Nonlinear integral equations play a significant role in describing numerous real-word events [6, 9, 13, 21]. In nonlinear analysis, we are looking for conditions guaranteeing the existence of solutions of integral equations in various function spaces [6, 13, 21]. The choice of a suitable function space generates the methods applied in the investigations of the solvability of the integral equations in question. On the other hand, we usually choose such a function space which admits a general form and which allows us to apply convenient tools of nonlinear analysis.
It is worthwhile mentioning that the fixed-point theory creates a powerful and convenient branch of nonlinear analysis which is very applicable in proving existence theorems for several types of operator equations (differential, integral, functional integral etc., cf. [3, 16, 19]). It seems that the use of fixed-point theorems, associated with the technique of measures of noncompactness, is very fruitful in the described investigations [1, 3, 4, 6]. It turns out that the application of the theory of measures of noncompactness depends strongly on the choice of a function space in which we are studying the solvability of a considered operator equation. Obviously, such a choice requires the application of a suitable measure of noncompactness, which makes our investigations more or less convenient.
The aim of the paper is to investigate an appropriate criterion for relative compactness in the space of the so-called regulated functions. To make our considerations transparent, we will here investigate the space of real functions defined and regulated on a bounded and closed interval .
The concept of a regulated function (called sometimes a regular function) was introduced in the middle of the twentieth century . Subsequently, some authors presented this concept from different points of view and indicated some of its applications [14, 15, 17, 18]. Especially the approach presented in  seems to be very clear, transparent and applicable.
A lot of essential results concerning the space of regulated functions were given in , where one can encounter also a criterion of relative compactness of bounded subsets in the space of regulated functions. This criterion depends on the use of one-sided limits of functions belonging to a given bounded subset of the space of regulated functions. As far as we know, it is the only criterion of relative compactness published up to now.
Unfortunately, the mentioned criterion is not convenient in practice, since its use requires to impose rather strong assumptions referring to one-sided limits. Consequently, a measure of noncompactness constructed on the basis of that criterion has also the indicated faults .
Our paper is dedicated to describe a criterion of relative compactness in the space of regulated functions based on the approach to the concept of one-sided limits associated with the classical Cauchy condition. Such an approach was discussed in , and in this paper, we are going to extend this direction of investigations. Namely, we formulate a criterion of relative compactness in the space of regulated functions based on the mentioned Cauchy condition. Subsequently, on the basis of that criterion we construct a measure of noncompactness in the space in question, and we prove that the constructed measure has properties handy in applications, i.e., the so-called regular measure of noncompactness. To show the applicability of the mentioned measure of noncompactness, we prove the existence of solutions of a quadratic Hammerstein integral equation in the space of regulated functions.
2 Regulated functions and auxiliary facts
In this section we collect auxiliary facts concerning regulated functions. First we establish the notation. By we will denote the set of real numbers and the symbol will stand for the set of natural numbers (positive integers). Moreover, we denote .
We will consider real functions defined on the interval . If is a given function, then for , we will write or to denote the left-hand limit of the function x at the point t. Similarly, if , then or stand for the right-hand limit of x at t.
We recall the classical concept of the one-sided Cauchy condition.
Let and let (resp. ). We say that the function x satisfies at the point t the left-hand Cauchy condition (resp. the right-hand Cauchy condition) if for all , there exists such that for all (resp. ), we have .
It is well known that the left-hand limit exists and is finite if and only if the function f satisfies at the point t the left-hand Cauchy condition. A similar statement holds for the existence of the finite right-hand limit.
In what follows we will denote by the Banach space of real functions bounded on the interval , equipped with the classical supremum norm
Obviously, the space , consisting of functions from the space which are continuous on , is a closed subspace of under the norm .
Another important subspace of is that consisting of the so-called step functions. Recall that the function is called a step function if there exists a finite sequence , with , such that the function x is constant on each interval , . The set of all step functions on the interval will be denoted by . Obviously, is a linear space and . Notice also that can be normed by the supremum norm but it is not a closed subspace of the space .
A function is said to be a regulated function if it has one-sided limits at every point and the limits and exist.
It can be shown, see , that the definition of a regulated function can be formulated equivalently in the following way.
A function is called regulated if for each , the limits and exist and are finite, and the limits and exist and are finite too.
In what follows we will denote by the set of all functions regulated on . Obviously, is a linear subspace of the space . Moreover, it can be shown, see  (cf. also ), that is a closed subspace of with respect to the norm . This means that is a Banach space with the norm . Moreover, the space of step functions forms a dense subspace of the space , see . For further properties of regulated functions, we refer to .
Assume that X is a subset of the space . We will say that the set X is equiregulated on the interval if the following two conditions are satisfied:
For all , there exists such that for all , and , we have .
For all , there exists such that for all , and , we have .
Now, we recall the result due to Fraňkova , which characterizes the relative compactness in the space .
Let X be a bounded subset of the space . The set X is relatively compact in if and only if X is equiregulated on the interval .
Let us notice that the above result is formulated as [15, Corollary 2.4].
In what follows we present a result which turns out to be equivalent to that contained in Theorem 2.5, but being more handy in applications.
Let X be a bounded subset of the space . The set X is relatively compact in if and only if the following two conditions are satisfied:
For all , there exists such that for all , and , we have .
For all , there exists such that for all , and , we have .
At first, let us assume that the set X is relatively compact. Then, according to Theorem 2.5, this means that X is equiregulated on . Further, fix arbitrarily and choose a number to the number pursuant to conditions (i) and (ii) of Definition 2.4. Next, take a number and choose arbitrary numbers . Then, in view of condition (i), we get
Hence, we infer that the set X satisfies condition (a). Similarly, we can show that the set X satisfies also condition (b).
Conversely, suppose that conditions (a) and (b) are satisfied. Fix a number and choose pursuant to conditions (a) and (b). Next, take an arbitrary number or . Assume, for example, that . Then, according to condition (a), for any function and for arbitrary numbers , we have
Now, let us take an arbitrary sequence such that and . Then, in view of (2.1), we obtain
for any . Hence, applying standard facts from classical analysis, we conclude that
for an arbitrary . This shows that for functions belonging to the set X condition (i) of Definition 2.4 is satisfied.
Similarly, we can prove that the set X satisfies also condition (ii).
Combining the above established facts with Theorem 2.5, we complete the proof. ∎
3 A measure of noncompactness in the space of regulated functions
Now, we are going to construct a measure of noncompactness in the space of regulated functions . To our knowledge, the first attempt to construct a measure of noncompactness in was made in . This measure of noncompactness was based on Theorem 2.5. Unfortunately, since the formula expressing it involved one-sided limits of functions belonging to a set on which it was defined, this measure is not handy in practice.
Indeed, in order to apply such a measure of noncompactness to the theory of functional integral equations, K. Cichoń, M. Cichoń and Metwali  were forced to impose assumptions depending on one-sided limits of functions being components of the mentioned equations.
A measure of noncompactness which we are going to describe in this section is based on Theorem 2.6 and in its construction we will not utilize one-sided limits of functions involved. In this regard, this measure seems to be rather convenient and handy in applications. Our aim is to show its applicability in proving existence theorems for functional integral equations.
We begin by introducing some notation needed in our considerations. Let E be a Banach space with the norm and the zero element θ. In our study, we will write instead of if this does not lead to misunderstanding. Next, by we denote the closed ball centered at x with radius r and by the ball . If X is a subset of the space E, we denote by the closure of X and we write to denote the closed convex hull of X. Moreover, the symbols , () stand for usual algebraic operations on sets.
In what follows, by we denote the family consisting of all nonempty and bounded subsets of E while denotes its subfamily consisting of all relatively compact sets.
The definition of the concept of a measure of noncompactness will be accepted according to .
A function will be called a measure of noncompactness in the Banach space E if it satisfies the following conditions:
The family is nonempty and .
If is a sequence of closed sets belonging to , with () and , then the set is nonempty.
The set described in axiom (1) is referred to as the kernel of the measure of noncompactness μ. Notice that if is the set appearing in axiom (5), then for any . This implies that for . Hence, we infer that . This simple observation plays a crucial role in our further considerations.
Further, let us recall that the measure of noncompactness μ is said to be sublinear if it satisfies the following additional conditions (cf. ):
If the measure μ satisfies the condition
then we say that it has the maximum property.
The measure of noncompactness μ such that will be called the full measure.
Finally, let us remind (cf. ) that the measure of noncompactness μ will be called regular if it is sublinear, has the maximum property and is full .
Let us pay attention to the fact that every regular measure of noncompactness has also some additional useful properties and it is very convenient in applications (cf. [1, 3, 4, 6]). On the other hand, the most convenient regular measure of noncompactness seems to be the so-called Hausdorff measure of noncompactness, see , which is defined in the following way:
We recall now a lemma which will be useful in our investigations.
Let be a function satisfying the following conditions:
The proof of this lemma may be found in . Let us notice that this simple lemma is very important in practice, since the mentioned axiom (5) of Definition 3.1 forms a generalization of the well known Cantor intersection theorem and, in general, it is rather difficult to verify whether a set function satisfies it. Obviously, such a function does not satisfies the conditions listed in Lemma 3.2 (cf. ).
The above lemma will be essentially exploited in our further considerations.
Now, we recall the fixed-point theorem of Darbo type [4, 12], which is formulated with help of the concept of a measure of noncompactness. This theorem is often applied in problems associated with the solvability of operator equations (functional, differential, integral equations, etc., see [1, 3, 4, 6], for details).
Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E, and let μ be a measure of noncompactness defined on E. Assume that is a continuous operator such that for any nonempty subset X of Ω, where is a constant. Then the operator F has at least one fixed point in the set Ω.
It can be shown that the set of all fixed points of the operator F in the set Ω is a member of the family .
Further on, we are going to present the construction of a regular measure of noncompactness in the space of regular functions described in Section 2. To this end, let us take a set . For the sake of simplicity, we will write instead of .
Next, fix a number and for an arbitrarily chosen and for a number , let us define the following quantity:
Similarly, for a fixed , we define
The above defined quantities , can be viewed as left-hand and right-hand-sided moduli of convergence of the function x at the point or , respectively.
Now, let us put
It is easily seen that the functions and are well defined, which is an immediate consequence of the definition of regulated functions and the relation between one-sided limits of a function and the Cauchy condition concerning the existence of one-sided limits (cf. Definition 2.1). Next, let us pay attention to the fact that the functions and are nondecreasing on the interval . Thus, the following limits exist and are finite:
Finally, let us define the quantity
We are ready to present the main result of the paper.
The function μ defined by formula (3.1) is a regular measure of noncompactness in the space .
First let us observe that, in view of Theorem 2.6, we deduce that the function μ satisfies condition (i) of Lemma 3.2. This implies immediately that the function μ satisfies axiom (1) of Definition 3.1 with . Further, let us take into account the fact that the functions and , being the components of the function μ defined by (3.1), are defined by using the supremum. Obviously this yields that the function μ satisfies axiom (2) of Definition 3.1 (or, equivalently, condition (ii) of Lemma 3.2). By the same reasoning, we conclude that the function μ has the maximum property, i.e., it satisfies condition (iii) of Lemma 3.2 (equivalently, axiom 8).
Next, let us observe that for arbitrary functions and for , we obtain
Hence, it is not difficult to deduce that for an arbitrary set , we have
Applying the same reasoning as above we can easily see that
for an arbitrary set , where the symbol stands for the convex hull of the set X. Combining the above inequality with the fact that μ satisfies axiom (2), we infer that
for an arbitrary set .
Finally, we show that
for . To this end, observe firstly that the inequality
is a consequence of axiom (2).
To show the converse inequality let us take an arbitrary function . This means that x is a limit of a sequence of functions belonging to the set X. Thus, we can write
uniformly on the interval . Further, let us fix arbitrarily . Then, for a fixed and for , we have
since the sequence is uniformly convergent to the function x on the interval .
The above established facts allows us to infer that
In the same way we can show also that
Combining the above inequalities with (3.1), we get
The second part of this axiom is a consequence of equality (3.2) and the fact that for an arbitrary set .
The proof is complete. ∎
Let us observe that in view of the result proved in , we have, for an arbitrary set , that the following inequality is satisfied:
where χ denotes the Hausdorff measure of noncompactness in the space and . It is not difficult to calculate that . Thus, from (3.5), we obtain the estimate
for . It is an open question whether there exists a constant such that
Notice that the answer to the above question has no influence on our further considerations concerning the applicability of the measure of noncompactness μ given by (3.1).
4 An application
This section is dedicated to present an application of the measure of noncompactness μ introduced in the previous section. We will investigate the solvability of a quadratic Hammerstein integral equation of the form
for . Our considerations will focus on the space of regulated functions on the interval (cf. Section 2).
We will consider equation (4.1) by imposing the following assumptions:
The function satisfies the Lipschitz condition with respect to the variable x, with the constant . Moreover, the function is regulated on the interval , locally uniformly with respect to the variable , i.e., for any , the function is regulated on for .
The function is continuous in s for any fixed . In addition, the function is regulated on the interval , uniformly with respect to .
The function is continuous on the set . Moreover, there exist a nonnegative constant c and a positive constant d such that for and .
Let us explain that the expression that the functions and are regulated with respect to t, (locally) uniformly with respect to the variables x and s, respectively (cf. the above formulated assumptions (ii) and (iii)), is understood in the sense of Definition 2.1.
To formulate our last assumption, let
In view of the fact the function is regulated on the interval (cf. assumption (iii)), we have that .
Moreover, we will denote by the constant defined as follows:
The fact that will be shown later on.
The following inequalities are satisfied:
where we set .
Now, we are prepared to state our existence result concerning equation (4.1).
Under assumptions (i)–(v), equation (4.1) has at least one solution in the space .
Let us consider the operator T associated with equation (4.1). This means that T is defined on the space by the formula
For further purposes, let us consider the operators G, F and K defined in the following way:
Then the operator T defined by (4.2) can be represented as
where the symbol is understood as the composition of the operators F and K.
Now, let us fix arbitrarily and . Then, for an arbitrary function and for arbitrary , on the basis of assumption (ii), we obtain
In view of assumption (ii), we infer that as . Particularly this implies that the operator G transforms the space into itself.
In a similar way, taking into account assumption (iii), we derive
Notice that, in view of assumption (iii), we have that (similarly, ) as for each . This implies that the function (or the function ) is regulated on the interval . Consequently, it follows that the function is bounded on the interval and simultaneously justifies the fact that , which was stated before.
Now, from (4.4), (4.5), assumption (iv) and taking into account representation (4.3), we conclude that the operator T transforms the space into itself. Obviously, the above reasoning can be repeated for any fixed and for , if we replace the quantity by .
Let us notice that by utilizing our assumptions, for an arbitrarily fixed and , we have
where we write in place of and the constants , L, , a, b, were defined previously or imposed in the assumptions.
The above inequality yields the estimate
Hence, keeping in mind assumption (v), we deduce that there exists a positive number such that for all functions , we have that , i.e., the operator T transforms the ball into itself.
Keeping in mind the convenience, we will further accept that
To prove the continuity of the operator T defined by (4.3) on the ball let us observe that, in view of assumption (ii), it is sufficient to prove the continuity of the operator on . Thus, fix arbitrarily and take such that . Then, for a fixed , we obtain
Observe that taking into account the uniform continuity of the function f on the set (cf. assumption (iv)), we deduce that as . Combining this fact with (4.8), we infer that the operator is continuous on the ball .
Further on, let us fix an arbitrary nonempty set and a number . Then, for and for , let us choose arbitrary numbers . Then we get the estimate
Next, keeping in mind assumption (i) and the properties of the functions , and , and letting , we obtain the estimate
In the same way, we can prove that
Observe that, in view of (4.7) and assumption (v), we obtain
Hence, keeping in mind estimate (4.13) and Theorem 3.3, we conclude that the operator T has at least one fixed point x in the ball . Obviously, the function is a desired solution of equation (4.1) belonging to the space . The proof is complete. ∎
Now, we provide an example illustrating the result contained in Theorem 4.2.
Let us fix a natural number and consider the function defined on the interval in the following way:
where stands for the characteristic function of the interval for . Obviously, the function , being the step function, is the regulated function on the interval . Moreover, . Particularly, this means that the function p satisfies assumption (i) of Theorem 4.2.
Next, consider the function defined by
where (similarly as above) the function denotes the characteristic function of the interval for and denotes the integer part of the number y. It is easily seen that the function is Lipschitzian with respect to x. Indeed, for an arbitrary and , we get
This means that the function satisfies the Lipschitz condition with the constant .
Now, taking into account the fact that the functions and are step functions on the interval I and keeping in mind that for each fixed , the function
Further, let us take the function defined on the set by
where denotes the characteristic function of the interval for and is a constant.
Observe that the function satisfies assumption (iii) of Theorem 4.2. Moreover, we have
for any , so we can accept that , where the constant was defined previously.
Further, let us consider the quadratic Hammerstein integral equation
Obviously, the function is continuous on the set and
for and . Thus, the function satisfies assumption (iv) of Theorem 4.2 with and .
Summing up, we see that assumptions (i)–(iv) of Theorem 4.2 are satisfied. In addition, we have that
Thus, we can take .
To verify assumption (v), let us note that the first inequality in (v) has the form
Hence, we see that for each fixed , we can choose a number such that (4.18) is satisfied.
Further, let us take into account the second inequality in assumption (v). It has the form
R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators (in Russian), “Nauka”, Novosibirsk, 1986. Google Scholar
G. Aumann, Reelle Funktionen, Grundlehren Math. Wiss. 68, Springer, Berlin, 1954. Google Scholar
J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl. 99, Birkhäuser, Basel, 1997. Google Scholar
J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes Pure Appl. Math. 60, Marcel Dekker, New York, 1980. Google Scholar
J. Banaś and M. Kot, On regulated functions, J. Math. Appl. 40 (2017), 21–36. Google Scholar
J. Banaś and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 2014. Google Scholar
J. Banaś, D. Szynal and S. A. W’edrychowicz, On existence, asymptotic behaviour and stability of solutions of stochastic integral equations, Stochastic Anal. Appl. 9 (1991), no. 4, 363–385. CrossrefGoogle Scholar
S. Chandrasekhar, Radiative Transfer, Oxford University Press, London, 1950. Google Scholar
K. Cichoń, M. Cichoń and M. M. A. Metwali, On some parameters in the space of regulated functions and their applications, Carpathian J. Math. 34 (2018), no. 1, 17–30. Google Scholar
G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova 24 (1955), 84–92. Google Scholar
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. Google Scholar
J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1969. Google Scholar
D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), no. 1, 20–59. Google Scholar
K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math. 28, Cambridge University Press, Cambridge, 1990. Google Scholar
C. S. Hönig, Volterra Stieltjes-Integral Equations, North-Holland, Amsterdam, 1975. Google Scholar
C. S. Hönig, Équations intégrales généralisées et applications, Publ. Math. Orsay 83-01 (1983), Expose No. 5. Google Scholar
A. Jeribi and B. Krichen, Nonlinear Functional Analysis in Banach Spaces and Banach Algebras, Monogr. Res. Notes Math., CRC Press, Boca Raton, 2016. Google Scholar
P. P. Zabrejko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, L. S. Rakovschik and J. Stetsenko, Integral Equations, Nordhoff, Leyden, 1975. Google Scholar
About the article
Published Online: 2018-06-13
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1099–1110, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0024.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0