#### Abstract

We discuss some existence results for various types of functional, differential, and integral equations which can be obtained with the help of argumentations based on compactness conditions. We restrict ourselves to some classical compactness conditions appearing in fixed point theorems due to Schauder, Krasnosel’skii-Burton, and Schaefer. We present also the technique associated with measures of noncompactness and we illustrate its applicability in proving the solvability of some functional integral equations. Apart from this, we discuss the application of the mentioned technique to the theory of ordinary differential equations in Banach spaces.

#### 1. Introduction

The concept of the compactness plays a fundamental role in several branches of mathematics such as topology, mathematical analysis, functional analysis, optimization theory, and nonlinear analysis [1–5]. Numerous mathematical reasoning processes depend on the application of the concept of compactness or relative compactness. Let us indicate only such fundamental and classical theorems as the Weierstrass theorem on attaining supremum by a continuous function on a compact set, the Fredholm theory of linear integral equations, and its generalization involving compact operators as well as a lot of fixed point theorems depending on compactness argumentations [6, 7]. It is also worthwhile mentioning such an important property saying that a continuous mapping transforms a compact set onto compact one.

Let us pay a special attention to the fact that several reasoning processes and constructions applied in nonlinear analysis depend on the use of the concept of the compactness [6]. Since theorems and argumentations of nonlinear analysis are used very frequently in the theories of functional, differential, and integral equations, we focus in this paper on the presentation of some results located in these theories which can be obtained with the help of various compactness conditions.

We restrict ourselves to present and describe some results obtained in the last four decades which are related to some problems considered in the theories of differential, integral, and functional integral equations. Several results using compactness conditions were obtained with the help of the theory of measures of noncompactness. Therefore, we devote one section of the paper to present briefly some basic background of that theory.

Nevertheless, there are also successfully used argumentations not depending of the concept of a measure on noncompactness such as Schauder fixed point principle, Krasnosel’skii-Burton fixed point theorem, and Schaefer fixed point theorem.

Let us notice that our presentation is far to be complete. The reader is advised to follow the most expository monographs in which numerous topics connected with compactness conditions are broadly discussed [6, 8–10].

Finally, let us mention that the presented paper has a review form. It discusses some results described in details in the papers which will be cited in due course.

#### 2. Selected Results of Nonlinear Analysis Involving Compactness Conditions

In order to solve an equation having the form where is an operator being a self-mapping of a Banach space , we apply frequently an approach through fixed point theorems. Such an approach is rather natural and, in general, very efficient. Obviously, there exists a huge number of miscellaneous fixed point theorems [6, 7, 11] depending both on order, metric, and topological argumentations.

The most efficient and useful theorems seem to be fixed point theorems involving topological argumentations, especially those based on the concept of compactness. The reader can make an acquaintance with the large theory of fixed point-theorems involving compactness conditions in the above, mentioned monographs, but it seems that the most important and expository fixed point theorem in this fashion is the famous Schauder fixed point principle [12]. Obviously, that theorem was generalized in several directions but till now it is very frequently used in application to the theories of differential, integral, and functional equations.

At the beginning of our considerations we recall two well-known versions of the mentioned Schauder fixed point principle (cf. [13]). To this end assume that is a given Banach space.

Theorem 1. * Let be a nonempty, bounded, closed, and convex subset of and let be a completely continuous operator (i.e., is continuous and the image is relatively compact). Then has at least one fixed point in the set (this means that the equation has at least one solution in the set ).*

Theorem 2. *If is a nonempty, convex, and compact subset of and is continuous on the set , then the operator has at least one fixed point in the set .*

Observe that Theorem 1 can be treated as a particular case of Theorem 2 if we apply the well-known Mazur theorem asserting that the closed convex hull of a compact subset of a Banach space is compact [14]. The basic problem arising in applying the Schauder theorem in the version presented in Theorem 2 depends on finding a convex and compact subset of which is transformed into itself by operator corresponding to an investigated operator equation.

In numerous situations, we are able to overcome the above-indicated difficulty and to obtain an interesting result on the existence of solutions of the investigated equations (cf. [7, 15–17]). Below we provide an example justifying our opinion [18].

To do this, let us denote by the real line and put . Further, let .

Next, fix a function defined and continuous on with positive real values. Denote by the space consisting of all real functions defined and continuous on and such that It can be shown that forms the Banach space with respect to the norm For our further purposes, we recall the following criterion for relative compactness in the space [10, 15].

Theorem 3. *Let be a bounded set in the space . If all functions belonging to are locally equicontinuous on the interval and if uniformly with respect to , then is relatively compact in .*

In what follows, if is an arbitrarily fixed function from the space and if is a fixed number, we will denote by the modulus of continuity of on the interval ; that is,

Further on, we will investigate the solvability of the nonlinear Volterra integral equation with deviated argument having the form where . Equation (5) will be investigated under the following formulated assumptions.(i) is a continuous function and there exist continuous functions such that for all and .

In order to formulate other assumptions, let us put Next, take an arbitrary number and consider the space , where . Then, we can present other assumptions.(ii) The function is continuous and there exists a nonnegative constant such that for . (iii) There exists a constant such that for . (iv) is a continuous function satisfying the condition for , where is a constant. (v) and for all .

Now, we can formulate the announced result.

Theorem 4. *Under assumptions (i)–(v), (5) has at least one solution in the space such that for .*

We give the sketch of the proof (cf. [18]). First, let us define the transformation on the space by putting In view of our assumptions, the function is continuous on .

Further, consider the subset of the space consisting of all functions such that for . Obviously, is nonempty, bounded, closed, and convex in the space . Taking into account our assumptions, for an arbitrary fixed and , we get This shows that transforms the set into itself.

Next we show that is continuous on the set .

To this end, fix and take such that . Next, choose arbitrary . Using the fact that the function is uniformly continuous on the set , where , for , we obtain where is a continuous function with the property .

Now, take . Then we get Hence, for sufficiently large, we have for . Linking (12) and (14), we deduce that is continuous on the set .

The next essential step in our proof which enables us to apply the Schauder fixed point theorem (Theorem 1) is to show that the set is relatively compact in the space . To this end let us first observe that the inclusion and the description of imply the following estimate: This yields that uniformly with respect to the set .

On the other hand, for fixed and for such that , in view of our assumptions, for , we derive the following estimate: where we denoted Taking into account the fact that , we infer that functions belonging to the set are equicontinuous on each interval . Combining this fact with (16), in view of Theorem 3, we conclude that the set is relatively compact. Applying Theorem 1, we complete the proof.

Another very useful fixed point theorem using the compactness conditions is the well-known Krasnosel’skii fixed point theorem [19]. That theorem was frequently modified by researchers working in the fixed point theory (cf. [6, 7, 20]), but it seems that the version due to Burton [21] is the most appropriate to be used in applications.

Below we formulate that version.

Theorem 5. * Let be a nonempty, closed, convex, and bounded subset of the Banach space and let and be two operators such that*(a)* is a contraction, that is, there exists a constant such that for , *(b)* is completely continuous, *(c)* for all . **Then the equation has a solution in . *

In order to show the applicability of Theorem 5, we will consider the following nonlinear functional integral equation [22]: where . Here we assume that and are given functions.

The above equation will be studied in the space consisting of all real functions defined, continuous, and bounded on the interval and equipped with the usual supremum norm Observe that the space is a special case of the previously considered space with for . This fact enables us to adapt the relative compactness criterion contained in Theorem 3 for our further purposes.

In what follows we will impose the following requirements concerning the components involved in (19):(i)The function is continuous and there exist constants such that for all and for all , , . (ii)The function is bounded on with . (iii)The functions are continuous and as . (iv)The function is continuous. (v)The function is continuous and there exist continuous functions and such that for all and . Moreover, we assume that

Now, let us observe that based on assumption (v), we conclude that the functions defined by the formulas are continuous and bounded on . Obviously this implies that the constants defined as: are finite.

In order to formulate our last assumption, let us denote , where the constants appear in assumption (i).(vi) . Then we have the following result [22] which was announced above.

Theorem 6. * Under assumptions (i)–(vi), (19) has at least one solution in the space . Moreover, solutions of (19) are globally attractive.*

*Remark 7. * In order to recall the concept of the global attractivity mentioned in the above theorem (cf. [22]), suppose that is a nonempty subset of the space and is an operator. Consider the operator equation
We say that solutions of (26) are *globally attractive* if for arbitrary solutions of this equation, we have that
Let us mention that the above-defined concept was introduced in [23, 24].

*Proof of Theorem 6. * We provide only the sketch of the proof. Consider the ball in the space centered at the zero function and with radius . Next, define two mappings and by putting
for . Then (19) can be written in the form

Notice that in view of assumptions (i)–(iii), the mapping is well defined, and for arbitrarily fixed function , the function is continuous and bounded on . Thus transforms into itself. Similarly, applying assumptions (iii)–(v), we deduce that the function is continuous and bounded on . This means that transforms the ball into .

Now, we check that operators and satisfy assumptions imposed in Theorem 5. To this end take . Then, in view of assumption (i), for a fixed , we get
This implies that , and in view of assumption (vi), we infer that is a contraction on .

Next, we prove that is completely continuous on the ball . In order to show the indicated property of , we fix and we take with . Then, taking into account our assumptions, we obtain
Hence, keeping in mind assumption (v), we deduce that there exists such that for . Combining this fact with (31), we get
for .

Further, for an arbitrary , in the similar way, we obtain
where we denoted and
Keeping in mind the uniform continuity of the function on the set , we deduce that as . Hence, in view of (32) and (33), we conclude that the operator is continuous on the ball .

The boundedness of the operator is a consequence of the inequality
for . To verify that the operator satisfies assumptions of Theorem 3 adapted to the case of the space , fix arbitrarily . In view of assumption (v), we can choose such that for . Further, take an arbitrary function . Then, keeping in mind (35), for , we infer that
Next, take arbitrary numbers with . Then we obtain the following estimate:
where we denoted
From estimate (37), we get
where , , , and denotes the usual modulus of continuity of the function on the interval .

Now, let us observe that in view of the standard properties of the functions , , we infer that and as . Hence, taking into account the boundedness of the image and estimates (36) and (39), in view of Theorem 3, we conclude that the set is relatively compact in the space ; t is completely continuous on the ball .

In what follows fix arbitrary and assume that the equality holds for some . Then, utilizing our assumptions, for a fixed , we get
Hence we obtain
On the other hand, we have that . Thus or, equivalently, . This shows that assumption (c) of Theorem 5 is satisfied.

Finally, combining all of the above-established facts and applying Theorem 5, we infer that there exists at least one solution of (19).

The proof of the global attractivity of solutions of (19) is a consequence of the estimate
which is satisfied for arbitrary solutions , of (19).

Hence we get
which implies that . This means that the solutions of (19) are globally attractive (cf. Remark 7).

It is worthwhile mentioning that in the literature one can encounter other formulations of the Krasnosel’skii-Burton fixed point theorem (cf. [6, 7, 20, 21]). In some of those formulations and generalizations, there is used the concept of a measure of noncompactness (both in strong and in weak sense) and, simultaneously, the requirement of continuity is replaced by the assumption of weak continuity or weak sequential continuity of operators involved (cf. [6, 20], for instance).

In what follows we pay our attention to another fixed point theorem which uses the compactness argumentation. Namely, that theorem was obtained by Schaefer in [25].

Subsequently that theorem was formulated in other ways and we are going here to present two versions of that theorem (cf. [11, 26]).

Theorem 8. * Let be a normed space and let be a continuous mapping which transforms bounded subsets of onto relatively compact ones. Then either*(i)* the equation has a solution**or *(ii)* the set is unbounded. *

The below presented version of Schaefer fixed point theorem seems to be more convenient in applications.

Theorem 9. *Let be a Banach space and let be a continuous compact mapping (i.e., is continuous and maps bounded subsets of onto relatively compact ones). Moreover, one assumes that the set
**
is bounded. Then has a fixed point.*

It is easily seen that Theorem 9 is a particular case of Theorem 8.

Observe additionally, that Schaefer fixed point theorem seems to be less convenient in applications than Schauder fixed point theorem (cf. Theorems 1 and 2). Indeed, Schaefer theorem requires a priori bound on utterly unknown solutions of the operator equation for . On the other hand, the proof of Schaefer theorem requires the use of the Schauder fixed point principle (cf. [27], for details).

It is worthwhile mentioning that an interesting result on the existence of periodic solutions of an integral equation, based on a generalization of Schaefer fixed point theorem, may be found in [26].

#### 3. Measures of Noncompactness and Their Applications

Let us observe that in order to apply the fundamental fixed point theorem based on compactness conditions, that is, the Schauder fixed point theorem, say, the version of Theorem 2, we are forced to find a convex and compact subset of a Banach space which is transformed into itself by an operator . In general, it is a hard task to encounter a set of such a type [16, 28]. On the other hand, if we apply Schauder fixed point theorem formulated as Theorem 1, we have to prove that an operator is completely continuous. This causes, in general, that we have to impose rather strong assumptions on terms involved in a considered operator equation.

In view of the above-mentioned difficulties starting from seventies of the past century mathematicians working on fixed point theory created the concept of the so-called measure of noncompactness which allowed to overcome the above-indicated troubles. Moreover, it turned out that the use of the technique of measures of noncompactness allows us also to obtain a certain characterization of solutions of the investigated operator equations (functional, differential, integral, etc.). Such a characterization is possible provided we use the concept of a measure of noncompactness defined in an appropriate axiomatic way.

It is worthwhile noticing that up to now there have appeared a lot of various axiomatic definitions of the concept of a measure of noncompactness. Some of those definitions are very general and not convenient in practice. More precisely, in order to apply such a definition, we are often forced to impose some additional conditions on a measure of noncompactness involved (cf. [8, 29]).

By these reasons it seems that the axiomatic definition of the concept of a measure of noncompactness should be not very general and should require satisfying such conditions which enable the convenience of their use in concrete applications.

Below we present axiomatics which seems to satisfy the above-indicated requirements. That axiomatics was introduced by Banaś and Goebel in 1980 [10].

In order to recall that axiomatics, let us denote by the family of all nonempty and bounded subsets of a Banach space and by its subfamily consisting of relatively compact sets. Moreover, let stand for the ball with the center at and with radius . We write to denote the ball , where is the zero element in . If is a subset of , we write , to denote the closure and the convex closure of , respectively. The standard algebraic operations on sets will be denoted by and , for .

As we announced above, we accept the following definition of the concept of a measure of noncompactness [10].

*Definition 10. * A function is said to be a *measure of noncompactness* in the space if the following conditions are satisfied. The family is nonempty and . . . for . If is a sequence of closed sets from such that for and if , then the set is nonempty.

Let us pay attention to the fact that from axiom we infer that for any . This implies that . Thus belongs to the family described in axiom . The family is called the *kernel of the measure of noncompactness *.

The property of the measure of noncompactness mentioned above plays a very important role in applications.

With the help of the concept of a measure of noncompactness, we can formulate the following useful fixed point theorem [10] which is called the fixed point theorem of Darbo type.

Theorem 11. * Let be a nonempty, bounded, closed, and convex subset of a Banach space and let be a continuous operator which is a contraction with respect to a measure of noncompactness ; that is, there exists a constant , , such that for any nonempty subset of the set . Then the operator has at least one fixed point in the set .*

In the sequel we show an example of the applicability of the technique of measures of noncompactness expressed by Theorem 11 in proving the existence of solutions of the operator equations.

Namely, we will work on the Banach space consisting of real functions defined and continuous on the interval and equipped with the standard maximum norm. For sake of simplicity, we will assume that , so the space on question can be denoted by .

One of the most important and handy measures of noncompactness in the space can be defined in the way presented below [10].

In order to present this definition, take an arbitrary set . For and for a given , let us put Next, let us define It may be shown that the function is the measure of noncompactness in the space (cf. [10]). This measure has also some additional properties. For example, and provided and [10].

In what follows we will consider the nonlinear Volterra singular integral equation having the form where , is a continuous function, and , are operators acting continuously from the space into itself. Apart from this, we assume that the function has the form where is continuous and is monotonic with respect to the first variable and may be discontinuous on the triangle .

Equation (47) will be considered in the space under the following assumptions (cf. [30]).(i) is a continuous function. (ii)The function is continuous and there exists a nonnegative constant such that for any and for all .(iii)The operator transforms continuously the space into itself and there exists a nonnegative constant such that for any set , where is the measure of noncompactness defined by (46). (iv)There exists a nondecreasing function such that for any . (v)The operator acts continuously from the space into itself and there exists a nondecreasing function such that for any . (vi) has the form (48), where the function is continuous. (vii)The function occurring in the decomposition (48) is monotonic with respect to (on the interval []), and for any fixed , the function is Lebesgue integrable over the interval . Moreover, for every , there exists such that for all with and the following inequalities are satisfied:

The main result concerning (47), which we are going to present now, will be preceded by a few remarks and lemmas (cf. [30]). In order to present these remarks and lemmas, let us consider the function defined by the formula In view of assumption (vii), this function is well defined.

Lemma 12. * Under assumption (vii), the function is continuous on the interval .*

For proof, we refer to [30].

In order to present the last assumptions needed further on, let us define the constants , , by putting The constants and are finite in view of assumptions (vi) and (ii), while the fact that is a consequence of assumption (vii) and Lemma 12.

Now, we formulate the announced assumption.(viii) There exists a positive solution of the inequality such that .

For further purposes, we definite operators corresponding to (47) and defined on the space in the following way: for . Apart from this, we introduce two functions and defined on by the formulas Notice that in view of assumption (vii), we have that and as .

Now, we can state the following result.

Lemma 13. * Under assumptions (i)–(vii), the operator transforms continuously the space into itself.*

*Proof. * Fix a function . Then , which is a consequence of the properties of the so-called superposition operator [9]. Further, for arbitrary functions , in virtue of assumption (ii), for a fixed , we obtain
This estimate in combination with assumption (i) yields
Hence we conclude that acts continuously from the space into itself.

Next, fix and . Take such that . Without loss of generality, we may assume that . Then, based on the imposed assumptions, we derive the following estimate:
where we denoted
Since the function is assumed to be monotonic, by assumption (vii) we get
This fact in conjunction with the above-obtained estimate yields
where the symbol denotes the modulus of continuity of a function .

Further observe that as , which is an immediate consequence of the uniform continuity of the function on the triangle . Combining this fact with the properties of the functions and and taking into account (62), we infer that . Consequently, keeping in mind that and assumption (iii), we conclude that the operator is a self-mapping of the space .

In order to show that is continuous on , fix arbitrarily and . Next, take such that . Then, for a fixed , we get
On the other hand, we have the following estimate:
Similarly, we obtain
and consequently
Next, linking (63)–(66) and (58), we obtain the following estimate:
Further, taking into account assumptions (iv) and (v), we derive the following inequality
Finally, in view of the continuity of the operators and (cf. assumptions (iii) and (v)), we deduce that operator is continuous on the space . The proof is complete.

Now, we can formulate the last result concerning (47) (cf. [30]).

Theorem 14. * Under assumptions (i)–(viii), (47) has at least one solution in the space .*

*Proof. * Fix and . Then, evaluating similarly as in the proof of Lemma 13, we get

Hence, we obtain
From the above inequality and assumption (viii), we infer that there exists a number such that the operator maps the ball into itself and . Moreover, by Lemma 13, we have that is continuous on the ball .

Further on, take a nonempty subset of the ball and a number . Then, for an arbitrary and with , in view of (66) and the imposed assumptions, we obtain
where we denoted
Hence, in virtue of (62), we deduce the estimate
Finally, taking into account the uniform continuity of the function on the set and the properties of the functions , , , and and keeping in mind (46), we obtain
Linking this estimate with assumption (iii), we get
The use of Theorem 11 completes the proof.

#### 4. Existence Results Concerning the Theory of Differential Equations in Banach Spaces

In this section, we are going to present some classical results concerning the theory of ordinary differential equations in Banach space. We focus on this part of that theory in a which the technique associated with measures of noncompactness is used as the main tool in proving results on the existence of solutions of the initial value problems for ordinary differential equations. Our presentation is based mainly on the papers [31, 32] and the monograph [33].

The theory of ordinary differential equations in Banach spaces was initiated by the famous example of Dieudonné [34], who showed that in an infinite-dimensional Banach space, the classical Peano existence theorem is no longer true. More precisely, Dieudonné showed that if we consider the ordinary differential equation with the initial condition where and is an infinite-dimensional Banach space, then the continuity of (and even uniform continuity) does not guarantee the existence of solutions of problem (76)-(77).

In light of the example of Dieudonné, it is clear that in order to ensure the existence of solutions of (76)-(77), it is necessary to add some extra conditions. The first results in this direction were obtained by Kisyński [35], Olech [36], and Ważewski [37] in the years 1959-1960. In order to formulate those results, we need to introduce the concept of the so-called Kamke comparison function (cf. [38, 39]).

To this end, assume that is a fixed number and denote , . Further, assume that is a nonempty open subset of and is a fixed element of . Let be a given function.

*Definition 15. *A function (or ) is called a *Kamke comparison function* provided the inequality
for and (or ), together with some additional assumptions concerning the function , guarantees that problem (76)-(77) has at most one local solution.

In the literature, one can encounter miscellaneous classes of Kamke comparison functions (cf. [33, 39]). We will not describe those classes, but let us only mention that they are mostly associated with the differential equation with initial condition or the integral inequality for with initial condition . It is also worthwhile recalling that the classical Lipschitz or Nagumo conditions may serve as Kamke comparison functions [39].

The above-mentioned results due to Kisyński et al. [35–37] assert that if is a continuous function satisfying condition (78) with an appropriate Kamke comparison function, then problem (76)-(77) has exactly one local solution.

Observe that the natural translation of inequality (78) in terms of measures of noncompactness has the form
where denotes an arbitrary nonempty subset of the ball . The first result with the use of condition (79) for ( is a constant) was obtained by Ambrosetti [40]. After the result of Ambrosetti, there have appeared a lot of papers containing existence results concerning problem (76)-(77) (cf. [41–44]) with the use of condition (79) and involving various types of Kamke comparison functions. It turned out that generalizations of existence results concerning problem (76)-(77) with the use of more and more general Kamke comparison functions are, in fact, only apparent generalizations [45], since the so-called Bompiani comparison function is sufficient to give the most general result in the mentioned direction.

On the other hand, we can generalize existence results involving a condition like (79) taking general measures of noncompactness [31, 32]. Below we present a result coming from [32] which seems to be the most general with respect to taking the most general measure of noncompactness.

In the beginning, let us assume that is a measure of noncompactness defined on a Banach space . Denote by the set defined by the equality
The set will be called the *kernel set* of a measure . Taking into account Definition 10 and some properties of a measure of noncompactness (cf. [10]), it is easily seen that is a closed and convex subset of the space .

In the case when we consider the so-called *sublinear measure of noncompactness* [10], that is, a measure of noncompactness which additionally satisfies the following two conditions: , for , then the kernel set forms a closed linear subspace of the space .

Further on, assume that is an arbitrary measure of noncompactness in the Banach space . Let be a fixed number and let us fix . Next, assume that (where ) is a given uniformly continuous and bounded function; say, .

Moreover, assume that satisfies the following comparison condition of Kamke type:
for any nonempty subset of the ball and for almost all .

Here we will assume that the function is continuous with respect to for any and measurable with respect to for each . Apart from this, and the unique solution of the integral inequality
such that , is .

The following formulated result comes from [32].

Theorem 16. * Under the above assumptions, if additionally , where and is a positive constant such that , the initial value problem (76)-(77) has at least one local solution such that for .*

The proof of the above theorem is very involved and is therefore omitted (cf. [32, 33]). We restrict ourselves to give a few remarks.

At first, let us notice that in the case when is a sublinear measure of noncompactness, condition (81) is reduced to the classical one expressed by (79) provided we assume that . In such a case, Theorem 16 was proved in [10].

The most frequently used type of a comparison function is this having the form , where is assumed to be Lebesgue integrable over the interval . In such a case comparison, condition (81) has the form An example illustrating Theorem 16 under condition (83) will be given later.

Further, observe that in the case when is a sublinear measure of noncompactness such that , condition (83) can be written in the form Condition (84) under additional assumption , with some nonnegative constants and , is used frequently in considerations associated with infinite systems of ordinary differential equations [46–48].

Now, we present the above-announced example coming from [33].

*Example 17. * Consider the infinite system of differential equations having the form