Weak compactness and the Eisenfeld-Lakshmikantham measure of nonconvexity

Abstract

In this article, weakly compact subsets of real Banach spaces are characterized in terms of the Cantor property for the Eisenfeld-Lakshmikantham measure of nonconvexity. This characterization is applied to prove the existence of fixed points for condensing maps, nonexpansive maps, and isometries without convexity requirements on their domain.

Mathematics Subject Classification 2010: Primary 47H10; Secondary 46B20, 47H08, 47H09.

1. Introduction

Throughout this article, (X, ∥ · ∥) will denote a real Banach space.

Definition 1.1. The Eisenfeld-Lakshmikantham measure of nonconvexity (E-L measure of nonconvexity, for short) of a bounded subset A of X is defined by

$$\mu \left(A\right)=\underset{x\in \overline{\text{co}}A}{\text{sup}}\underset{a\in A}{\text{inf}}∥x-a∥=H\left(A,\overline{\text{co}}A\right),$$

where$\overline{\text{co}}A$denotes the closed and convex hull of A and H(C, D) is the Hausdorff-Pompeiu distance between the bounded subsets C and D of X.

The E-L measure of nonconvexity was introduced in [1]. The following properties of μ can be derived in a fairly straightforward manner from its definition. Here, A, B ⊂ X are assumed to be bounded and $\overline{A}$ denotes the closure of A.

• (i) μ(A) = 0 if, and only if, $\overline{A}$ is convex.

• (ii) μ(λA) = |λ|μ(A) (λ ∈ ℝ).

• (iii) μ(A + B) ≤ μ(A)+μ(B).

• (iv) |μ (A) - μ(B)| ≤ μ (A - B).

• (v) $\mu \left(\overline{A}\right)=\mu \left(A\right)$.

• (vi) μ(A) ≤ δ(A), where

  $$\delta \left(A\right)=\underset{x,y\in A}{\text{sup}}∥x-y∥$$

is the diameter of A.

• (vii) |μ(A) - μ(B)| ≤ 2H(A, B).

The following result was obtained in [2].

Lemma 1.2 ([2, Lemma 2.4]). Let${\left\{A_n\right\}}_{n=1}^{\infty }$be a decreasing sequence of nonempty, closed, and bounded subsets of a Banach space X with

$$\underset{n\to \infty }{\text{lim}}\mu \left(A_n\right)=0,$$

where μ is the E-L measure of nonconvexity of X, and let$A_{\infty }={\bigcap }_{n=1}^{\infty }{A}_n$. Then$A_{\infty }={\bigcap }_{n=1}^{\infty }\overline{\text{co}}{A}_n$.

Definition 1.3. Let Y be a nonempty and closed subset of the Banach space X. The E-L measure of nonconvexity μ of X is said to have the Cantor property in Y if for every decreasing sequence${\left\{A_n\right\}}_{n=1}^{\infty }$of nonempty, closed, and bounded subsets of Y such that$\underset{n\to \infty }{\text{lim}}\mu \left(A_n\right)=0$, the closed and bounded (and, by Lemma 1.2, convex) set$A_{\infty }={\bigcap }_{n=1}^{\infty }{A}_n$is nonempty.

Theorem 1.4 ([2, Theorem 2.5]). For a Banach space X, the following statements are equivalent:

• (i) X is reflexive.

• (ii) The E-L measure of nonconvexity of X satisfies the Cantor property in X.

In Section 2 below we prove a result (Theorem 2.1), more general than Theorem 1.4, which characterizes weak compactness also in terms of the Cantor property for the E-L measure of nonconvexity. As an application of this characterization, we show that the convexity requirements can be dropped from the hypotheses of a number of fixed point theorems in [3–5] for condensing maps (see Section 3.1), nonexpansive maps (see Section 3.2) and isometries (see Section 4).

2. A characterization of weak compactness

Theorem 2.1. Let X be a Banach space with E-L measure of nonconvexity μ, and let C be a nonempty, weakly closed, and bounded subset of X. The following statements are equivalent:

• (i) C is weakly compact.

• (ii) The measure μ satisfies the Cantor property in $\overline{\text{co}}C$.

• (iii) For every decreasing sequence ${\left\{A_n\right\}}_{n=1}^{\infty }$ of nonempty and closed subsets of $\overline{\text{co}}C$ such that $\underset{n\to \infty }{\text{lim}}\mu \left(A_n\right)=0$, the set $A_{\infty }={\bigcap }_{n=1}^{\infty }{A}_n$ is nonempty.

Proof. Part (iii) is just a rephrasement of part (ii).

Suppose (i) holds. By the Krein-Šmulian theorem [6, Theorem V.6.4], $\overline{\text{co}}C$ is weakly compact. Let ${\left\{A_n\right\}}_{n=1}^{\infty }$ be a decreasing sequence of nonempty and closed subsets of $\overline{\text{co}}C$ with lim_n→∞μ(A _n ) = 0. By Lemma 1.2, $A_{\infty }={\bigcap }_{n=1}^{\infty }\overline{\text{co}}{A}_n$, where ${\left\{\overline{\text{co}}{A}_n\right\}}_{n=1}^{\infty }$ is a decreasing sequence of nonempty closed, and convex subsets of the weakly compact and convex set $\overline{\text{co}}C$. The Šmulian theorem [6, Theorem V.6.2] then allows us to conclude that A_∞ is nonempty.

Conversely, assume (iii). If we take any decreasing sequence ${\left\{C_n\right\}}_{n=1}^{\infty }$ of nonempty, closed, and convex subsets of the bounded and convex set $\overline{\text{co}}C$, then μ(C _n ) = 0 (n ∈ ℕ), and therefore $C_{\infty }\ne \varnothing$. Appealing again to the Šmulian theorem [6, Theorem V.6.2] we find that the convex set $\overline{\text{co}}C$ is weakly compact. Finally, being a weakly closed subset of $\overline{\text{co}}C$, the set C itself is weakly compact.

Note that Theorem 1.4 can be easily derived from Theorem 2.1. For the sake of completeness, we give a proof of this fact.

Corollary 2.2. For a Banach space X with E-L measure of nonconvexity μ, the following statements are equivalent:

• (i) X is reflexive.

• (ii) The closed unit ball B _X of X is weakly compact.

• (iii) For every decreasing sequence ${\left\{A_n\right\}}_{n=1}^{\infty }$ of nonempty and closed subsets of B _X such that lim_n→∞ μ(A _n ) = 0, the set $A_{\infty }={\bigcap }_{n=1}^{\infty }{A}_n$ is nonempty and convex.

• (iv) The measure μ satisfies the Cantor property in X.

Proof. The equivalence of (i) and (ii) is well known [6, Theorem V.4.7]. To see that (ii) and (iii) are equivalent, take C = B _X in Theorem 2.1, bearing in mind that $\overline{\text{co}}C=B_X$. For the proof that (iii) implies (iv), let ${\left\{A_n\right\}}_{n=1}^{\infty }$ be a decreasing sequence of nonempty, closed, and bounded subsets of X such that lim_n→∞μ(A _n ) = 0. Since A₁ is bounded and ${\left\{A_n\right\}}_{n=1}^{\infty }$ is decreasing, there exists λ > 0 such that

$$B_n=\lambda A_n\subset B_X\left(n\in \mathbb{N}\right).$$

Now ${\left\{B_n\right\}}_{n=1}^{\infty }$ is a decreasing sequence of nonempty, closed, and bounded subsets of B _X with

$$\underset{n\to \infty }{\text{lim}}\mu \left(B_n\right)=\lambda \underset{n\to \infty }{\text{lim}}\mu \left(A_n\right)=0.$$

Therefore $A_{\infty }={\lambda }^{-1}{B}_{\infty }\ne \varnothing$, as asserted. Finally, it is apparent that (iv) implies (iii).

3. Fixed points for condensing and nonexpansive maps

Definition 3.1 ([2, Definition 4.3]). Let Y be a nonempty, closed, and bounded subset of a Banach space X. A map f : Y → Y is said to have property (C) if lim_n→∞μ(Y _n ) = 0, where μ is the E-L measure of nonconvexity in X and${\left\{Y_n\right\}}_{n=1}^{\infty }$is the decreasing sequence of nonempty, closed, and bounded subsets of X defined by

$$Y_1=\overline{f\left(Y\right)},Y_{n+1}=\overline{f\left(Y_n\right)}\left(n\in \mathbb{N}\right).$$

Proposition 3.2. Let Y be a nonempty and weakly compact subset of a Banach space X, and let f : Y → Y be a map with property (C). Then Y contains a nonempty, closed, and convex (hence, weakly compact) set K such that f(K) ⊂ K.

Proof. Let ${\left\{Y_n\right\}}_{n=1}^{\infty }$ be as above. Since f has property (C), we have

$$\underset{n\to \infty }{\text{lim}}\mu \left(Y_n\right)=0.$$

Theorem 2.1 yields that $K=Y_{\infty }={\bigcap }_{n=1}^{\infty }{Y}_n$ is nonempty, closed, and convex. Clearly, f(K) ⊂ K. Closed convex sets are weakly closed [6, Theorem V.3.18] and therefore K is weakly compact, as claimed.

As an application of Proposition 3.2, some fixed point theorems for condensing and nonexpansive maps will be proved.

3.1. Condensing maps

Definition 3.3. Let Y be a nonempty and bounded subset of a Banach space X, and let γ denote some measure of noncompactness in X, in the sense of [7, Definition 3.2]. A map f : Y → Y is called γ-condensing provided that

$$\gamma \left(f\left(B\right)\right)<\gamma \left(B\right)$$

for every B ⊂ Y with f(B) ⊂ B and γ(B) > 0.

The following result is an extension of [3, Theorem 4]. It can be also viewed as a version of Sadovskii's theorem [8].

Theorem 3.4. Let γ be a measure of noncompactness in a Banach space X and let Y be a nonempty and closed subset of X such that$\overline{co}Y$is weakly compact. Assume that the map f:Y →Y is continuous, γ-condensing and has property (C). Then f has at least one fixed point in Y.

Proof. Arguing as in the proof of Proposition 3.2 we get a nonempty, closed, and convex set K ⊂ Y such that f(K) ⊂ K. The required conclusion follows from [7, Corollary 3.5].

3.2. Nonexpansive maps

Definition 3.5. Let A ⊂ X be bounded. A point x ∈ A is a diametral point of A provided that sup_y∈A∥x - y∥ = δ(A). The set A is said to have normal structure if for each convex subset B of A containing more than one point, there exists some x ∈ B which is not a diametral point of B.

The following is a version of Kirk's seminal theorem (cf. [4, Theorem 4.1]) which does not require the convexity of the domain.

Theorem 3.6. Let Y be a nonempty and weakly compact subset of a Banach space X. Suppose Y has normal structure. If f : Y → Y has property (C) and is nonexpansive, that is, satisfies

$$∥f\left(x\right)-f\left(y\right)∥\le ∥x-y∥\left(x,y\in Y\right),$$

then f has a fixed point.

Proof. The asserted conclusion can be derived from Proposition 3.2 and [4, Theorem 4.1].

4. Fixed points for isometries

Definition 4.1. Let Y be a nonempty and weakly compact subset of a Banach space X. We say that Y has the fixed point property, FPP for short, if every isometry f:Y → Y has a fixed point. The set Y is said to have the hereditary FPP if every nonempty, closed, and convex subset of Y has the FPP.

Definition 4.2. Given a nonempty, closed, and bounded subset Y of a Banach space X, let

$$\begin{array}{ll}\hfill r\left(x\right)& =r\left(x,Y\right)=\underset{y\in Y}{\text{sup}}∥x-y∥\left(x\in X\right),\\ r\left(Y\right)=\underset{x\in Y}{\text{inf}}r\left(x\right),\end{array}$$

and

$$\tilde{Y}=\left\{x\in Y:r\left(x\right)=r\left(Y\right)\right\}.$$

The number r(Y) and the members of Ỹ are respectively called Chebyshev radius and Chebyshev centers of Y. Further, define

$$\begin{array}{ll}\hfill {\tilde{Y}}_n& =\left\{x\in Y:r\left(Y\right)\le r\left(x\right)\le r\left(Y\right)+\frac{1}{n}\right\}\\ =\bigcap _{y\in Y}\left[y+\left(r\left(Y\right)+\frac{1}{n}\right)B_X\right]\cap Y\left(n\in \mathbb{N}\right).\end{array}$$

We say that Y has property (S) provided that lim_n→∞μ(Ỹ _n ) = 0, where μ is the E-L measure of nonconvexity in X.

Lemma 4.3. Let Y be a nonempty and weakly compact subset of a Banach space X. If Y has property (S), then Ỹ is nonempty, closed, and convex.

Proof. Note that ${\left\{{Ỹ}_n\right\}}_{n=1}^{\infty }$ is a decreasing sequence of nonempty and closed subsets of Y, with lim_n→∞μ(Ỹ _n ) = 0. From Theorem 2.1, the set of Chebyshev centers

$$\tilde{Y}={\tilde{Y}}_{\infty }=\bigcap _{n=1}^{\infty }{\tilde{Y}}_n$$

is nonempty closed, and convex.

Theorem 4.4. Let Y be a nonempty and weakly compact subset of a Banach space X. Assume further that Y has both property (S) and the hereditary FPP. Then every isometry f:Y → Y such that f(Ỹ) ⊂ Ỹ has a fixed point in Ỹ.

Proof. From Lemma 4.3, Ỹ is nonempty, closed, and convex. It suffices to invoke the hereditary FPP of Y.

Definition 4.5. Let Y be a nonempty, closed, and bounded subset of a Banach space X. Given an isometry f:Y → Y, let us consider

$$R_{f,0}\left(x\right)=r\left(x,Y\right)=\underset{z\in Y}{\text{sup}}∥x-z∥\left(x\in X\right),$$

$$\begin{array}{c}{R}_{f,m}\left(x\right)=r\left(x,Y_m\right)=\underset{z\in Y_m}{\text{sup}}∥x-z∥\\ =r\left(x,f^m\left(Y\right)\right)=\underset{y\in Y}{\text{sup}}∥x-f^m\left(y\right)∥\left(x\in X,m\in \mathbb{N}\right),\end{array}$$

$$\begin{array}{c}{R}_f\left(x\right)=\underset{m\to \infty }{\text{lim}}{R}_{f,m}\left(x\right)=\underset{m\in {\mathbb{Z}}_{+}}{\text{inf}}{R}_{f,m}\left(x\right)\left(x\in X\right),\\ R_f\left(Y\right)=\underset{x\in Y}{\text{inf}}{R}_f\left(x\right),\end{array}$$

and

$${\hat{Y}}_f=\left\{x\in Y:R_f\left(x\right)=R_f\left(Y\right)\right\}.$$

The number R _f (Y) and the set Ŷ _f are respectively called asymptotic Chebyshev radius and asymptotic Chebyshev center of${\left\{Y_m\right\}}_{m=0}^{\infty }={\left\{f^m\left(Y\right)\right\}}_{m=0}^{\infty }$with respect to Y. Further, define

$$\begin{array}{ll}\hfill {\hat{Y}}_{f,n}& =\left\{x\in Y:R_f\left(Y\right)\le R_f\left(x\right)\le R_f\left(Y\right)+\frac{1}{n}\right\}\\ =\bigcap _{m\in {\mathbb{Z}}_{+}}\bigcap _{z\in Y_m}\left[z+\left(R_f\left(Y\right)+\frac{1}{n}\right)B_X\right]\cap Y\left(n\in \mathbb{N}\right).\end{array}$$

We say that f has property (A) provided that lim_n→∞μ(Ŷ _f,n ) = 0, where μ is the E-L measure of nonconvexity in X.

Lemma 4.6. Let Y be a nonempty and weakly compact subset of a Banach space X, and let f:Y → Y be an isometry with property (A). Then Ŷ _f is nonempty, closed, and convex.

Proof. Note that ${\left\{{\hat{Y}}_{f,n}\right\}}_{n=1}^{\infty }$ is a decreasing sequence of nonempty and closed subsets of Y, with ${\text{lim}}_{n\to \infty }\mu \left({\hat{Y}}_{f,n}\right)=0$. From Theorem 2.1, the asymptotic Chebyshev center

$${\hat{Y}}_f={\hat{Y}}_{f,\infty }=\bigcap _{n=1}^{\infty }{\hat{Y}}_{f,n}$$

is nonempty closed, and convex.

Lemma 4.7. Let Y be a nonempty and weakly compact subset of a Banach space X, and let f : Y → Y be an isometry. Assume c ∈ Ŷ _f is such that f(c) = c. then c ∈Ỹ.

Proof. We argue as in the proof of [5, Theorem 2]. Since f is an isometry and f(c) = c, we have

$$R_{f,m}\left(c\right)=R_{f,m}\left(f\left(c\right)\right)=R_{f,m-1}\left(c\right)\left(m\in \mathbb{N}\right),$$

whence

$$R_{f,m}\left(c\right)=R_{f,0}\left(c\right)\left(m\in \mathbb{N}\right).$$

From Definition 4.5 and the hypothesis that c ∈ Ŷ _f , it follows that

$$r\left(c,Y\right)=R_{f,0}\left(c\right)=\underset{m\to \infty }{\text{lim}}{R}_{f,m}\left(c\right)=R_f\left(c\right)=R_f\left(Y\right).$$

Now, for any x ∈ Y we get

$$r\left(c,Y\right)=R_f\left(Y\right)\le \underset{m\in {\mathbb{Z}}_{+}}{\text{inf}}{R}_{f,m}\left(x\right)\le R_{f,0}\left(x\right)=r\left(x,Y\right),$$

which proves that c ∈ Ỹ.

Theorem 4.8. Let Y be a nonempty and weakly compact subset of a Banach space X. Suppose Y has the hereditary FPP. Then every isometry f:Y → Y with property (A) has a fixed point in Ỹ.

Proof. Let f : Y → Y be an isometry with property (A). From Lemma 4.6, Ŷ _f is nonempty, closed, and convex. Moreover, f(Ŷ _f ) ⊂ Ŷ _f (cf. [5, Proposition 3]). The hereditary FPP of Y then yields c ∈ Ŷ _f such that f(c) = c, and Lemma 4.7 ensures that c ∈ Ỹ.

Corollary 4.9 ([5, Theorem 2]). Let Y be a nonempty, weakly compact, and convex subset of a Banach space X. Suppose Y has the hereditary FPP. Then every isometry f:Y → Y has a fixed point in Ỹ.

Proof. Since Y is convex, every isometry f : Y → Y has property (A). Theorem 4.8 completes the proof.

The following is an extension of Kirk's theorem [4, Theorem 4.1] for isometries.

Theorem 4.10. Let Y be a nonempty and weakly compact subset of a Banach space X. Assume further that Y has normal structure. Then every isometry f:Y → Y with property (A) has a fixed point in Ỹ.

Proof. Let f : Y → Y be an isometry with property (A). From Lemma 4.6, Ŷ _f is nonempty, closed, and convex. Moreover, f(Ŷ _f ) ⊂ Ŷ _f (cf. [5, Proposition 3]). Kirk's theorem [4, Theorem 4.1] along with Lemma 4.7 yield c ∈ Ỹ such that f(c) = c.

Corollary 4.11 ([5, Corollary 1]). Let Y be a nonempty, weakly compact, and convex subset of a Banach space X. Assume further that Y has normal structure. Then every isometry f:Y → Y has a fixed point in Ỹ.

Proof. The convexity of Y guarantees that every isometry f : Y → Y satisfies property (A). The desired conclusion follows from Theorem 4.10.