Abstract
Functions of bounded variations form important transition between absolute continuous and singular functions. With Bainov’s introduction of impulsive differential equations having solutions of bounded variation, this class of functions had eventually entered into the theory of differential equations. However, the determination of existence of solutions is still problematic because the solutions of differential equations is usually at least absolute continuous which is disrupted by the solutions of bounded variations. As it is known, if is of bounded variation then is the sum of an absolute continuous function and a singular function where the total variation of generates a singular measure and is absolute continuous with respect to . In this paper we prove that a function of bounded variation has two representations: one is which was described with an absolute continuous part with respect to the Lebesgue measure , while in the other an integral with respect to forms the absolute continuous part and defines the singular measure. Both representations are obtained as parameter transformation images of an absolute continuous function on total variation domain .
1. Introduction
Bainov, Laksmikantam, and Semionov introduced a new class of differential equations [1] which, in addition to the usual dynamics, had impulse effects, acting on the system in zero time periods. Hence, the dynamics was composed of the usual interactions and singular interactions. This led to solutions of bounded variations. As recent publications show, finding solutions of integral equations in the spaces of bounded variations in various fields is coming to the focus of attention of international research in mathematics like solution of Volterra-Hammerstein integral equations. This research was supported among others by the Venezuelan Central Bank [2–4].
The time dependence with the described composition of the dynamics led to difficulties in formulating efficient existence theorems, handling delay systems, and many other issues. The nature of bounded variation of the solutions was in contrast with such functions that constituted the solutions of differential equations (while not even being continuous).
We want to establish a presentation of the functions of bounded variation which makes it possible to handle the solutions of bounded variations in terms of absolute continuous functions. By the origin of the problem outlined, we will focus on -valued functions of bounded variations. The combination of functions of bounded variation and singular functions in differential equations appears in the literature [5, 6].
It is well known that a function of bounded variation on an interval is the sum of an absolute continuous component and a component singular with respect to the Lebesgue measure [7, 8]. The total variation of is then an ascending singular function.
Let denote the measure defined by which is singular with respect to . Now we form a measure with respect to which both & are absolute continuous. The measure is generated by the strictly ascending function . The strictly ascending function has strictly ascending left and a right continuous versions and , respectively [9], and maps the interval into where .
The transformation of into is defined as follows. The function , being strictly ascending on , has a countable set of discontinuity points. Therefore, the inequality holds, and is continuous in the rest of the points. Hence we define the transformation as follows: if (see details in the paper).
These three transformations enable us to formulate and prove presentations stated in the main result. The theorem states that a function is of bounded variation if and only if there exists an absolute continuous function with such thatThe same main theorem states that a function of bounded variation has another representation where measure gives the absolute continuous component and is an ascending function singular with respect to .
2. Integral Representation of Functions of Bounded Variation
2.1. Summary of Known Results
See the details of this section in [8, 10].
Definition 1. A function is a function of bounded variation if where .
The main target of this paper is the study of n-dimensional vector valued functions of bounded variation for some . We will introduce two types of variations for vector valued functions:
Definition 2. A function is a function of bounded variation if
Theorem 3. If is a function () then , .
We will focus our investigations on the properties of ascending functions; therefore we will now turn to the integral properties of ascending functions of bounded variation (equivalent with boundedness on the interval under consideration).
2.2. Total Variation of a Function of Bounded Variation
We will first establish that ascending functions play major role in our integral representation of functions of bounded variations.
Theorem 4. Let be a function of bounded variation. Then the function , fulfils ; hence is an ascending function and .
Thus any function of bounded variation is differentiable almost everywhere and can be written as while is the singular part of f. Hence .
The singular function is of bounded variation; hence assume the following.
Definition 5. , .
The singular function is absolute continuous with respect to ; hence and thereforeWe introduce which is a strictly ascending function and with . It follows from Definition 5 of that
The function defines a measure , such that both are absolute continuous with respect to :Using this in (5), we get an integral representation of the function of bounded variation in terms of a single measure:
2.2.1. Properties of the Radon-Nikodym Derivatives
The integral in (8) is a reformulation of (5) with the sum of two measures: . This is a measure generated from a right continuous ascending function . As stated in (7), both measures are absolute continuous with respect to ; therefore both can be written as an integral of the Radon-Nikodym derivatives [10] . These have the following important properties.
Lemma 6. Let , and then(1);(2);(3).
Proof. We call our attention to the following relation: let . Then the singularity of with respect to leads to the following relations:since is singular with respect to (if the fifth integral would be positive then A would be a set of positive -measure with positive measures which contradicts the singularity of with respect to ). Hence almost everywhere with respect to . Therefore holds almost everywhere with respect to .
2.2.2. The Relationship between the t-Scale and the Total Variations Scale
This section presents the main construction of this paper which connects the structure of the domain of the function of bounded variation and the range of the sum of the time parameter and the total variation of the singular component of defined in Definition 5.
The Structure from the Domain of a Function of Bounded Variation. The starting point here is the domain of the function of bounded variation. The strictly ascending function equivalently with maps the domain of to the range of total variation scale. Both are absolute continuous with respect to and the setspartition .
The function as a strictly ascending on has its left and right continuous versions.
Definition 7.
Definition 8. We assign a closed interval to each point of : to each .
Definition 9. Let and let ( denote the power set of ).
Lemma 10. Both are strictly ascending and & by . Moreover both are continuous on .
The mapping defined in Definition 8 assigns a closed bounded interval to each point in such that and .
Proof. (1)Both are strictly ascending and map by . Let be such that . is an ascending function; hence that proves the first statement.(2)By the proved strictly ascending property of , holds. Hence implies that the proposed interval is the smallest interval enclosing the -image of point . Moreover, if either or holds. Assume the first option. Then (3)If then is continuous and is continuous since ; hence is continuous at s.(4)Let then or . In the first case such that . In the second case by Definition 9; hence such that .
Definition 11.
Lemma 12. The mapping is constant on , i. e., . Moreover is continuous, ascending, and onto.
Proof. By (13) for , it follows that . The onto property follows from the proof of point in Lemma 10.
Lemma 13. The relationships between & are as follows:However, exchanging the two mappings will not give . Indeed,
Proof. We prove it for the first case of (14) only. Let . Then if . since (when is a continuity point, is a one point set by Definition 11).
If then and the left endpoint of which is (by the definition of in Lemma 10). Similarly, if then and the right endpoint of which is (by the definition of in Lemma 10). Hence the first statement in (14) of the lemma is proved.
We now prove the first statement of (15). Let . Then since ; hence for a continuity point of . And . This proves the lemma.
The partitioning of into the union can be transformed into the partitioning of as follows.
Definition 14.
The Structure from the Total Variation. In this section we will construct functions of bounded variations from the scale and show that the time interval , is obtainable from the structure defined below.
Definition 15. Let be a closed bounded interval and let be the Lebesgue measure defined by on the Borel sets of . Let be Borel subsets of such thatSince the conditions prescribed for time and in Definition 15 are symmetric, we will construct two representations with mappings below.
Definition 16. The definition of basic mappings for time representation is as follows: (1)let the time scale interval be with ;(2), ;(3), ;(4), .
Definition 17. The definition of the -based mappings for -representation is as follows: (1)let the -scale interval be with ;(2), ;(3); ;(4), . We will treat the time case only since the other is the same word for word.
(1) The Time, Singular Time and (1)If by Definition 16 (if the integral on an interval is positive then the length of the interval is positive.).(2)By the axiom of choice, is a strictly ascending function such that . Then which follows from the continuity of .(3)The ascending function has a countable set of discontinuity points . Hence is continuous.(4)By ascending continuity of (absolute continuous for being an indefinite integral) is closed bounded interval by being preimage of a closed singleton . Moreover and .(5)The integral representation of : using Definition 16 of , and the definition of in point gives the right continuous version of , i.e., Here we used the relations in (17) and defined as ascending mapping.(6) The ascending function is singular with respect to and vice versa, and the ascending function is singular with repect to . Let . Then(7)Hence for any is a function of bounded variation.
With this we proved that the two representations mutually determine each other and what is more there are two representations of any function of bounded variation which is not absolute continuous.
2.2.3. -Integrable Functions on Time and -Scales
In this section we will construct everything for the scale. All steps applied here can be done for the scale also.
Since the ascending functions are noncontinuous functions of , the measures have to be defined from the semiring of left closed, right open intervals with continuity endpoints.
Definition 18. Let the semiring of left closed, right open intervals with endpoints of continuity in be Let the smallest -algebra containing the semiring be
Definition 19. Let the semiring of left closed right open intervals with endpoints of continuity in beLet the smallest -algebra containing the semiring be
Definition 20. Let the semiring of left closed right open intervals with endpoints from in for beand the semiring of left closed right open intervals with endpoints from in for be
The corresponding -algebras are defined as follows.
Definition 21. Let the smallest -algebra containing the semiring be and the smallest -algebra containing the semiring be
Now we can define the measures from the ascending function and from the ascending function .
Definition 22.
Definition 23.
An important relationship is presented in the following lemma.
Lemma 24. From it follows that .
From it follows that .
Moreover and .
Proof. Let . By the statement of this theorem . ThenLet us select a sequence from the covers of constructed in (32) as follows:Lemma 25. Letand let. Thensuch that.
Proof. Let us work in the smallest ring containing the semiring . Since . Let , such that and , where we used the standard technique of changing an infinite union of ring elements into infinite union of pairwise disjoint ring elements [8, 11].□
Corollary 26. By the representation theorem of, a pairwise disjoint systemsuch that. Hence. Thereforeis obtained as a countable pairwise disjoint union of left closed right open intervals with continuity endpoints. Henceand.
Let us apply these propositions to the sequence defined in (33).
Let and let , . By the conditions of the theorem, , , , .
Then let andwhere we used the definition of the covering intervals in (33). Hence , . Recalling that any set of outer measure is measurable we get , hence with .
(1) The Relationship between or Structures and -Ones. We will show that and are isomorphic semirings, respectively, and what is more, the isomorphisms preserve the respective measures.
Lemma 27. by .
Conversely, let and thenby the identity in (15) and Definition 23.
Proof. Essentially the lemma contains the references from which the statements of the lemma follow.
2.2.4. The Relationships between the Semirings and Their Extensions
The lemma below describes the relationship between the continuity points of and the points in .
Lemma 28. , such that (1)if ; otherwise if then and if ; otherwise if then ;(2) for and for ;(3) for and for .
Proof. (1)Let , then , . Similarly, if then , .(2)Let . Let .(3)Since is countable and the open intervals (continuity points).(4)Let .(5)Let and .(6)Go to .(7)This process will stop after generating the sequences of continuity points which are strictly ascending/descending sequences converging to .By the strictly ascending property of on the sequences converge to left and right limits of at , respectively. This completes the proof.
Lemma 29. (1) the following statements hold:(a);(b).(2) and is an atom which means either or . Equivalently a measurable set does not cut an atom: (3)If is -measurable then , where is a constant depending on A.(4)An interval is in if and only if .
Proof. (1)By Lemma 28, , , such that the following applies.(a)If then is strictly ascending and for . For the sequence is constant .(b)If then is strictly descending and for . For the sequence is constant .(c)By it follows that (similarly for the intervals with closed ending).(d) and .(e)Hence and by (d) of this proof. Hence from being -algebra it follows that(2)Let for a . Hence by point of this proof. We show now that if then is not measurable. Let for a . If by . Hence . Then holds for the outer measure of . Hence H is not measurable if by Caratheodory’s condition for outer measures ( ) with : proves the lemma.(3)Let be measurable with respect to . Let us assume that and such that . Then with the sets are disjoint and hence proper nonempty subsets of the measurable set ; hence by statement 1 of this lemma, the sets are not measurable; hence f is not measurable. If is measurable and then each component of f is measurable and hence constant on the elements of ; hence the statement follows.(4)(a)Let . Then such that . Then by Definition 16 point , . Hence , almost everywhere. By statements and of this lemma, the measurable function const on follows in ; hence in .(b)Conversely, let and let . Then . Hence by Definition 16 point , . Hence proves that .
Lemma 30.
Case (1). Let , , and let fulfil if and otherwise. Then .
Case (2). Let and let fulfil if and otherwise. Then .
Proof.
Case (1). . Applying Lemma 13 and the set of equations (14) to the left and right sides of the inequalities, respectively, leads to the following:Hence = has been proved and therefore case holds.
Case (2). Let . Then . Applying Lemma 13 and the set of equations (15) leads to the following:Hence has been proved and therefore case holds.
Notation 31. Let
2.2.5. Transpose of the Transformations between t and Total Variation Scales
Let us consider the transpose of a transformation of the domains of function spaces.
Definition 32. Let be sets and let with and let . Then letNote that against .
Notation 33. Notations for the lemma about the transpose mappings. Let and let . Let be a set of functions.
Let ,
Lemma 34. Let and be two mappings as defined above. Then the following relations hold:Moreover & are linear isomorphisms.
Proof. (1)Let and let be the transpose of . Then (2)Let and let be the transpose of . Then (3)Let us consider now the composition : By the first identity in (15) we discovered that holds. If then for a ; hence ; hence (with f being constant on ). By (13) = = = = completes the proof.(4)Let us now consider the composition : by (14). Hence . (5)If is a vector space then are vector spaces for the point-wise addition and multiplication by scalar. Equation (45) gives the linear isomorphism.
Corollary 35. Let the notations of Lemma 34 be changed as follows: Then Lemma 34 with and holds. Precisely Hence the mappings & are linear isomorphisms between the two measurable function spaces.
Proof. The two continuous ascending mappings → → between the two domains are the same as in Lemma 34.
Let . be measurable and is continuous; henceLet be measurable and is continuous; hence
. By the properties proved in Lemma 34 is constant on all . Hence the statements of Lemma 34 are true (the two identities in (45)).
Let and let
. Then the following applies.
Theorem 36. By Corollary 35 it follows thatWhat is more,
Proof. (1)By Definition 22, let be a measurable set and let its characteristic function be . Then By Definition 23 let be a measurable set and let its characteristic function be . Then(2)By Lemma 34 for any subset and holds for the outer measures . Conversely such that either or and then the relation holds: . From this it follows that for measurable sets in both -algebras the outer measures define the extended measures and therefore the measures on the -algebras are fulfilled. This is By the linearity of the transpose mappings, the identities are valid for all nonnegative step-functions and from there for all integrable function. This completes the proof of the theorem.
Theorem 37. Let & . ThenLet & and :
Proof. (1)Let . Then where the first statement in (53) of Theorem 36 was applied and the statement of case in Lemma 30 gave the last equality from which finally we obtain the statement: This proves the first statement.(2)Let : Now we will apply the statement of case in Lemma 30 to prove the final result:
2.2.6. The Main Result
In Theorem 37, we established the relationship between absolute continuous indefinite integrals of members of and the absolute continuous indefinite integrals of members of .
Our next step is to establish the relationship between the indefinite integrals of the members of and the indefinite integrals of the members of .
We prove the following statements.(1)Let and let , . Let , .(2)From it follows that . From it follows that .(3)The signed measure is absolute continuous with respect to by . The signed measure is absolute continuous with respect to by .(4)By the absolute continuity stated in point mentioned above, the -measurable Radon Nikodym derivative exists: By the absolute continuity stated in point mentioned above, the -measurable Radon Nikodym derivative exists:
With the statements mentioned above we have proved the following result.
Theorem 38. A function is of bounded variation on if and only if the following conditions are fulfilled. (1)There exists a closed interval partitioned as union of two disjoint measurable sets such that is onto (see the details in Definition 15, equation (17)).(2)There exists an absolute continuous function with such that the following applies.(a) On the t-scale with the Radon - Nikodym derivative defined in (64) is expressed by(b) The scale representation of with the Radon - Nikodym derivative defined in (65) is expressed bywhere we used the notations and functions defined in Definition 15, equation (17) and the mappings defined in Definitions 16 and 17.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.