
Before we present our result, we need the theorem below. Even though this theorem is a direct adaptation of Theorem 5 from [8] for the modular ρ, we include its proof for the sake of completeness.
We shall use a simple version of Ramsey Lemma. By N we denote the set of nonnegative integers, [N] the collection of its infinite subsets and for Since m is arbitrary, we obtain ψ({z n n }) ≤ ψ({z n }) ≤ ψ({z n n }) and the claim is proved.
Choose now an arbitrary ε > 0 and a subsequence {y n } of {x n } satisfying the property in the claim. Taking a subsequence (which "a fortiori" satisfies the same property) we can assume ψ({y n }) + ε ≥ ρ(y n − y m ) for every n, m. Define the following function from  Assume that X ρ is a φ-UNC. Let C be a nonempty closed bounded convex subset of the space X ρ and let {x n } be a sequence in C which is regular relative to C. Then there exists λ < 1 depending on r ρ (C, {x n }) such that Proof. We are going to prove the result for φ = α. Denote r = r ρ (C, {x n }) and A = A ρ (C, {x n }). According to Theorem 5.2, we can find a subsequence {y n } of {x n } such that the limit lim n =m ρ(y n − y m ) exists. Take z ∈ ∩ ∞ n=1 co{y k : k ≥ n} which is a nonempty set due to property (R). Since {x n } is regular relative to C, we have r = r ρ (C, {y n }) and from Lemma 5.1, we obtain Hence, α({y n }) ≥ r. Thus, α(co{y k : k ≥ n}) ≥ r, for every n ∈ N. Assume x lies in A. Consider λ r, 1 2 < 1 and η(r, 1 2 ) > 0 given by Lemma 4.8. Since r = lim sup n ρ(y n − x), for every 0 < ε < η(r, 1 2 ) there exists n 0 ∈ N such that ρ(y n − x) < r + ε for all positive integer n greater than or equal to n 0 . Hence, the sequence {y n } n≥n0 is contained in the closed ball B ρ (x, r + ε) and α(co{y k : k ≥ n}) ≥ (r + ε) r r+ε . Therefore, for all n ≥ n 0 In view of Theorem 4.9, we can find z n ∈ co{y k : k ≥ n}) such that Again by Theorem 4.10, ∩ ∞ n=n0 co{z k : k ≥ n} = ∅. For each w ∈ ∩ ∞ n=n0 co{z k : k ≥ n} and for all n ≥ n 0 , we have Therefore, Since this inequality is true for every ε > 0 and for every x ∈ A, we obtain the inequality in the statement.

Fixed point results
Our fixed point results for multivalued ρ-nonexpansive mappings rely on the following proposition.
Proof. Since C is separable, according to Lemma 2.13 there exists a subsequence {z n } of {x n } which is regular and asymptotically uniform with respect to C. Denote r = r ρ (C, {z n }).
From the compactness of the set T z n , we can find a sequence {u n } in C such that u n ∈ T z n and lim n→∞ ρ(z n − u n ) = 0. Take any x ∈ A and v n ∈ T x such that Because of the ρ-compactness of T x, we can assume, by passing through a subsequence, that {v n } converges to a point v ∈ T x. From Lemma 2.9, we obtain lim sup This shows that v ∈ A, and so T x ∩ A = ∅. Now we are ready to prove an analogous result to the Kirk-Massa's theorem [25] in modular spaces. Theorem 6.2. Let ρ be a convex modular satisfying the Δ 2 -type condition. Assume that C is a nonempty closed bounded convex subset of the space X ρ and T : C → KC ρ (C) a ρ-nonexpansive mapping. Suppose that each sequence in C has a nonempty and compact asymptotic center relative to C. Then T has a fixed point.
Proof. Since the Δ 2 -type condition is satisfied and T is a continuous compact valued self-mapping, we can construct a closed convex subset of C which is separable and invariant under T (see [26]). Thus, we can suppose that C is separable.
For a fixed element x 0 ∈ C and for each n ≥ 1, define the mapping We have that T n is a multivalued ρ-contraction and it has a fixed point x n ∈ C by Theorem 2.14. It is easily seen that lim n d ρ (x n , T x n ) = 0. Without loss of generality, we may assume that {x n } is regular and asymptotically uniform with respect to C.
According to the previous proposition, we can also assume that Now we define the mappingT : Since T is continuous, from Proposition 2.45 in [19], we know that the mappingT is upper semicontinuous. Since T x ∩ A is a compact convex set we can apply the Kakutani-Bohnenblust-Karlin Theorem (see [16]) to obtain a fixed point forT and so for T .
It should be pointed out that a modular ρ satisfying (UUC1) has the property (R) (see [1] and [21,Theorem 4.2]). If, in addition, the modular ρ is uniformly continuous on bounded sets, it is easy to check that the asymptotic center of a sequence is nonempty and singleton. In view of this result, we can deduce the following corollary from Theorem 6.2. Corollary 6.3. Let ρ be a (UUC1) convex modular satisfying the Δ 2 -type condition. Assume that C is a nonempty closed bounded convex subset of the space X ρ and T : C → KC ρ (C) a ρ-nonexpansive mapping. Then T has a fixed point. The following theorem states a relationship between ρ-contractive and ρχ C −contractive mappings. In the framework of a Banach space, an analogous result was proved in [13]. Theorem 6.5. Let ρ be a convex modular satisfying the Δ 2 -type condition and C a nonempty closed subset of X ρ . Assume that T : Proof. Let A a bounded subset of C, ε > 0 and μ > 1. Since μ > 1 we can take α ∈ (0, 1) such that αω( 1 α ) ≤ μ. Now, we choose a number ε > 0 such that ε .
By definition of χ C (A) there exists a 1 , ...a N ∈ C such that . On the other hand, since T is compact valued, there exists y 1 , ...
Let x ∈ A and i ∈ {1, ..., N } be such that x ∈ B ρ (a i , χ C (A)+ε ). Taking y ∈ T x, from the compactness of T a i , we can find z i ∈ T a i such that Since z i ∈ T a i , we have ρ(z i − y j ) ≤ ε for some j ∈ {1, ..., n}. Property (iii) of the modular and the definition of the growth function ω give Thus ρ(y − y j ) ≤ μkχ C (A) + 2ε. Hence, χ C (T (A)) ≤ μkχ C (A) + 2ε. Since ε is arbitrary we have χ C (T (A)) ≤ μkχ C (A) for all μ > 1. Thus, χ C (T (A)) ≤ kχ C (A), obtaining the desired result. Theorem 6.6. Let ρ be a convex modular satisfying the Δ 2 -type condition and C a nonempty closed convex subset of X ρ . Assume that T : Proof. According to Theorem 6.5, the mapping T is ρ-χ C -contractive. Denote A 1 = A and assume that we have defined a finite decreasing sequence of closed convex sets A n ⊂ A n−1 ⊂ ... ⊂ A 1 such that T x ∩ A j = ∅ for every x ∈ A j and χ C (A j ) ≤ kχ C (A j−1 ) for all j = 1, ..., n. Define A n+1 = [co T (A n )] ∩ A n . Then, A n+1 is a closed convex subset of A n . Furthermore, for every x ∈ A n+1 , we have T x ∩ A n = ∅. Since T x ⊂ T (A n ), we obtain that T x ∩ A n+1 is nonempty. On the other hand, Let x ∈ A ∞ and take a n ∈ T x∩A n which is nonempty. Since χ C ({a n }) = χ C ({a k : k ≥ n}) ≤ χ C (A n ), the sequence {a n } has some cluster point a ∈ A ∞ . On the other hand, the sequence {a n } lies in T x which is a compact set implying that a belongs to T x. Thus, T x ∩ A ∞ = ∅. Since A ∞ is compact convex we apply Kakutani-Bohnenblust-Karlin Theorem to obtain that T has a fixed point in A ∞ ⊂ A.
We state now the main fixed point result in this paper.
Taking limit in m in both sides we obtain lim m d ρ (x m , T x m ) = 0. In a similar way we can prove that d ρ (x, T x) = 0, i.e. x ∈ T x. Indeed, we have and letting m → ∞ we get the desired result.
The above theorem extends the Kirk-Massa theorem in modular spaces, namely Theorem 6.2, in the sense that we do not need the compactness of asymptotic center of a sequence. We illustrate this fact by means of the following example. It is clear that σ is a convex modular and ( pn , σ) satisfies the Δ 2 -type condition. Furthermore, which implies that the Luxemburg norms · σ and · ρ are equivalent. (Nominally, where P = lim sup n p n ). Thus, ( pn , · σ ) is a reflexive Banach space and, as in Example 4.7, any weakly null sequence converges to zero coordinatewise.
Assume that A is a convex subset of the closed ball B σ (0, r) such that χ(A) > εr. As in Example 4.7, we find sequences {x n } in A and {u n }, {v n } such that sep σ (x n ) ≥ εr, {x n } is weakly convergent, say to x, and lim n σ(x n − x) =: lr does exist; lim n σ(x − u n ) = 0, lim n σ(x n − x − v n ) = 0, supp u n < supp v n and supp v n < supp v n+1 for every n ∈ N. For an arbitrary η > 0, we can choose n, m large enough such that (1.1)-(1.4) in Example 4.7 are satisfied (replacing ρ by σ). The same argument as in Example 4.7 proves that εr − η < σ(v n − v m ) ≤ 2lr + 2η.
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