The Concept of Measures of Noncompactness in Banach Spaces

Abstract

This article is a survey, and its aim is to provide a concise introduction to the topic of measures of noncompactness and measures of weak noncompactness in Banach spaces. These measures constitute a useful tool in nonlinear analysis, for example, in studies on the existence of solutions to nonlinear differential and integral equations. Recently, they have also been applied to the analysis of infinite systems of such equations. Throughout the paper, particular attention is given to highlighting the symmetry that exists between these concepts. Some open problems are also included at the end of the paper.

      Dedicated to Professor Józef Banaś

      on the occasion of his 75th birthday

1. Introduction

What a wonderful property compactness is!—this is how Klaus Jänich describes compactness in his well-known book Topology. Indeed, the compactness of a set or an operator plays a key role in the proofs of many important theorems, such as Schauder’s and Tikhonov’s fixed-point theorems. On the other hand, in many problems—e.g., those concerning operator equations—the considered sets or operators are not compact, and for studying such problems, techniques related to measures of noncompactness are very useful. In many cases, they are the only known analytical tool. As a result, measures of noncompactness find wide applications in studies concerning nonlinear differential and integral equations, as well as in fixed-point theory. Recently, they have been successfully used in investigations of the existence of solutions to infinite systems of such equations in various Banach spaces (see, e.g., [1,2,3,4]). Intuitively, measures of noncompactness are tools that quantify how far a set is from being compact.

The concept of a measure of noncompactness (in metric spaces) was introduced by K. Kuratowski [5] in 1930. In 1955, using Kuratowski’s measure of noncompactness, G. Darbo [6] generalized the previously mentioned Schauder theorem to the case of sets that are not compact.

The concept of the Kuratowski measure of noncompactness was modified in 1957 [7] (see also [8]) by I. T. Gohberg, L. S. Goldenstein, and A. S. Markus, who introduced the so-called Hausdorff measure of noncompactness. In 1972, V. I. Istrăţescu [9] defined another interesting measure of noncompactness. All these measures have the same fundamental properties, which are essential in applications. This fact suggested the idea of an axiomatic approach to defining measures of noncompactness. The first who went this way was B. N. Sadovskii [10]. In monographs [11,12,13], three different ways of axiomatically introducing measures of noncompactness are presented. In this paper, we accept the one that was introduced in 1980 by J. Banaś and K. Goebel [13].

In addition to measures of noncompactness (sometimes referred to as measures of noncompactness in the strong sense), one also considers measures of weak noncompactness. This notion was introduced by F. S. De Blasi [14] in 1977. Moreover, an analogue of the already mentioned Darbo theorem, using De Blasi’s measure of weak noncompactness, was proved by De Blasi [14] in the case of separable spaces and by G. Emmanuele [15] for an arbitrary Banach space. An axiomatic approach to the notion of a measure of weak noncompactness—which is very similar to the approach associated with the notion of a measure of noncompactness in the strong sense—was proposed in 1988 by J. Banaś and J. Rivero [16].

The structure of the paper is as follows: Section 2 presents the axiomatic approach to the concept of measures of noncompactness in Banach spaces, proposed by J. Banaś and K. Goebel in their well-known and frequently cited monograph [13]. It also introduces the definition of the measure of weak noncompactness in Banach spaces, proposed by J. Banaś and J. Rivero in [16]. The relationships between the axioms defining measures of noncompactness and those defining measures of weak noncompactness are discussed as well. Section 3 provides examples of measures of noncompactness and weak noncompactness (including those of Kuratowski, Hausdorff, Istrăţescu, and De Blasi), along with a discussion of selected properties and interrelations among these measures. Special attention is given to the high usefulness of the Hausdorff measure of noncompactness in applications. Section 4 presents two generalizations of the classical Schauder fixed-point theorem (namely, the theorems of Darbo and Sadovskii). Section 5 mentions some open problems and current trends in the theory of measures of noncompactness.

The aim of this paper is to gather in one place the most important information concerning measures of noncompactness and measures of weak noncompactness in Banach spaces, with particular emphasis on the symmetry that exists between these concepts. This article has a survey character and is based on the papers listed in the references.

Throughout this paper, we assume that all considered Banach spaces are over the field of real numbers.

2. The Axiomatic Approach of Banaś and Goebel to the Concept of Measures of Noncompactness

Assume that E is a (real) Banach space. We will denote by ${𝔐}_E$ the family of all nonempty and bounded subsets of E and by ${𝔑}_E,{𝔑}_E^w$ its subfamilies consisting of all relatively compact and relatively weakly compact sets, respectively. The symbol $\overline{X}$ stands for the closure of a set X, while ${\overline{X}}^w$ denotes its weak closure. By $\mathrm{Conv}X$, we denote the closure of the convex hull of X.

We start with the definitions of a measure of noncompactness [13] and a measure of weak noncompactness [16].

Definition 1.

A mapping $\mu :{𝔐}_E\to [0,+\infty )$ is called a measure of noncompactness (in the strong sense) in the space E if it satisfies the following conditions:

S1°

The family $\ker \mu =\{X\in {𝔐}_E:\mu (X)=0\}$ is nonempty and $\ker \mu \subset {𝔑}_E$. This family is called the kernel of the measure of noncompactness μ. The set $E_{\mu }=\{x\in E:\left\{x\right\}\in \ker \mu \}$ is called the kernel set of the measure of noncompactness μ.

S2°

If $X,Y\in {𝔐}_E$ and $X\subset Y$, then $\mu (X)⩽\mu (Y).$

S3°

$\mu (X)=\mu (\mathit{Conv}X),X\in {𝔐}_E.$

S4°

$\mu (\lambda X+(1-\lambda )Y)⩽\lambda \mu (X)+(1-\lambda )\mu (Y),X,Y\in {𝔐}_E,\lambda \in [0,1].$

S5°

If $(X_n)$ is a sequence of closed sets belonging to ${𝔐}_E$, such that $X_{n+1}\subset X_n$ for $n=1,2,\dots ,$ and if ${lim}_{n\to \infty }\mu (X_n)=0,$ then the intersection $X_{\infty }={\bigcap }_{n=1}^{\infty }{X}_n$ is nonempty.

Definition 2.

A mapping $\mu :{𝔐}_E\to [0,+\infty )$ is called a measure of weak noncompactness in the space E if it satisfies the following conditions:

W1°

The family $\ker \mu =\{X\in {𝔐}_E:\mu (X)=0\}$ is nonempty and $\ker \mu \subset {𝔑}_E^w$. This family is called the kernel of the measure of weak noncompactness μ. The set $E_{\mu }=\{x\in E:\left\{x\right\}\in \ker \mu \}$ is called the kernel set of the measure of weak noncompactness μ.

W2°

If $X,Y\in {𝔐}_E$ and $X\subset Y$, then $\mu (X)⩽\mu (Y).$

W3°

$\mu (X)=\mu (\mathit{Conv}X),X\in {𝔐}_E.$

W4°

$\mu (\lambda X+(1-\lambda )Y)⩽\lambda \mu (X)+(1-\lambda )\mu (Y),X,Y\in {𝔐}_E,\lambda \in [0,1].$

W5°

If $(X_n)$ is a sequence of weakly closed sets belonging to ${𝔐}_E$, such that $X_{n+1}\subset X_n$ for $n=1,2,\dots ,$ and if ${lim}_{n\to \infty }\mu (X_n)=0,$ then the intersection $X_{\infty }={\bigcap }_{n=1}^{\infty }{X}_n$ is nonempty.

Remark 1.

Conditions S2°, S3°, and S4° (W2°, W3°, and W4°) are called monotonicity, invariance under passage to the closed convex hull, and convexity, respectively. The property of the function μ described in S5° (W5°) is called the Cantor intersection property, since it is a generalization of the classical Cantor intersection theorem.

In what follows, we define a class of measures having additional good properties.

Definition 3.

Let μ be a measure of noncompactness (a measure of weak noncompactness) in E and let $X,Y\in {𝔐}_E$, $x\in X$. We say that

• μ is full if $\ker \mu ={𝔑}_E(\ker \mu ={𝔑}_E^w).$
• μ has maximum property if
  $6^{\circ }\hspace{1em}\mu (X\cup Y)=max\{\mu (X),\mu (Y)\}$.
• μ is sublinear if
  $7^{\circ }\hspace{1em}\mu (X+Y)⩽\mu (X)+\mu (Y)(\mathit{subadditive}\mathit{property}),$ and
  $8^{\circ }\hspace{1em}\mu (\lambda X)=\left|\lambda \right|\mu (X),(\lambda \in \mathbb{R})(\mathit{homogeneous}\mathit{property}).$
• μ is regular if it is full, has maximum property, and it is sublinear.
• μ has weak maximum property if
  $9^{\circ }\hspace{1em}\mu (X\cup \left\{x\right\})=\mu (X).$
• μ is invariant under translations if
  ${10}^{\circ }\hspace{1em}\mu (X+x)=\mu (X).$

Let us note some useful observations that follow from our definitions.

Remark 2.

(a) 

Let us assume that the mapping μ satisfies conditions S2° and S3° (W2° and W3°). Then

$$\mu (X)\le \mu (\overline{X})\le \mu (\mathit{Conv}X)=\mu (X)$$

$$(\mu (X)\le \mu ({\overline{X}}^w)\le \mu (\mathit{Conv}X)=\mu (X))$$

and consequently $\mu (X)=\mu (\overline{X})(\mu (X)=\mu ({\overline{X}}^w)),$ so μ is invariant under passage to the closure (weak closure).

(b) 

If the mapping μ satisfies conditions S1°, S2°, and S5° (W1°, W2°, and W5°), then the set $X_{\infty }\in \ker \mu$ and is compact (weakly compact). This follows directly from inequalities $0\le \mu (X_{\infty })\le \mu (X_n),$ which hold for $n=1,2,\dots$, and the properties of closed (weakly closed) sets.

(c) 

Let us assume that $\ker \mu ={𝔐}_E$. Then condition S3° (W3°) is satisfied for relatively compact sets (relatively weakly compact sets). This follows from the well-known Mazur’s theorem and its “weak” counterpart.

(d) 

Conditions 7° and 8° imply condition 4°.

(e) 

Condition 6° implies condition S2° (W2°).

(f) 

Condition S2° (W2°) implies that $\mu (X\cap Y)\le min\{\mu (X),\mu (Y)\}.$

(g) 

Let us assume that $x\in E_{\mu }$ for any $x\in E$ and condition 6° is satisfied. Then condition 9° is fulfilled.

Moreover, we have the following (see [12,16]):

Theorem 1.

Let $\mu :{𝔐}_E\to [0,+\infty )$ be a mapping satisfying conditions S1°, S2° and 9° (W1°, W2° and 9°). Then μ satisfies condition S5° (W5°).

Proof. 

Let $(X_n)$ be a sequence of sets from ${𝔐}_E$ such that

$$X_{n+1}\subset X_n,{\overline{X}}_n=X_n({\overline{X}}_n^w=X_n),n=1,2,\dots ,$$

and

$$\underset{n\to \infty }{lim}\mu (X_n)=0.$$

Further, take an arbitrary sequence $(x_n)$ such that $x_n\in X_n$ for $n\in \mathbb{N}$. Then, we have

$$0\le \mu (\{x_1,x_2,\dots \})\le \mu (\{x_n,x_{n+1},\dots \})\le \mu (X_n),n\ge 1.$$

Thus, we have

$$\mu (\{x_1,x_2,\dots \})=0,$$

and consequently the set

$$\{x_1,x_2,\dots \}$$

is relatively compact (relatively weakly compact). Let $\overline{x}$ be the limit of a subsequence of $(x_n)$. Obviously, $\overline{x}\in X_n$ for $n\in \mathbb{N}$, and hence, $X_{\infty }\ne \varnothing .$ □

Definition 4.

Let ${\mu }_1$ and ${\mu }_2$ be measures of noncompactness or measures of weak noncompactness in $E.$ We say that ${\mu }_1$ and ${\mu }_2$ are:

(a) 

comparable if there exists positive number $\alpha ,$ such that

$${\mu }_1(X)\le \alpha {\mu }_2(X),X\in {𝔐}_E.$$

(b) 

equivalent if there exist two positive numbers $\alpha ,\beta ,$ such that

$$\alpha {\mu }_1(X)\le {\mu }_2(X)\le \beta {\mu }_1(X),X\in {𝔐}_E.$$

Let us also note that by modifying certain conditions in Definition 1, one can define a measure of noncompactness under the assumption that the underlying space is a complete metric space.

Definition 5.

Let E be a complete metric space. The function $\mu :{𝔐}_E\to [0,+\infty )$ is said to be a measure of noncompactness in E if it satisfies the following conditions:

M1°

The family $\ker \mu =\{X\in {𝔐}_E:\mu (X)=0\}$ is nonempty and $\ker \mu \subset {𝔑}_E$.

M2°

If $X,Y\in {𝔐}_E$ and $X\subset Y$, then $\mu (X)⩽\mu (Y).$

M3°

$\mu (X)=\mu (\overline{X}),X\in {𝔐}_E.$

M4°

If $(X_n)$ is a sequence of closed sets belonging to ${𝔐}_E$, such that $X_{n+1}\subset X_n$ for $n=1,2,\dots ,$ and if ${lim}_{n\to \infty }\mu (X_n)=0,$ then the intersection $X_{\infty }={\bigcap }_{n=1}^{\infty }{X}_n$ is nonempty.

In the remainder of this paper, we will consider measures of noncompactness only in Banach spaces.

3. Examples of Measures of Noncompactness

In this section, we provide some examples of measures of noncompactness and measures of weak noncompactness (see [11,12,13,16,17,18,19,20]). We start with measures of noncompactness (in the strong sense).

Example 1.

Let $X\in {𝔐}_E$. We define the functions

$$\alpha (X)=inf\{\epsilon >0:X\mathit{has}a\mathit{finite}\mathit{covering}\mathit{by}\mathit{sets}\mathit{with}\mathit{diameter}\le \epsilon \},$$

$$\chi (X)=inf\{\epsilon >0:X\mathit{has}a\mathit{finite}\epsilon –\mathit{net}\},$$

$$\beta (X)=inf\{\epsilon >0:X\mathit{has}\mathit{no}\mathit{infinite}\epsilon –\mathit{discrete}\mathit{subset}\},$$

$${\mu }_1(X)=\mathit{diam}(X),$$

$${\mu }_2(X)=\parallel X\parallel =sup\{\parallel x\parallel :x\in X\},$$

$${\mu }_3(X)={(\mathit{diam}(X))}^2.$$

Recall that the set $A\subset X\subset E$ is said to be ε–discrete if $\parallel x-y\parallel \ge \epsilon$ for all $x,y\in A$ with $x\ne y$.

The function α is the Kuratowski measure of noncompactness (or set measure of noncompactness) [5], the second one is the Hausdorff measure of noncompactness (or ball measure of noncompactness) [7], and the third one is the Istrăţescu measure of noncompactness (or lattice measure of noncompactness) [9]. As noted in the introduction, the function α was the first measure of noncompactness introduced in mathematics.

Theorem 2.

(a) 

The functions $\alpha ,\chi$, and β are regular measures of noncompactness and they are invariant under translations. Moreover,

$$\chi (X)\le \beta (X)\le \alpha (X)\le 2\chi (X),X\in {𝔐}_E,$$

so all these three measures of noncompactness are equivalent.

(b) 

The functions ${\mu }_1,{\mu }_2$, and ${\mu }_3$ are measures of noncompactness which are not regular. The kernels of the measures ${\mu }_1$ and ${\mu }_3$ are the families of all singletons, while the kernel of the measure ${\mu }_2$ consists only of the set $\left\{0\right\}$. Moreover, the measure ${\mu }_1$ is sublinear but does not have the maximum property, while the measure ${\mu }_2$ is sublinear with the maximum property. The measure ${\mu }_3$ is not sublinear and does not have the maximum property.

For the proof of this theorem, see [17] and the references therein; see also [11,12,13].

Now, we give an example of a measure of weak noncompactness.

Example 2.

Let $X\in {𝔐}_E$. We define the function

$$\omega (X)=inf\{ϵ>0:X\mathit{has}\mathrm{a}\mathit{weakly}\mathit{compact}\epsilon \text{–}\mathit{net}\}.$$

The function ω is the De Blasi measure of weak noncompactness [14].

Theorem 3.

The function ω is a regular measure of weak noncompactness, and it is invariant under translations. Moreover,

$$\omega (X)\le \chi (X),X\in {𝔐}_E,$$

so the De Blasi measure of weak noncompactness and the Hausdorff measure of noncompactness are comparable.

For the proof of this theorem, see [11].

Providing the exact value of a measure of noncompactness or of weak noncompactness for a given (nonempty and bounded) subset of a space E is usually not easy. For example, determining the Kuratowski measure of noncompactness of a ball in an infinite-dimensional Banach space was a difficult problem. It was not until 1970 (i.e., forty years (!) after the publication of Kuratowski’s paper [5]) that R. D. Nussbaum [21], and independently M. Furi and A. Vignoli [22], solved this problem.

Theorem 4.

Assume that $B(a,r)\subset E$ is a closed (or open) ball with center a and radius r.

(a) 

If $dimE<\infty$, then $\alpha (B(a,r))=\chi (B(a,r))=\beta (B(a,r))=0$.

(b) 

If $dimE=\infty$, then $\alpha (B(a,r))=2r$, $\chi (B(a,r))=r$, and $\beta (B(a,r))\ge r$.

(c) 

If E is a reflexive space, then $\omega (B(a,r))=0;\mathit{even}\omega (X)=0\mathit{for}X\in {𝔐}_E.$

(d) 

If E is a nonreflexive space, then $\omega (B(a,r))=r$.

The proof of this theorem can be found in [11,12,13] for parts (a) and (b), and in [11] for parts (c) and (d); note that part (c) is obvious.

Remark 3.

The Istrăţescu measure of noncompactness of the ball $B(a,r)$ is not the same for all Banach spaces but depends on the geometry of the space E.

The Hausdorff measure of noncompactness is much more convenient in applications (i.e., in the theory of integral equations) than the Kuratowski and Istrăţescu measures of noncompactness, because in some Banach spaces, we are able to give useful formulas expressing the Hausdorff measure. On the other hand, such formulas for the Kuratowski and Istrăţescu measures are not known.

Theorem 5.

(a) 

In the Banach space $c_0$ we have

$$\chi (X)=\underset{n\to \infty }{lim}\left\{\underset{x\in X}{sup}\left\{max\left\{\right|x_k|:k\ge n\}\right\}\right\},X\in {𝔐}_{c_0}.$$

(b) 

In the Banach space $C([a,b])$ we have

$$\chi (X)={\displaystyle \frac{1}{2}}\underset{\epsilon \to 0^{+}}{lim}\left\{\underset{x\in X}{sup}\left\{sup\left\{|x(t)-x(s)|:t,s\in [a,b],|t-s|\le \epsilon \right\}\right\}\right\}$$

for $X\in {𝔐}_{C([a,b])}$.

(c) 

In the Banach space c the situation is more complicated. In this space, we do not know a formula expressing the Hausdorff measure of noncompactness χ, but it can be shown that the quantity

$$\mu (X)=\underset{n\to \infty }{lim}\left\{\underset{x\in X}{sup}\left\{sup\left\{\right|x_p-x_q|:p,q\ge n\}\right\}\right\},X\in {𝔐}_c,$$

defines a regular measure of noncompactness that is equivalent to the Hausdorff measure χ.

(d) 

In the Banach space $l^p$, $p\in [1,\infty )$ we have

$$\chi (X)=\underset{n\to \infty }{lim}\left\{\underset{x\in X}{sup}{\left(\sum _{k=n}^{\infty }{\left|x_k\right|}^p\right)}^{1/p}\right\},X\in {\mathfrak{M}}_{l^p}.$$

(e) 

In the Banach space $L^p(a,b)$, $p\in [1,\infty )$ we do not know a formula for the Hausdorff measure χ, but (similar to the case of the space c) we can provide a formula expressing a regular measure of noncompactness that is equivalent to the Hausdorff measure χ.

For the proof of this theorem, see [18] and the references therein.

Remark 4.

Observe that in the definition of the Hausdorff measure of noncompactness of the set X, it is not assumed that the centers of the balls covering X belong to X. If these centers are in X, we have the following quantity:

$${\chi }_i(X)=inf\{\epsilon >0:X\mathit{has}\mathrm{a}\mathit{finite}\epsilon –\mathit{net}\mathit{in}X\},X\in {𝔐}_E.$$

However, this quantity, called the inner Hausdorff measure of noncompactness, is not a measure of noncompactness in the sense of Definition 1 because, for example, it is not invariant under passage to the convex hull, which is a property of great importance in applications.

Now, we present a few examples of measures of noncompactness that are not regular. Such measures are mainly constructed in those Banach spaces in which we do not know the necessary and sufficient conditions for relative compactness of sets. Our examples will be given in the Banach space $BC({\mathbb{R}}_{+})$ consisting of all functions $x:[0,+\infty )\to \mathbb{R}$ which are continuous and bounded with the classical supremum norm.

To this end, fix $\epsilon >0$, $T>0$, $X\in {𝔐}_{BC({\mathbb{R}}_{+})}$ and $x\in X$. Let us put

$${\omega }^T(x,\epsilon )=sup\left\{\right|x(t)-x(s)|:t,s\in [0,T],|t-s|\le \epsilon \}$$

and

$${\omega }^T(X,\epsilon )=sup\{{\omega }^T(x,\epsilon ):x\in X\}.$$

Clearly the function $\epsilon \to {\omega }^T(X,\epsilon )$ is nondecreasing, so there exists a finite limit

$${\omega }_0^T(X)=\underset{\epsilon \to 0^{+}}{lim}{\omega }^T(X,\epsilon ).$$

Now we put

$${\omega }_0(X)=\underset{T\to \infty }{lim}{\omega }_0^T(X).$$

The above quantity is not a measure of noncompactness in $BC({\mathbb{R}}_{+})$ [23]. In order to define measures of noncompactness in this space with the help of ${\omega }_0(X)$, let us define

$$a(X)=\underset{T\to \infty }{lim}\left\{\underset{x\in X}{sup}\{sup\{\left|x(t)\right|:t\ge T\left\}\right\}\right\},$$

$$b(X)=\underset{T\to \infty }{lim}\left\{\underset{x\in X}{sup}\{sup\{|x(t)-x(s)|:t,s\ge T\left\}\right\}\right\},$$

$$c(X)=\underset{t\to \infty }{lim\; sup}\mathrm{diam}X(t),$$

where $X(t)=\{x(t):x\in X\}$.

Theorem 6.

The functions

$${\mu }_a(X)={\omega }_0(X)+a(X),$$

$${\mu }_b(X)={\omega }_0(X)+b(X),$$

$${\mu }_c(X)={\omega }_0(X)+c(X)$$

are measures of noncompactness in the space $BC({\mathbb{R}}_{+})$ which are not regular. Moreover, for any $X\in {𝔐}_{BC({\mathbb{R}}_{+})}$ we have

$$\chi (X)\le {\mu }_b(X)\le 2{\mu }_a(X),$$

$$\chi (X)\le {\mu }_c(X)\le 2{\mu }_a(X),$$

so all these three measures of noncompactness are comparable with the Hausdorff measure of noncompactness.

The proof of this theorem can be found in [23].

It is rather difficult to express the De Blasi measure of weak noncompactness with the help of a convenient formula in a concrete Banach space. Such a formula is only known in the space $L^1(I)$, where I is a bounded interval in $\mathbb{R}$ and has been given by J. Appell and E. De Pascale [24].

Theorem 7.

In the Banach space $L^1(I)$, where $I\subset \mathbb{R}$ is a bounded interval we have

$$\omega (X)=\underset{\epsilon \to 0^{+}}{lim}\left\{\underset{x\in X}{sup}\left\{sup\left\{{\int }_D\left|x(t)\right|dt:D\subset I,m(D)\le \epsilon \right\}\right\}\right\},X\in {𝔐}_{L^1(I)}.$$

The construction of a useful measure of weak noncompactness in the space $L^1({\mathbb{R}}_{+})$ is a bit more complicated and has been carried out by J. Banaś and Z. Knap [25].

To do this, take $X\subset {𝔐}_{L^1({\mathbb{R}}_{+})}$ and define

$$c(X)=\underset{\epsilon \to 0^{+}}{lim}\left\{\underset{x\in X}{sup}\left\{sup\left\{{\int }_D\left|x(t)\right|dt:D\subset {\mathbb{R}}_{+},m(D)\le \epsilon \right\}\right\}\right\},$$

$$d(X)=\underset{T\to \infty }{lim}\left\{\underset{x\in X}{sup}{\int }_T^{\infty }\left|x(t)\right|dt\right\}.$$

Finally, we put

$$\gamma (X)=c(X)+d(X).$$

Theorem 8.

The function γ is a regular measure of weak noncompactness in the space $L^1({\mathbb{R}}_{+})$. Moreover,

$$\omega (X)\le \gamma (X)\le 2\omega (X),X\in {𝔐}_{L^1({\mathbb{R}}_{+})},$$

so the measures γ and De Blasi are equivalent.

The proof of this theorem can be found in [25].

4. The Darbo and Sadovskii Theorems

By using the concept of measures of noncompactness or measures of weak noncompactness, one can relatively easily generalize the classical Schauder fixed-point theorem. In this generalization, neither the compactness of the set nor the compactness of the operator is assumed. As we noted in the introduction, the first such generalization was formulated by G. Darbo [6], who, in his considerations, used the Kuratowski measure of noncompactness. However, the reasoning he presented also holds in a more general setting. Indeed, we have the following:

Theorem 9.

(Darbo). Let Ω be a nonempty, closed, bounded, and convex subset of a Banach space E, and let $\mu :{𝔐}_E\to [0,+\infty )$ be a mapping satisfying conditions S1°, S2°, S3°, and S5° (W1°, W2°, W3°, and W5°). Assume that $T:\Omega \to \Omega$ is μ—contraction, i.e., T is a continuous (weakly continuous) mapping such that there exists a constant $k\in [0,1)$ satisfying the condition

$$\begin{array}{c}\mu (T(X))\le k\mu (X)\end{array}$$

(1)

for any nonempty subset $X\subset \Omega$. Then, T has at least one fixed point in Ω and the set of all fixed points of T in Ω belongs to $\ker \mu$, i.e., $\mu (\{x\in \Omega :T(x)=x\})=0.$

Proof. 

Consider the sequence of sets $(X_n)$ defined as follows:

$$X_0=\Omega ,X_{n+1}=ConvT(X_n),n\ge 0.$$

By condition $S{3}^{\circ }$ ($W{3}^{\circ }$) we have

$$\mu (X_{n+1})=\mu (ConvT(X_n))=\mu (T(X_n))\le k\mu (X_n),n\ge 0.$$

Using the inequality above and applying the principle of mathematical induction, we obtain

$$0\le \mu (X_n)\le k^n\mu (X_0),n\ge 0$$

and consequently ${lim}_{n\to \infty }\mu (X_n)=0$.

Applying the induction principle again, we easily see that $T(X_n)\subset X_n$ for $n\ge 0$. Hence, it follows that $X_{n+1}\subset X_n$ for $n\ge 0$.

Using condition $S{5}^{\circ }$ ($W{5}^{\circ }$) and Remark 2 (b), we find that the set $X_{\infty }={\bigcap }_{n=1}^{\infty }{X}_n$ is nonempty and compact (weakly compact). Of course, it is also convex. Moreover, we have

$$T(X_{\infty })=T\left(\bigcap _{n=1}^{\infty }{X}_n\right)\subset \bigcap _{n=1}^{\infty }T(X_n)\subset \bigcap _{n=1}^{\infty }{X}_n=X_{\infty }.$$

From Schauder’s (Tikhonov’s) fixed-point theorem applied to the mapping $T_{{|}_{X_{\infty }}}:X_{\infty }\to X_{\infty }$, we find that T has at least one fixed point in the set $X_{\infty }\subset \Omega$. The last statement follows directly form (1). □

Remark 5.

For $k=1$, the above theorem does not hold even in Hilbert spaces. A corresponding example can be found, for instance, in [12,13,17].

In 1967, in [26], B. N. Sadovskii, using the Hausdorff measure of noncompactness, slightly generalized the above Darbo theorem. In his proof, Sadovskii made use of the axiom of choice (more precisely, he applied the principle of transfinite induction). Nowadays, various proofs of this theorem are known in the literature, but it seems that the most beautiful and elementary proof was given by G. Emmanuele [15].

Theorem 10.

(Sadovskii). Let Ω be a nonempty, closed, bounded, and convex subset of a Banach space E, and let $\mu :{𝔐}_E\to [0,+\infty )$ be a mapping satisfying conditions $S{1}^{\circ }$, $S{3}^{\circ }$, and $9^{\circ }$ ($W{1}^{\circ }$, $W{3}^{\circ }$, and $9^{\circ }$). Assume that $T:\Omega \to \Omega$ is μ—condensing, i.e., T is a continuous (weakly continuous) mapping such that for any nonempty subset $X\subset \Omega$, if $\mu (X)>0$, then

$$\begin{array}{c}\mu (T(X))<\mu (X).\end{array}$$

(2)

Then T has at least one fixed point in Ω and the set of all fixed points of T in Ω belongs to $\ker \mu$, i.e., $\mu (\{x\in \Omega :T(x)=x\})=0.$

Proof. 

Let $\mathcal{A}$ denote the set of all closed, convex subsets X of $\Omega$ for which $x_0\in X$ and $T(X)\subset X$. Let us denote $X_{\infty }={\bigcap }_{X\in \mathcal{A}}X$ and put $Y=Conv(T(X_{\infty })\cup \left\{x_0\right\})$. Obviously $X_{\infty }\in \mathcal{A}$. Since $x_0\in X_{\infty }$ and $T(X_{\infty })\subset X_{\infty }$, then $Y\subset X_{\infty }$. On the other hand, $T(Y)\subset T(X_{\infty })\subset Y$. Since $x_0\in Y$, it follows that $Y\in \mathcal{A}$. Therefore, $X_{\infty }\subset Y$, which implies

$$X_{\infty }=Conv(T(X_{\infty })\cup \left\{x_0\right\}).$$

The properties of $\mu$ now imply that

$$\mu (X_{\infty })=\mu \left(T(X_{\infty })\cup \left\{x_0\right\}\right)=\mu \left(T(X_{\infty })\right),$$

and consequently $\mu (X_{\infty })=0$. Thus, the set $X_{\infty }$, due to its closure (weak closure), is compact (weakly compact). From Schauder’s (Tikhonov’s) fixed-point theorem applied to the mapping $T_{{|}_{X_{\infty }}}:X_{\infty }\to X_{\infty }$, we find that T has at least one fixed point in the set $X_{\infty }\subset \Omega$. The last statement follows directly form (2). □

Remark 6.

(a) 

Observe that the continuity of the mapping T in both of these theorems only needs to be assumed on the set $X_{\infty }\subset \Omega$.

(b) 

The fact that $\mu (\{x\in \Omega :T(x)=x\})=0$ is very important in applications, for example, in the theory of integral equations, since by choosing an appropriate mapping $\mu :{𝔐}_E\to [0,+\infty )$, we not only obtain the existence of a solution to the investigated equation, but also some information about the structure of its solutions.

(c) 

Let us note that conditions (1) and (2) only need to be verified for arbitrary nonempty, closed, and convex subsets of the set Ω.

(d) 

The Sadovskii theorem has minor significance in applications, since it is usually difficult to verify whether a given operator is μ - condensing.

(e) 

Some generalizations of Darbo’s and Sadovskii’s theorems can be found in [27].

5. Discussion

The article presents results related to measures of noncompactness and measures of weak noncompactness in Banach spaces. One of the pioneers of this theory is Professor Józef Banaś, to whom this work is dedicated. This paper may be of interest and great assistance to those wishing to explore this subject.

Let us note several open problems related to the theory of measures of noncompactness and to issues concerning infinite systems of differential and integral equations, in which measures of noncompactness may be useful.

• It is relatively easy to show that in every Banach space, any regular measure of noncompactness is comparable with the Hausdorff measure. For many years, an unsolved problem was the following: Does there exist, in every infinite dimensional Banach space, a regular measure of noncompactness that is not equivalent to the Hausdorff measure? Partial results in this area were obtained by J. Banaś and A. Martinon in [28], and by J. Mallet-Paret and R.D. Nussbaum in [29]. They showed that in certain Banach spaces, such measures do exist. The problem was fully solved in 2019 by E. Ablet, L. Cheng, Q. Cheng, and W. Zhuang [30]. The counterpart of the Hausdorff measure in the case of weak measures of noncompactness is the De Blasi measure.
  Open problem: Does there exist, in every nonreflexive Banach space, a regular weak measure of noncompactness that is not equivalent to the De Blasi measure?
  In 1995, J. Banaś and A. Martinon showed in [31] that this is indeed the case for certain (sequence) Banach spaces.
• In some Banach spaces (see Section 3), explicit formulas are known that express the Hausdorff measure of noncompactness (or at least a regular measure of noncompactness that is equivalent to the Hausdorff measure) in terms of the structure of the given space. For the De Blasi measure of weak noncompactness, such a formula is known in the space $L^1(I)$, where I is a bounded interval in $\mathbb{R}$ (see Theorem 7). In the space $L^1({\mathbb{R}}_{+})$, no formula is known for expressing the De Blasi measure, but a formula is known for a regular measure of weak noncompactness that is equivalent to the De Blasi measure (see Theorem 8).
  Open problem: Find effective formulas for the De Blasi measure of weak noncompactness (or at least for a regular measure of weak noncompactness that is equivalent to the De Blasi measure) in other (nonreflexive) Banach spaces.
• In a recently published paper [1], the authors J. Banaś, A. Chlebowicz, and B. Rzepka pose several questions concerning infinite systems of differential and integral equations. Solving any of them represents a significant challenge. Moreover, it seems that valuable results concerning infinite systems of differential and integral equations obtained using measures of weak noncompactness are still lacking. Obtaining any result in this area would be highly significant.