Distributional, Differential and Integral Problems: Equivalence and Existence Results

We are interested in studying the matter of equivalence of the following problems: Dx = f (t, x)Dg x(0) = x 0 (1) where Dx and Dg stand for the distributional derivatives of x and g, respectively; x g (t) = f (t, x(t)), m g-a.e. x(0) = x 0 (2) where x g denotes the g-derivative of x (in a sense to be specified in Section 2) and m g is the variational measure induced by g; and x(t) = x 0 + t 0 f (s, x(s))dg(s), (3) where the integral is understood in the Kurzweil–Stieltjes sense. We prove that, for regulated functions g, (1) and (3) are equivalent if f satisfies a bounded variation assumption. The relation between problems (2) and (3) is described for very general f , though, more restrictive assumptions over the function g are required. We provide then two existence results for the integral problem (3) and, using the correspondences established with the other problems, we deduce the existence of solutions for (1) and (2).

Dx = f (t, x)Dg x(0) = x 0 (1) where Dx and Dg stand for the distributional derivatives of x and g, respectively; x g (t) = f (t, x(t)), m g -a.e.
x(0) = x 0 (2) where x g denotes the g-derivative of x (in a sense to be specified in Section 2) and m g is the variational measure induced by g; and where the integral is understood in the Kurzweil-Stieltjes sense.We prove that, for regulated functions g, (1) and ( 3) are equivalent if f satisfies a bounded variation assumption.The relation between problems (2) and ( 3) is described for very general f , though, more restrictive assumptions over the function g are required.We provide then two existence results for the integral problem (3) and, using the correspondences established with the other problems, we deduce the existence of solutions for (1) and (2).

Introduction
This paper deals with three types of equations aiming to investigate the equivalence of their solvability, that is, whether the existence of solutions to one of the equations leads to the existence of solutions to the other two.Among the problems to be studied here, the distributional differential equations of the form certainly represent a very general formulation of differential problems.Evidently, when g is absolutely continuous, then its distributional derivative coincides with the usual derivative and we retrieve the classical differential equation.Besides, recalling that the distributional derivative of a function of bounded variation originates a Borel measure, it is clear that measure-driven equations can be regarded as a particular case of (1); see [3], [31] and the references therein.Accordingly, equation (1) covers a broad range of problems for the theory of measure differential equations has been an effective tool in the study of impulsive systems, retarded equations and equations on time scales (e.g.[13], [14] and [25]).
A novel feature in the present study is that the function g in the distributional problem (1) is not assumed to be of bounded variation, but only regulated.To treat such a problem we will make use of the regulated primitive integral introduced in [38].This integral somehow inverts the distributional derivatives of regulated functions allowing us to convert a distributional equation to an integral equation.This method has been used in many papers recently; see, for instance, [20], [21] and [22].In our approach, though, we take advantage of the connection between the regulated primitive integral and the Kurzweil-Stieltjes integral (cf.[38,Definition 12] or Theorem 2.15).Therefore, we investigate problem (1) by reducing it to an integral equation (3).It is important to remark that, to avoid paradoxes, extra attention is required when defining solutions for (1) as functions satisfying (3); see [19] for more details.
The study of derivatives with respect to functions and its connection with integrals is not exactly new in analysis (cf.[41] and [4]).A rather recent idea, though, is presented in [27] together with an interesting applicability of such a differentiation process.In [27], the authors consider derivatives with respect to non-decreasing left-continuous functions; however, nothing really prevents the study of such a notion in a more general setting.Besides, for monotone g, in most cases we can reduce the differentiation with respect to g to ordinary differentiation.This motivated us to define g-derivative for left-continuous regulated functions g.The generality of such a derivative asks for a notion of measure which can be meaningfully applied to more general functions, thus the use of a variational measure in the present paper (see Definition 2.4).In the case when g is the identity, it is known even in the abstract setting that the equivalence between (2) and ( 3) is always possible by appropriately choosing the integration process and respectively the type of derivative (see [2]).In our case, the investigation of g-differentiation problems of the type (2) via integral equations (3) is due to new versions of the Fundamental Theorems of Calculus we proved under quite weak assumptions.
At last, we provide two existence results for the integral problem (3) which, unlike other results available in the literature (cf.[13,Theorem 5.3]), do not rely on the assumption of g being monotone.We conclude the paper by using the correspondences established with the other problems to deduce the existence of solutions for (1) and (2).

Preliminary results
Recall that a function g : [a, b] → R is regulated if the one sided-limits exist, more precisely: It is well-known that regulated functions are bounded and they can have at most a countable number of points of discontinuity (see [23,Corollary I.3.2]).The space G([a, b]) of real-valued regulated functions on [a, b] is a Banach space when endowed with the norm Moreover, the set of all left-continuous regulated functions on [a, b] and right-continuous at a is a closed subspace of G([a, b]) and it will be denoted by The following notion is important when investigating compactness in the space of regulated functions.

Definition 2.1 ([15]). A set F ⊂ G([a, b]
) is said to be equiregulated if for every ε > 0 and every t 0 ∈ [a, b] there exists δ > 0 such that, for all x ∈ F we have: If the set { f n : n ∈ N} is equiregulated, then f n converges uniformly to f .
For regulated functions, the analogous to Arzelà-Ascoli theorem reads as follows.
Given a gauge δ on A, i.e. δ : The set of all of δ-fine systems on A will be denoted by S(A, δ).
A partition of the interval [a, b] is a system S = {(c j , [a j , b j ]) : j = 1, . . ., m} satisfying b j = a j+1 , j = 1, . . ., m, where a 1 = a and a m+1 = b.We remark that for an arbitrary gauge δ on [a, b] there always exists a δ-fine partition of [a, b].This is stated by the Cousin lemma (see [32,Lemma 1.4]).
Throughout this paper, λ(E) denotes the Lebesgue measure of E, for Lebesgue measurable sets E ⊂ R. The following definition corresponds to the notion of variational measure which figures in the problem (2).
Note that m g is actually the Thomson's variational measure S 0 -µ g defined in [39] (see [7, Proposition 4.2 (xiv)]).In the case when g is the identity function, the definition above leads to the Lebesgue outer measure (see [10,Proposition 3.4] for details).The next proposition summarizes some of the properties of m g and ensures that it defines a metric outer measure (see [39, p. 87] and [10,Proposition 3.3] for the proofs).
Proposition 2.5 (v) shows that the variational measure over singletons provides information on the 'size' of the discontinuity of the function at a point.More important, a function g is continuous at c if and only if m g ({c}) = 0. Remark 2.6.Regarding the outer measure m g , we will say that a property holds m g -almost everywhere (shortly, m g -a.e.) if it is valid except for a set N ⊂ [a, b] with m g (N) = 0.
Note that, given A ⊂ [a, b], for a fixed gauge γ : A → R + we have Thus, in order to prove that a set A has m g -measure zero, it is enough to show that given ε > 0, there exists γ ε : A → R + such that W g (S) < ε for every S ∈ S(A, γ ε ).
The definition above was presented in [11] for functions g which are continuous and BVG.In the particular case when g is the identity function (and consequently m g is the Lebesgue outer measure) the notion of g-normal coincides with the so-called (strong) Lusin condition (see [29] or [33]).The interested reader can find more details on the relation between these two notions in [12,Section 5] and [9,Section 4].
The following result is a particular case of [ Enclosing this subsection we will discuss two other general notions of variation and some of their properties.
Definition 2.9 ([12]).Let g : [a, b] → R. We say that: It is easy to see that any function g of bounded variation on [a, b] We can draw an analogy connecting the concept of BVG • functions and σ-finite measure.Indeed, if g is BVG • , this means that the outer measure m g is σ-finite on [a, b].Thus, in view of [39,Theorem 40.1], the relation between BVG • and the notion of generalized bounded variation, VBG * in the sense of Saks [29], reads as follows: a function is BVG • if and only if it is bounded and VBG * .
From the remarks above, we can see that a BVG • function is bounded; moreover, it is not hard to show that such a function has at most countably many points of discontinuity (see [39, p. 93]).Although the class BVG • encompasses the functions of bounded variation, a BVG • function need not even be regulated.A simple example of this fact is the function g : [0, 1] → R given by g(1/n) = 1 for n ∈ N, and g(t) = 0 otherwise.
The following proposition provides a useful estimate for BV • functions.Remark 2.11.It is worth emphasizing that in Proposition 2.10 the increasing function H can be chosen left-continuous when g is supposed to be left-continuous.Indeed, let us recall from the proof of Lemma 3.5 in [12] that where the gauge δ on E is chosen so that W g (S) < m g (E) + 1 for every δ-fine system S on E. It is not hard to see that for any t ∈ [a, b] and ε > 0 Thus, assuming that g is left-continuous, the left-continuity of H at an arbitrary point t ∈ (a, b] holds once we show that lim ε→0+ sup{W g (S) : S ∈ S(E, δ), S ⊂ (t − ε, t)} = 0.
To prove this fact we follow the method of the proof of Lemma 16 at page 140 in [5].More precisely, reasoning by contradiction, suppose that there exists η > 0 such that for every ε > 0, there exists n 1 and let S 2 ∈ S(E, δ) be such that S 2 ⊂ (t − ε 2 , t) and W g (S 2 ) ≥ η.If we proceed in this way, we obtain a decreasing sequence of positive numbers ε k , k ∈ N, and systems In view of the above remark, next assertion provides a characterization of BVG • functions borrowed from [12,Lemma 3.6].Proposition 2.12.Let g : [a, b] → R be given.Then, g is BVG • if and only if there exists a strictly increasing function H If, in addition, g is left-continuous, then H can be chosen left-continuous.
The following result will be useful later.

Integrals and derivatives
This subsection is devoted to the notions of integrals and their related derivatives which will be used in our work.We recall some of their basic properties and prove a few new ones which, to our knowledge, are not available in the literature.As problem ( 1) is related to the theory of distributions, we will begin with a short introduction into this setting (see [35,36] for more details).
A distribution on [a, b] is a linear continuous functional on the topological vector space D of test functions, namely, functions φ : R → R which have continuous derivative φ (j) of any order j ∈ N vanishing on R \ (a, b).The space D is endowed with the topology induced by the following convergence of sequences: The distributional derivative of a distribution G, denoted by DG, is itself a distribution defined by DG, φ = − G, φ for every φ ∈ D.
In particular, if f : [a, b] → R is a left-continuous BV-function, then its distributional derivative corresponds to the Stieltjes measure associated to f , defined by and then extended to all Borel subsets of [a, b] in the standard way (for details, see [28,Example 6.14]).
To deal with the problem (1) we will make use of the notion of regulated primitive integral introduced in [38].Hence, we will restrict ourselves to distributions which correspond to the distributional derivative of a regulated function, i.e., distributions g on [a, b] These distributions are called RP-integrable in the sense to be specified in the following definition.
Definition 2.14.Let g be a distribution on [a, b] and G ∈ G − ([a, b]) be such that g = DG.The regulated primitive integral of g is defined by and we say that g is RP-integrable with primitive G.The space of RP-integrable distributions on We remark that the definition above can be regarded as a particular case of the notion introduced in [38] -which is concerned with distributions on the extended real line.It is shown in [38] that the RP-integral is more general than Riemann, Lebesgue and Henstock-Kurzweil integrals.Moreover, A R := A R (R) is a Banach space when endowed with the Alexiewicz norm and, consequently, the completion of the space of signed Radon measures (see [38,Theorem 4]).
In the sequel we borrow some of the results presented in [38] with an obvious adaptation to compact intervals.In this regard, it is also worth mentioning [26] where, along with a discussion on the product of distributions, we find the following identity for the case when f is a BV-function, G is regulated and both functions are assumed to be right-continuous.
The connection between the RP-integral and the distributional derivative is described in the following Fundamental Theorem of Calculus.
Now we present a short overview on Kurzweil-Stieltjes integral, which is the integral found in problem (3).For a more comprehensive study of this topic, see [32] or [34] for instance.

Definition 2.18. A function
Notice that when g(t) = t, t ∈ [a, b], the definition above reduces to the notion of Henstock-Kurzweil integral (for which the reader is referred to [16], see also [24]).Recall that such an integral generalizes the Lebesgue integral and integrates all derivatives.When it comes to Stieltjes-type integrals, it is known that, for integrators of bounded variation, Lebesgue-Stieltjes integrability implies Kurzweil-Stieltjes integrability (cf.[29, Theorem VI.8.1]), while the equivalence relies on stronger assumptions (see [6,Theorem 2.71]).
is regulated and satisfies If, in addition, g is a BV-function and f is bounded, then F is a BV-function.
In order to investigate some properties of the Kurzweil-Stieltjes integral regarding the variational measure defined previously, we recall the following lemma.

Lemma 2.20 (Saks-Henstock lemma
Next proposition presents two additional properties of the indefinite Kurzweil-Stieltjes integral (for a similar result in the framework of functions VBG * see [37, Lemma 3.12]).
Then, F is g-normal.
If, in addition, g is a BVG Given ε > 0 and fixed n ∈ N, there is a gauge γ 1 : for every S ∈ S(A n , γ 1 ).
Let γ 2 : [a, b] → R + be a gauge as in the Saks-Henstock lemma (Lemma 2.20) and consider the gauge γ(t) = min{γ 1 (t), γ 2 (t)}, t ∈ A n .Bearing all these in mind, for any system S ∈ S(A n , γ), with S = {(c j , [a j , b j ]) : j = 1, . . ., m}, we have Therefore, W F (S) < 2 ε for every S ∈ S(A n , γ), which implies that The second statement can be proved in a similar way observing that: if The following result is contained in [12, Proposition 2.9].Convergence theorems are essential when working with integral equations.In our study we will need a result based on the following notion.
The proof of the following theorem follows the same approach used in [32, Theorem 1.28] (see also [18,Theorem 3.28]).
In [27] a notion of differentiability connected to Stieltjes-type integral was introduced for non-decreasing left-continuous functions g.In this work, we will consider the g-derivative as defined in [27], but assuming simply that g is regulated and left-continuous.
if g is discontinuous at t, provided the limit exists.In this case, we say that Given a function g ∈ G − ([a, b]), we consider the following sets: It is not hard to see that for t ∈ J + g , the g-derivative f g (t) exists if and only if f (t + ) exists.In particular, we have the following proposition.Proposition 2.27.If f , g ∈ G([a, b]) and g is left-continuous, then f is g-differentiable at the points of J + g .
In [12], a notion of differentiation with respect to another function is defined in terms of limit superior and limit inferior.It is worth highlighting that, at the points of continuity of g, our Definition 2.26 coincides with [12, Definition 3.1].
As it was observed in [27], for the points t ∈ [a, b] in which g is continuous, the definition above has sense only if t ∈ C g .However, the next theorem shows that this set is rather 'small', not representing a real drawback to our purposes in this work.
Proof.Since C g is open, it can be writen as a countable union of disjoint open intervals.Hence, due to Proposition 2.5 (iii), it is enough to prove that m g ((u, v)) = 0, where (u, v) is assumed to be one of those open intervals.
For n ∈ N, consider the interval . By the fact that g is constant on (u, v), it follows that W g (S) = 0 for any system S ∈ S(J n , γ), and consequently m g (J n ) ≤ inf δ≤γ sup{W g (S) : S ∈ S(J n , δ)} = 0, Since (u, v) = n∈N J n , it follows from Proposition 2.5 (iii) that m g ((u, v)) = 0.
The following is a direct consequence of Proposition 2.5 (v).
Remark 2.30.In view of Propositions 2.28 and 2.29, whenever a property holds m g -almost everywhere in some set E ⊆ [a, b], without loss of generality, we can assume that it holds excluding also the sets C g and N g , that is, m g -almost everywhere in E \ (C g ∪ N g ).
The following proposition is the corresponding to [27, Lemma 6.1].
i) If t 0 ∈ J + g , then for every ε > 0 there exists ρ(t 0 ) > 0 such that ii) If t 0 ∈ J + g ∪ N g , then for every ε > 0 there exists ρ(t 0 ) > 0 such that In order to give conditions ensuring the differentiability with respect to increasing functions, we will need the following classical result from real analysis.(2.3) The following result contains a variant of [11,Proposition 4] as well as an analogous to [8, Lemma 5.2] in the case of functions BV • .Proposition 2.33.Let H : [a, b] → R be a strictly increasing and left-continuous function.
The key point in the proof of this assertion is the fact that H is continuous in [a, b] \ J + H and BVG • (due to its motonicity); therefore, the H-differentiability of F can be understood as the differentiability in the sense of [12,Definition 3.1].In view of this and recalling that F is BVG • , we can apply [12,Proposition 3.10] H can be written as a union U = U 1 ∪ U 2 , with m H (U 1 ) = 0 and U 2 at most countable.
This shows that: g is continuous at t if and only if H is continuous at t. Therefore, if t is a point of continuity of g, g H (t) .On the other hand, for t ∈ A\ C g ∪ Z such that t ∈ J + g , we have that t is a point of discontinuity of H and g H (t) .Let us prove that m g (Z) = 0. Fixed an arbitrary ε > 0, by Proposition 2.31, for t ∈ Z \ J + H , there exists ρ(t) > 0 such that Put Z ∩ J + H = {τ i : i ∈ Γ}, where Γ ⊆ N, with τ i = τ j for i = j.The left-continuity of g implies that, for each Given S ∈ S(A n , γ ε ), with S = {(c j , [a j , b j ]) : j = 1, . . ., k}, using the inequalities above we obtain This, together with Remark 2.6, proves that m g (Z) = 0.
In [27], we find two Fundamental Theorems of Calculus (Theorems 6.2 and 6.5) connecting the KS-integral w.r.t.g with the g-derivative in the case when g is non-decreasing leftcontinuous.In Subsection 2.2 we will provide similar results for the case when g is a function in G − ([a, b]) which is BVG • .

Fundamental Theorem of Calculus
A descriptive characterization of the Kurzweil-Henstock integral in terms of variational measures is given in [33].Concerning Stieltjes-type integral, we can mention the results in [12]; though, some continuity assumption is required.The content of this subsection, devoted to Fundamental Theorem of Calculus, somehow provides a descriptive characterization of the Kurzweil-Stieltjes integral.
The first Fundamental Theorem of Calculus to be presented extends the result from [27,Theorem 6.5] to a more general class of functions g, namely, regulated functions which are BVG • .The passage to a BVG • integrator is based on the notion of g-normal function, in connection with some elements from [12] and [11].We mention that this result also generalizes [12,Corollary 4.8] proved for continuous BVG • functions.
28 and Proposition 2.34), it remains to show that m g (U) = 0.
Recalling that H, H 1 and H 2 are increasing, it is not hard to see that for any system S on N we have

19). Let us prove the equality for points
→ R be a gauge as in Saks-Henstock lemma (Lemma 2.20).Using Proposition 2.31, we can choose 0 < ρ(t 0 ) < δ(t 0 ) so that Without loss of generality, assume t 0 < t.Thus, applying Saks-Henstock lemma together with the inequality above we obtain Since ε is arbitrary, we conclude that F g (t 0 ) = f (t 0 ) and the result follows.
Also connecting g-derivatives and the KS-integral, next Fundamental Theorem of Calculus somehow generalizes a similar result given for non-decreasing left-continuous functions in [27,Theorem 6.2].The method of proof combines ideas from [27] and [12]. Then, where h(s) = F g (s) for s ∈ [a, b]\N and h(s) = 0 otherwise.
By Proposition 2.10, for each n ∈ N, there exists a strictly increasing function Let ε > 0 be given.Without loss of generality, by Remark 2.30, we can assume that N g ⊂ N. Since F is g-normal, we have m F (N) = 0 and we can choose a gauge γ : N → R + such that W F (S) < ε for every S ∈ S(N, γ). ( Recalling that g has at most a countable number of points of discontinuity we can write J + g = {τ i : i ∈ Γ}, Γ ⊆ N, with τ i = τ j for i = j.Due to the left continuity of the functions F and g, for each i ∈ N there exists η i > 0 such that Given t ∈ [a, b]\N, we know that t ∈ Z n for some n ∈ N and we have then two cases to consider: if t ∈ J + g or not.If t ∈ Z n ∩ J + g , by Proposition 2.31, there exists (2.8) (where the series is actually a sum with finitely many terms).In view of (2.6), it follows that ∑ c j ∈N F(t j ) − F(t j−1 ) < ε.In order to analyse the remaining sum, let us fix an arbitrary n ∈ N. If Z n ∩ {c j : j = 1, . . ., } = ∅ there is nothing to be proved, otherwise, at least one of the sets Λ n = {j ∈ {1, . . ., } : c j ∈ Z n \J + g } Γ n = {j ∈ {1, . . ., } : c j = τ i j ∈ Z n for some i j ∈ Γ} is non-empty.It is not hard to see that the sum over c j ∈ Z n is obtained by combining the sums over Λ n and Γ n .Clearly, by (2.9) we obtain On the other hand, (2.8) together with (2.7) imply In the definition of a solution of the problem (2), we can always assume that C g ⊂ N (see Remark 2.30).
To convert a distributional differential equation to an integral equation in the space of primitive functions, the Fundamental Theorem of Calculus relatively to the regulated primitive integral found in [38,Theorem 6] is a very useful tool.This approach appears, for instance, in [20] and [21].In order to obtain an equivalence between problems (1) and ( 3), besides the aforementioned result we take into account the relation described in Remark 2.16.
Then x : [0, 1] → R is a solution of problem (1) if and only if it is a solution of problem (3).
Proof.Suppose that x is a solution of (1).Since h x (t) = f (t, x(t)), t ∈ [0, 1], is a function of bounded variation and Dg ∈ A R ([0, 1]), by Proposition 2.15 we have h x Dg ∈ A R ([0, 1]) and (where the integral on the right hand side is the Kurzweil-Stieltjes integral).Moreover, Theorem 2.17 implies that for all t ∈ [0, 1]: Combining these two facts we conclude that x is a solution of (3).
Let now x be a solution of (3).Using the equality found in Proposition 2.15 we get where h x (s) = f (s, x(s)), s ∈ [0, 1].Thus, by Theorem 2.17, the distributional derivative of x is Dx = f (t, x)Dg and the result follows.
Remark 3.4.The superposition assumption on f in Theorem 3.3 ensures the equivalence of the mentioned problems for a very large class of functions g, namely for every left-continuous regulated function.We remark that this assumption on f could be weakened if we require stronger assumptions on g; that is, if g is a left-continuous BV-function.Indeed, let us recall that [38,Definition 11] introduces the product h x Dg as the distributional derivative of the function defined in [38,Proposition 10] as follows: where {c n , n ∈ N} denotes the set of common discontinuity points of g and h x .It can be seen that Ξ is also BV when g and h x are BV, thus h x Dg is in this case the distributional derivative of a BV-function.As by Definition 3.1.1.a solution x of problem (1) satisfies the equality Dx = h x Dg, our solutions x will be in the space of BV-functions.It turns out that in the case when g is BV, the main assumption in the previous equivalence result can be replaced with the following (weaker) assumption: ) is a BV-function for every left-continuous BV-function x.
Conditions ensuring this property of the superposition operator can be found, for instance, in [1].
On the other hand, recalling that for left-continuous functions of bounded variation the distributional derivative is a Borel measure, this result ensures the equivalence between measure differential equations and g-differential equations.
In [27], the authors briefly illustrate the applicability of the g-derivative showing that ordinary differential equations, dynamic equations on a time scale and impulsive equations can be regarded as a g-differential problem (2).Next theorem is concerned with the relation between problem (2) and an integral equation, allowing us to explore other aspects of the g-differential equation.
Proof.Let x be a solution of (2) and, without loss of generality, assume that C g ⊂ N (see Remark 2.30).This means that x is g-normal and x g (t) = f (t, x(t)) for t ∈ [0, 1]\N.Therefore, by the second Fundamental Theorem of Calculus, Theorem 2.36, where f is the function given by Conversely, assume that x is a solution of problem (3), that is Using the Fundamental theorem of calculus, Theorem 2.35, we obtain that x is g-differentiable m g -a.e. and that Note that a solution of ( 3) is necessarily a fixed point of the operator T. We will prove that the assumptions of the nonlinear alternative, Theorem 3.7, are satisfied for Step Step 2. We claim that T(B) satisfies that assumptions of Lemma 2.3.Indeed, assumption (iii) states that {Tx(t) : x ∈ B R 0 } is bounded for each t ∈ [0, 1], so only the equiregularity of the family {Tx : x ∈ B} remains to be proved.As before, this is a consequence of Proposition 2.24, using (ii) and the fact that, due to the continuity in the second argument, for a fixed t ∈ [0, 1], there is Combining steps 1 and 2, we can see that T : B → G − ([0, 1]) is a compact mapping.Since assumption (iv) asserts that the alternative in Theorem 3.7 is not possible, we conclude that the operator F = T − x 0 has a fixed point.
In the case when g is a BVG • function, we can deduce another existence result under weaker assumptions on f .Theorem 3.9.Let g ∈ G − ([0, 1]) be a BVG • function, x 0 ∈ R and f : [0, 1] × R → R satisfy the assumptions (i), (ii) and (iv) in Theorem 3.8.Then, the integral problem (3) has at least one solution.
Proof.We will prove that if g ∈ G − ([0, 1]) is a BVG • function, then the assumption (iii) in Theorem 3.8 is a consequence of assumptions (i) and (ii).Let R > 0 be given and consider ).Note that assumption (ii) together with Saks-Henstock lemma imply that there exists a gauge γ : for every γ-fine system (ξ j , [α j−1 , α j ]) and for all x ∈ B. Since g is a BVG • function, the interval [0, 1] can be written as a disjoint countable union of sets E n such that m g (E n ) < ∞, n ∈ N.For each n, by the definition of m g , there exists a gauge δ n : E n → R + (which we can assume to be bounded from above by γ) such that W g (S) < m g (E n ) + 1 for every S ∈ S(E n , δ n ). (3.2) By the continuity of f in the second argument, assumption (i), for each t ∈ [0, for every x ∈ B and |s − t| < δ n (t).
We use now (3.3) and the compactness of [0, 1] in order to get the boundedness property (iii) of Theorem 3.8.
Remark 3.10.Both existence results, Theorem 3.8 and 3.9, might throw a new light in the study of measure functional differential equations in the sense of [13].Actually, problem (3) is an example of the so-called measure differential equations; however, unlike the theory which has been developed up to now, here we deal with a more general class of integrators.
In view of the equivalence stated in Theorem 3.3, from our first existence result we derive the following.

Definition 2 . 4 .
Let g : [a, b] → R. For each A ⊆ [a, b],we define the g-outer measure of A by

Proposition 2 . 15 .Remark 2 . 16 .
The multipliers of the space A R([a, b]) are the functions of bounded variation.Moreover, if f : [a, b] → R is a BV-function and G ∈ G − ([a, b]), the RP-integral of the product f DG is defined by r b a f DG = b a f (t) dG(t),where the integral on the right-hand side is the Kurzweil-Stieltjes integral (seeDefinition 2.18).The expression of the product presented in Proposition 2.15, defined via the integration by parts formula[38, Definition 12], agrees with Definition 11 in the same paper.

Lemma 2 . 22 .
Let g : [a, b] → R, and assume that f : [a, b] → R is null, except on a set N ⊂ [a, b] with m g (N) = 0. Then f is KS-integrable w.r.t.g and t a f (s)dg(s) = 0 for every t ∈ [a, b].

Definition 2 . 23 .
Let g : [a, b] → R and let F be a family of real functions defined in[a, b].We say that F is equiintegrable with respect to g if for every ε > 0 there exists a gauge δ on [a, b] such that

Proposition 2 .
24 ([30, Proposition 3.4]).Let g ∈ G([a, b]) and assume that F is a family of real functions defined in [a, b] equiintegrable w.r.t.g.If for each t ∈ [a, b], the set {
see Propositions 2.19 and 2.21).Let H 1 , H 2 : [a, b] → R be strictly increasing left-continuous functions which exist by Proposition 2.12 for F and g, respectively.Defining H = H 1 + H 2 , from Proposition 2.33 we know that the derivatives F H and g H exist on [a, b]\U, where U ⊂ [a, b] and m H (U) = 0. Applying Proposition 2.34 for A 1.Let us check that the operator T is continuous.To this end, consider a sequence (x n ) n converging uniformly to x in B. Hypothesis (i) implies that ( f (•, x n (•))) n converges pointwisely to f (•, x(•)) while assumption (ii) ensures the equiintegrability of the sequence.lim n→∞ Tx n (t) = Tx(t).On the other hand, it yields from the convergence of ( f (•, x n (•))) n that the sequence is pointwisely bounded.Since, it is equiintegrable by assumption (ii), an application of Proposition 2.24 brings us to the equiregularity of the family of their primitives, that is {Tx n : n ∈ N} is equiregulated.Then, by Lemma 2.2 Tx n converges to Tx uniformly on [0, 1], which implies the continuity of T at x ∈ B.