Existence and Uniqueness of Solution for Stieltjes Differential Equations with Several Derivators

In this paper, we study some existence and uniqueness results for systems of differential equations in which each of the equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have one solution under the Osgood condition, or even, the Montel–Tonelli condition. We also explore some results guaranteeing the existence of solution under these conditions. Along the way, we obtain some interesting properties for the Lebesgue–Stieltjes integral associated with a finite sum of nondecreasing and left-continuous maps, as well as a characterization of the pseudometric topologies defined by this type of maps.


Introduction
Stieltjes differential equations have gained popularity in the recent years. The main difference with respect to regular differential problems is the presence of the Stieltjes derivative, a modification of the usual derivative on the real line through a nondecreasing and left-continuous map. This change allows us to study impulsive differential equations and equations on time scales in a unified framework, see for example [6,9], or even [3] for the corresponding integral formulation counterpart.
The usual setting for Stieltjes differential equations in the literature involves a single derivator either in its theoretical-see for example [4,6,10,12,15,16,22]-or numerical studies [4,5]. This is also the case for other differential problems involving Stieltjes derivatives such as in [14,17,21], or even the corresponding integral counterparts. Nevertheless, it is the new setting of differential problems with Stieltjes derivatives that offers the possibility 181 Page 2 of 31 I. M. Albés and F. A. F. Tojo MJOM of a new type of problems: systems of differential equations in which each of the components is differentiated with respect to a different nondecreasing and left-continuous function. This was the case in papers such as [11,13], or even [7], where the authors considered differentiation with respect to functions that are not necessarily monotonous. Here, we aim to improve the work along this line regarding systems of equations. Specifically, we will consider maps g i : R → R n , i = 1, 2, . . . , n, such that each g i is nondecreasing and left-continuous, and we will discuss some existence and uniqueness results for the system x gi (t) = f i (t, x(t)), x i (t 0 ) = x 0,i , i = 1, 2, . . . , n, (1.1) where x gi denotes the Stieltjes derivative of x with respect to g i in the sense presented in [9]. In this setting, we build on the work in [11], adapting some of the results in [6,15] to the context of Stieltjes differential equations with several derivators. The paper is structured as follows: first, in Sect. 2, we present the basic tools for the Lebesgue-Stieltjes integration on the real line defined in terms of a nondecreasing and left-continuous map. We also obtain some interesting results regarding the Lebesgue-Stieltjes outer measure, as well as a fundamental property regarding the Lebesgue-Stieltjes measure for the finite sum of nondecreasing and left-continuous functions-more information on the Lebesgue-Stieltjes measure can be found in [1,2,6,8,9,25] and for the more general Kurzweil-Stieltjes integral in [18]. Next, in Sect. 3, we introduce the Stieltjes derivative in the sense of [6] and we explore some concepts of continuity in a similar fashion to [6,11]. In particular, we discuss some of the limitations of the mentioned work. Furthermore, throughout this section, we obtain important information regarding the pseudometric topology defined by a nondecreasing and left-continuous function, showing that it can be fully characterized in terms of some interesting sets related to such map. Finally, in Sect. 4, we turn our attention to the study of problems of the form (1.1). First, we continue the study of everywhere solutions started in [11] and, later, we follow the arguments in [15] to obtain some existence and uniqueness results involving Osgood and Montel-Tonelli conditions.

The Lebesgue-Stieltjes Measure
Throughout this paper, we will make us of the Lebesgue-Stieltjes integral associated with nondecreasing and left-continuous functions. This integral is constructed as the integral with respect to a measure defined in terms of the mentioned map through the classical Carathéodory's extension theorem, see for example [1,2,19,20,23]. Specifically, given a nondecreasing and leftcontinuous map g : R → R, and denoting by P(R) the set of all subsets of R, we define the map μ * g : P(R) → [0, +∞] as MJOM Existence and Uniqueness of Solution for Stieltjes Page 3 of 31 181 [a n , b n ), {[a n , b n )} ∞ n=1 ⊂ C , (2.1) with C = {[a, b) : a, b ∈ R, a < b}. The map μ * g is an outer measure and, by considering its restriction to the following σ-algebra for all E ∈ P(R)}, we obtain the Lebesgue-Stieltjes measure associated with g, which we denote by μ g .
Remark 2.1. Every Borel set belongs to LS g . In particular, this means that C ⊂ LS g . Furthermore, we have that μ g ([a, b)) = g(b)−g(a) for any [a, b) ∈ C.
For simplicity, in what follows we will use the term "g-measurable" for a set or function to refer to μ g -measurability in the corresponding sense; and we will denote the integration with respect to μ g as X f (s) d g(s).
In a similar way, we will replace μ g by g in other expressions such as "P holds for μ g -a.a. x ∈ X" or "P holds μ g -a.e. in X". Along these lines, it is important to note that the set i.e., the set of points around which g is constant, has null g-measure, as pointed out in [9,Proposition 2.5]. Furthermore, observe that, by definition, C g is open.
The aim of this section is to show that the expression used to compute μ g (2.1) can be simplified, as well as to prove some interesting properties regarding the Lebesgue-Stieltjes measure associated to the sum of a finite family of nondecreasing and left-continuous functions.
We begin by showing that (2.1) can be simplified. Specifically, we will show that the infimum in that expression can be considered over an smaller set, namely, assuming that the families in C are pairwise disjoint. To that end, we introduce the following lemma. Lemma 2.2. Let g : R → R be a nondecreasing and left-continuous function.
Proof. Let U = n∈N [a n , b n ) and C U be the set of all connected components of U . First, note that the set C U is at most countable.
Second, observe that all the elements of C U are connected subsets of R. Thus, we have that they are intervals (including the whole R) or singletons. Nevertheless, observe that an element of C U cannot be a singleton as each point of U belongs to [a n , b n ) for some n 0 ∈ N. Furthermore, we claim that sup I ∈ I for any I ∈ C U bounded from above. Indeed, let I ∈ C U be bounded from above and suppose that sup I ∈ I ⊂ U . In this conditions, there exists n 1 ∈ N, such that sup I ∈ [a n1 , b n1 ). Hence, we have that the set I ∪ [a n1 , b n1 ) is a connected set containing I, which is a contradiction with I ∈ C U . Therefore, C U is, by construction, an at most countable collection of pairwise disjoint sets of the form (a, b), For each I ∈ C U and define F I ⊂ C as follows: Proceeding this way, for each I ∈ C U , we find a countable pairwise disjoint family contained in C, F I , such that I = J∈FI J. Furthermore, for each I ∈ C U , it follows from Remark 2.1 and the fact that μ g is a measure that: Finally, using (2.2) and Remark 2.1, we have that where the last inequality follows from (2.1).
The following result contains a characterization of the outer measure μ * g . The result follows from Lemma 2.2 by considering the relations between the infima involved.
3 not only provides an easier way to compute the outer measure of sets, but it is also a fundamental tool for the proof of the following result which, to the best of our knowledge, is not available in the existing literature on the topic of Lebesgue-Stieltjes measures. Essentially, Proposition 2.4 guarantees that, given a finite family of nondecreasing and left-continuous functions, the measure defined as the sum of the corresponding Lebesgue-Stieltjes measures is, in fact, the Lebesgue-Stieltjes measure associated with the corresponding sum of nondecreasing and left-continuous functions.
Proposition 2.4. Let g 1 , g 2 , . . . , g n : R → R be a family of nondecreasing and left-continuous functions and define g : R → R as Then, for any E ∈ P(R) Proof. We shall only prove the result for n = 2, as the general case can be deduced from this. Let E ∈ P(R). Then, computing the corresponding outer measures as in (2.1), we have that For the reverse inequality, let ε > 0. It follows from (2.3) that there exist , each of them pairwise disjoint, such that: Observe that R is a countable set. Furthermore, given x ∈ E, since R 1 and R 2 are covers of E, there exist n 0 , m 0 ∈ N, such that x ∈ [a 1,n0 , b 1,n0 ) and x ∈ [a 2,m0 , b 2,m0 ), which ensures that x ∈ [a m0 n0 , b m0 n0 ). This guarantees that R is a countable cover of E. Therefore, Hence, it is enough to show that (n,m)∈I to conclude the result, since, in that case, (2.6) yields as ε > 0 was arbitrarily fixed. Let us show that (2.7) holds. For each n ∈ N, define I n = {m ∈ N : (n, m) ∈ I}. Note that I = ∞ n=1 I n , and so (n,m)∈I On the other hand, by definition, Now, (2.7) for i = 1 follows from (2.8). The case i = 2 is analogous and we omit it.
In particular, Proposition 2.4 ensures that every set which is g i -measurable for all i ∈ {1, 2, . . . , n} is also g-measurable and μ g (E) = μ g1 (E) + μ g2 (E) + · · · + μ gn (E) for all E ∈ n i=1 LS gi . The same can be said in regards to the measurability of maps. Furthermore, it follows that if a map f : This fact was implicitly used in [11] in the proof of the Theorem 4.3, but it was never discussed if such property was true. Here, we have shown that this is the case. This information will be fundamental for the work ahead. Furthermore, it is important to note that throughout this paper, the map g will denote the map defined as in (2.4).

The Stieltjes Derivative and the Concept of Continuity
In this section, we gather some information available in [6,9,11] regarding one of the fundamental tools for this paper: the Stieltjes derivative. Furthermore, we also include some information in those papers regarding different concepts of continuity there presented which are also required for the study of differential equations with several Stieltjes derivatives. In particular, we revisit those in [11] as the results there present some limitations. For the aims of this section, as well as the rest of the paper, we shall assume that R n is endowed with the maximum norm, that is We start by introducing the concept of Stieltjes derivator. From now on, we will refer to nondecreasing and left-continuous maps on R as derivators. Given a derivator g, we will denote by D g the set of all discontinuity points of g. Observe that, given that g is nondecreasing, we can write D g = {t ∈ R : , t ∈ R, and g(t + ) denotes the right handside limit of g at t. With this remark, we now have all the information required to introduce the following definition in [9]. Definition 3.1. Let g : R → R be derivator and f : R → R. We define the Stieltjes derivative, or g-derivative, of f at a point t ∈ R\C g as provided that the corresponding limits exist. In that case, we say that f is g-differentiable at t.
Observe that the points of C g are excluded from the definition of g-derivative. This is because the corresponding limit does not make sense in any neighborhood of these points. Nevertheless, as mentioned before, μ g (C g ) = 0, so, in most cases, this has no impact. Furthermore, note that for t ∈ D g , f g (t) exists if and only if f (t + ) exists and, in that case For more information on the Stieltjes derivatives, we refer the reader to [6,9]. Here, we will restrict ourselves to the information strictly necessary for the contents of this paper. Along these lines, we include a reformulation of [9,Theorem 5.4] with [9, Definition 5.1] added to its statement.
The following conditions are equivalent: 2. The function F satisfies the following conditions: Remark 3.4. A particularly interesting case of g-absolutely continuous function can be found in Theorem 2.4 and Proposition 5.2 in [9].
In the work ahead, we will consider systems of differential equations in R n where each component is differentiated with respect to a different derivator. Specifically, we will consider g : R → R n , g = (g 1 , g 2 , . . . , g n ), such that each g i , i ∈ {1, 2, . . . , n}, is a derivator, and we will be looking for solutions on the following set: That is, we will look for g-absolutely continuous functions (see [11,Definition 3.4]), or in other words, functions, such that the ith component is g i -absolutely continuous, i = 1, 2, . . . , n.
Remark 3.5. Observe that for the particular case in which g = (g, g, . . . , g) for some derivator g, we have that AC g ([a, b], R n ) = AC g ([a, b], R n ) in the sense presented in [6]. Note that, in particular, if To see that, it is enough to note that for each i ∈ Throughout this paper, we will use "component-by-component" notation for the derivatives and integrals, so that expressions such as the previous one can simply be reduced to The rest of this section is dedicated to the study of the concept of continuity with respect to a map g : R → R n , g = (g 1 , g 2 , . . . , g n ), such that each g i , i ∈ {1, 2, . . . , n}, is a derivator. Some research on this topic was carried out in [6] in the setting of a unique derivator, and in [11] in the general setting. Unfortunately, some of the results in the latter are incorrect. Here, we show their limitations and amend some of those errors. We start by introducing the following concept that contains [6, Definition 3.1]. Definition 3.6. Let g : R → R and g : R → R n , g = (g 1 , g 2 , . . . , g n ), be such that g, g i , i = 1, 2, . . . , n, are derivators; and consider f : We say that f is g- which, together with [6, Proposition 5.5], ensures that The following result, which can be directly deduced from [6, Proposition 3.2], contains some basic properties regarding g-continuous functions that will be useful in what follows.
Proposition 3.8 allows us to establish some relations between the continuity of a derivator with respect to another one and some of their characteristic sets, as presented in the next result.
On the other hand, if t ∈ R\D g2 , we have that g 2 is continuous at t. In that case, Proposition 3.8 yields that g 1 is continuous at t, and so, t ∈ R\D g1 . Thus, D g1 ⊂ D g2 .
Conversely, assume that C g2 ⊂ C g1 and D g1 ⊂ D g2 and let t ∈ R. Observe that showing that g 1 is g 2 -continuous at t is equivalent to showing that for each ε > 0, there exist δ 1 , δ 2 > 0, such that If A t is not bounded from above, we have that g 2 (s) = g 2 (t) for all s ∈ [t, +∞). In that case, (t, +∞) ⊂ C g2 ⊂ C g1 , which means that g 1 (s) = g 1 (t) for all s ∈ (t, +∞), and so, (3.1) holds for any δ 1 > 0. Otherwise, A t is bounded from above and we can define a t = sup A t ∈ [t, +∞). Observe that, by definition, we have that g 2 (s) = g 2 (t) for all s ∈ [t, a t ) and g 2 (s) > g 2 (t) for all s ∈ (a t , +∞). In particular, we have that (t, a t ) ⊂ C g2 ⊂ C g1 , which implies that g 1 (s) = g 1 (t) for all s ∈ [t, a t ). Now, since g 1 and g 2 are leftcontinuous, it follows that g 1 (s) = g 1 (t) and a t ], and so (3.1) follows (even when a t = t, as it becomes trivial). Otherwise, we have that a t ∈ D g1 which means that g 1 is continuous from the right at a t , so there exists δ > 0, such that Observe that it is enough to show that there exists δ 1 > 0, such that Finally, for (3.2), we proceed in a similar manner.
If B t is not bounded from below, an analogous reasoning to the one for A t shows that (3.2) holds trivially. Otherwise, B t is bounded from below and we can , t), and so, (3.1) follows (even when b t = t, as it becomes vacuous). Otherwise, b t ∈ D g2 which implies that b t ∈ D g1 . Observe that this ensures that (3.5) Furthermore, since g 1 is left-continuous at b t , there exists δ > 0, such that Now, an analogous reasoning to the one for (3.4) shows that, given that (3.5) holds, there exists δ 2 > 0 such that 0 < b t − s < δ for all s < b t satisfying that 0 < g 2 (b t ) − g 2 (s) < δ 2 , which is enough to conclude that (3.2) holds in a similar fashion to (3.1).
Remark 3.10. Condition (3.1) can be interpreted as "being g 2 -continuous at t from the right" and, similarly, (3.2) as "being g 2 -continuous at t from the left".
As pointed out in [6], the concept of g-continuity can be understood as continuity in the topological sense. Specifically, a map f : τ u ) is continuous, where τ u is the usual topology of R and τ g is the topology generated by the sets With this characterization, observe that g 1 is g 2 -continuous if and only if τ g1 ⊂ τ g2 . Indeed, first observe that g 1 is trivially g 1 -continuous. This means that g −1 1 (U ) ∈ τ g1 for any U ∈ τ u . Hence, if τ g1 ⊂ τ g2 , we have that g −1 1 (U ) ∈ τ g2 for any U ∈ τ u , or equivalently, g 1 is g 2 -continuous. Conversely, if g 1 is g 2continuous, and given that the corresponding topologies are generated by the sets in (3.6), it is enough to show that for any t ∈ R, r > 0 and s ∈ B g1 (t, r), there exists r s > 0, such that B g2 (s, r s ) ⊂ B g1 (t, r). Given t ∈ R, r > 0 and  r) and, since g 1 is g 2 -continuous, there exists r s > 0, such that , r), which finishes the proof of the equivalence. Combining this with Proposition 3.9, we obtain the following result.
Corollary 3.11. Let g 1 , g 2 : R → R be derivators. Then, the following statements are equivalent: As we will see later, this result will allow us to relate the concept of g-continuity with a similar type of continuity in [11]. Furthermore, Corollary 3.11 implies that the topologies of derivators can be classified in terms of the sets C g and D g as the following theorem shows. Note that 0, t \C is a closed set, thus Lebesgue-measurable. Hence, g C,D is well defined. Furthermore, g C,D is left-continuous and nondecreasing, D g = D and C g = C, which shows that Φ is surjective.  As mentioned before, in [11, Definition 3.1], we find a definition of continuity with respect to g alternative to the one provided in Definition 3.6. For completeness, we include [11, Definition 3.1] before comparing the two concepts.
Definition 3.14. Let g : R → R n , g = (g 1 , g 2 , . . . , g n ), be such that each If it is g-continuous at every point t ∈ A, we say that f is g-continuous on A.
In [11], the authors claimed that Definitions 3.6 and 3.14 are equivalent. Nevertheless, a careful reader might notice that the proof only shows that g-continuity implies g-continuity. Furthermore, the reverse implication is not true, as shown in the next example.
Example 3.15. Consider g, f : R → R 2 , g = (g 1 , g 2 ), f = (f 1 , f 2 ), defined as Note that f cannot be g-continuous as f 1 is not constant, see Proposition 3.8. However, f is g-continuous. Indeed, first, note that g(t) − g(s) = |g 2 (t) − g 2 (s)|, s, t ∈ R. Thus, showing that f is g-continuous is equivalent to showing that f is g 2 -continuous. Since we are considering the max-norm in R n , it suffices to show that f 1 and f 2 are g 2 -continuous. Now, f 1 is trivially g 2continuous as f 1 = g 2 , and [6, Example 3.3] shows that f 2 is g 2 -continuous.
It is important to note that the misinformation about the relations between Definitions 3.6 and 3.14 does not affect the existence and uniqueness results in [11]. However, it has some consequences when it comes to the study of solutions in the classical sense. Specifically, this affects Proposition 4.6 and Theorem 4.8 in [11]. In the next section, we discuss the implications of these facts. Nevertheless, in some contexts, those results remain true as a consequence of the following result. Let g : R → R n , g = (g 1 , g 2 , . . . , g n ), be such that each g i , i ∈ {1, 2, . . . , n}, is a derivator. Then, the following are equivalent: (i) Every g-continuous map is g-continuous.
The rest of the result is a consequence of Corollary 3.11.
Interestingly enough, it is possible to establish some other connections between the continuity in the sense of Definition 3.14 and [6, Definition 3.1] for an adequate choice of a derivator. Proposition 3.17. Let g : R → R n , g = (g 1 , g 2 , . . . , g n ), be such that each g i , i ∈ {1, 2, . . . , n}, is a derivator.

Indeed, it follows from the triangular inequality that | g(t) − g(s)| ≤ n g(t) − g(s) for any s ∈ A.
For the other inequality, we consider two cases. If t ≥ s, since g and each g i are nondecreasing, we have g(s) . On the other hand, if t < s, we proceed analogously to (3.9), interchanging the roles of t and s. Hence, (3.8) holds. Now, the equivalence between the two types of continuity follows.
Proposition 3.17 not only provides a simple condition to check if a map is g-continuous, but we can also deduce some interesting properties for this type of maps through the results in [6]. In particular, [6, Corollary 3.5] yields the following result that is fundamental for the uniqueness results in the following section. Remark 3.19. Since every Borel set is Lebesgue-Stieltjes measurable, it follows that every Borel measurable map is Lebesgue-Stieltjes measurable. In particular, we have that if A is a Borel set and f : A ⊂ R → R n is gcontinuous on A, then f is g j -measurable for all j ∈ {1, 2, . . . , n}.
Another type of continuity defined in terms of g that was introduced in [11] is what the authors called ( g ×Id)-continuity, which is a generalization of [6,Definition 7.7]. These concepts of continuity were introduced for the study of classical solutions in both papers. As mentioned earlier, the results in [11] are partially incorrect and we aim to correct them in this paper. To that end, we introduce the following definition, which is an alternative generalization of [6,Definition 7.7]. Definition 3.20. Let g : R → R and g : R → R n , g = (g 1 , g 2 , . . . , g n ), be such that g, g i , i = 1, 2, . . . , n, are derivators; and consider f : f 2 , . . . , f n ). We say that f is (g ×Id)-continuous at (t, x) ∈ A×B if for every ε > 0, there exists δ > 0, such that f (s, y) − f (t, x) < ε for all (s, y) ∈ A × B such that |g(s) − g(t)| < δ and y − x < δ.
We say that f is (g × Id)-continuous On the other hand, the corresponding definition in [11] reads as follows.
We say that f is ( The relations between Definitions 3.20 and 3.21 are analogous to the ones between Definitions 3.6 and 3.14. In particular, we have the following result. Proof . Fix ε > 0. Given i ∈ {1, 2, . . . , n}, we have that f i is (g i × Id)continuous at (t, x), so there exists δ i > 0, such that Once again, the reverse implication does not hold. To see that this is the case, it is enough to consider g and f as in (3.7) and F : R × R 2 → R 2 defined as F (t, (x, y)) = f (t), and note that F does not depend on (x, y), which implies that (g × Id) and ( g × Id)-continuity reduce to g and g-continuity. Furthermore, using a similar reasoning, we can obtain analogous results to Propositions 3.16 and 3.17 in the context of Definitions 3.20 and 3.21.
Interestingly enough, within the proof of [11,Theorem 4.8], the authors proved correctly the following superposition result involving Definitions 3.14 and 3.21. Remark 3.24. Lemma 3.23 guarantees that the composition of a g-continuous map with a (g × Id)-continuous one is g-continuous. Nevertheless, we cannot assure that the composition is g-continuous. Indeed, consider g : R → R 2 , g = (g 1 , g 2 ), be such that g 1 , g 2 are derivators satisfying Δg 1 (t 0 ) = 0 and Δg 2 (t 0 ) > 0 for some t 0 ∈ R. Let I be a neighborhood of t 0 and consider the maps x : It is clear that x is g-continuous at t 0 . Furthermore, observe that This implies that the map f (·, x(·)) is not g-continuous at t 0 , as f 1 (·, x(·)) is not g 1 -continuous at t 0 , see Proposition 3.8. However, the map f is (g × Id)-continuous at (t 0 , (g 1 (t 0 ), g 2 (t 0 ))). Indeed, we shall only show that f 1 is (g 1 × Id)-continuous at t 0 as the proof for f 2 being (g 2 × Id)-continuous is analogous. Let ε > 0 and take 0 < δ < ε/2. Denote u 0 = (g 1 (t 0 ), g 2 (t 0 )).

The Initial Value Problem with Several Derivativors
We now turn our attention to the study of initial value problems in the context of g-differential equations. That is, given g : R → R n , g = (g 1 , g 2 , . . . , g n ), such that each g i , i ∈ {1, 2, . . . , n}, is a derivator, we will study problems of the form . . , f n ). To that end, we introduce the concept of solution that is fundamental for the aims of this section. In what follows we denote by I σ = [t 0 , t 0 + σ), σ ∈ (0, T ], and I = [t 0 , t 0 + T ) and by I σ and I the corresponding closure sets with respect to the usual topology in R.

Definition 4.1.
A solution of (4.1) on an interval I σ , σ ∈ (0, T ], is a function If σ = T , we say that x is a global solution of (4.1); otherwise, i.e., if σ ∈ (0, T ), we say that x is a local solution of (4.1).
Before studying some existence and uniqueness results for (4.1), we will reflect on the concept of classical solution in [11]. When we talk about solutions in the classical sense, we mean solutions of (4.1) that solve the problem on their whole interval of definition and have continuous derivatives. Of course, given the nature of Definition 3.1, this is impossible unless C gi = ∅, i = 1, 2, . . . , n. Nevertheless, we can talk about "everywhere" solutions referring to solutions solving the problem on the biggest set possible, i.e., excluding the corresponding set C gi . With this idea in mind, we start exploring some basic properties of "everywhere" solutions that will culminate in Theorem 4.6. The first results concern the continuity of the derivative. x(t)), for all t ∈ I σ \C gi , i = 1, 2, . . . , n. Then, (x i ) gi is g-continuous on I σ \C gi .
Proof. Given that x is g-absolutely continuous on I σ , we have that x is gcontinuous on I σ ,which implies that it is g-continuous on I σ .Thus Lemma 3.23 yields that f (·, x(·)) is g-continuous. Hence, it follows that, for each i ∈ {1, 2, . . . , n}, the map f i (·, x(·)) is g-continuous and, since (x i ) gi is defined on I σ \C gi , the result follows. It follows from Proposition 3.17 that, under the hypotheses of Proposition 4.2, (x i ) gi is g-continuous on I σ \C gi . Furthermore, notice that if we replace the concept of g-continuity for g-continuity in the hypotheses of Proposition 4.2, we still obtain g-continuous solutions. Nevertheless, with a similar reasoning to the one used for Proposition 4.2, we can obtain the following result ensuring g-continuity.
Propositions 4.2 and 4.3 show that, under suitable conditions, the solutions of (4.1) have continuous derivatives in some sense. In particular, it is required that condition (4.3) is satisfied. Nevertheless, solutions in the sense of Definition 4.1 need not satisfy such condition. In the following results, we provide some conditions ensuring that (4.3) or similar conditions are satisfied.
Proof. Fix i ∈ {1, 2, . . . , n}. Since x is a solution of (4.1), we have that x(t)) for all t ∈ D gi , so it is enough to show that the corresponding equalities hold for all t ∈ I σ \(C gi ∪ D g ) for (i), and for all t ∈ I σ \(C gi ∪ D gi ) for (ii). We will first prove (ii) and then, by making small modifications to that proof, we will obtain (i). Fix t ∈ I σ \(C gi ∪ D gi ). Since g i is not constant on any neighborhood of t, we may have g i (s) < g i (t) for all s < t, g i (s) > g i (t) for all s > t, or both. If g i (s) < g i (t) for all s < t and t > t 0 , then, for all s ∈ [t 0 , t), the Fundamental Theorem of Calculus yields x(r)). On the other hand, if g i (s) > g i (t) for all s > t, then, following an analogous reasoning, we deduce that Now, for (ii), assume that f i (·, x(·)) is g i -continuous on I σ . In that case, since g i is continuous at t, we have that f i (·, x(·)) is continuous at t. Therefore, if t is such that g i (s) < g i (t) for all s < t, (4.4) implies that the following limit exists and: If g i (s) = g i (t) on some [t, t + δ], δ > 0, then the limit in (4.6) is (x i ) gi (t) and the proof is complete. Similarly, if t is such that g i (s) > g i (t) for all s > t, the continuity of f i (·, x(·)) at t and (4.5) ensure that the following limit exists and: This covers all of the remaining cases, so the proof is finished for f i (·, x(·)) g i -continuous, and subsequently, for f (·, x(·)) g-continuous.
On the other hand, if t ∈ I σ \(C gi ∪D g ), we have that t ∈ I σ \(C gi ∪D gi ). Hence, (4.4) holds if t is such that g i (s) < g i (t) for all s < t, and (4.5), if g i (s) > g i (t) for all s > t. Now, under the hypotheses of (i), we have that f (·, x(·)) is g-continuous, so it is g-continuous at t. As a consequence, f (·, x(·)) is continuous at t as g is continuous at that point. Hence, by making analogous reasonings, we can obtain (4.6) and (4.7) to finish the proof.
Essentially, the proof of Proposition 4.4 is a revision of the proofs of [6,Proposition 7.6] and [11,Proposition 4.6] in the context of (4.1). Of course, in the latter, the authors worked on the same framework as in this paper. Nevertheless, [11,Proposition 4.6] is not correct due to Definitions 3.6 and 3.20 not being equivalent to Definitions 3.14 and 3.21, respectively. The following result serves as a proper reformulation of [11,Proposition 4.6] based on Proposition 4.4. then Hence, for each i ∈ {1, 2, . . . , n}, such that (4.8) holds, we have that (x i ) gi (t) = f i (t, x(t)), t ∈ I σ \C gi . Now, Proposition 4.2 ensures that (x i ) gi is g-continuous in I σ \C gi , which finishes the proof.
Observe that (4.8) becomes vacuous when D gj = D g k for all j, k ∈ {1, 2, . . . , n} which, as stated by Proposition 3.16, is guaranteed to happen when Definitions 3.6 and 3.14 are equivalent. This justifies the statement of [11,Proposition 4.6], since the authors wrongly used both definitions equivalently. This, of course, affected [11,Theorem 4.8], where the authors guaranteed the existence of a local solution (and everything that follows from [11,Proposition 4.6]) under the assumption of an extra hypothesis: for each i ∈ {1, 2, . . . , n}, there exists h i ∈ L 1 gi (I, [0, +∞)), such that It is possible to obtain a correct formulation of [11,Theorem 4.8] by noting that the ( g × Id)-continuity of f together with condition (4.10) are enough to obtain the existence of a local solution through [11,Theorem 4.5]. After that, all that is left to do is to consider Theorem 4.5 to obtain the right version of [11,Theorem 4.8].
As a final note, we obtain an analogous result to Theorem 4.5 yielding g-continuity instead of g-continuity. This result follows from Proposition 4.3 combined with statement (ii) in Proposition 4.4. Theorem 4.6. Let x = (x 1 , x 2 , . . . , x n ) be a solution of (4.1) on I σ , σ ∈ (0, T ]. If f i (·, x(·)) is g i -continuous on I σ for some i ∈ {1, 2, . . . , n}, then and (x i ) gi is g i -continuous on I σ \C gi . In particular, if f (·, x(·)) is g-continuous on I σ , then (4.3) holds and (x i ) gi is g i -continuous on I σ \C gi for all i ∈ {1, 2, . . . , n}.

Uniqueness of Solution
We now continue the work in [11] by researching some uniqueness conditions for (4.1). In particular, we shall adapt the results in [15] to the context of differential equations with several Stieltjes derivatives. For this endeavour, as well as for the question of existence of solution, we can assume that g is continuous at t 0 , as pointed out in [11]. To see that this is the case, it is enough to use an analogous reasoning to that in [6,Section 5].
Our first uniqueness criterion for (4.1) is analogous to [15,Theorem 3.9] and it is inspired by the ideas of [24,Theorem 4.8]. To obtain the mentioned result we need the following reformulation of [24,Theorem 1.40] and where Ω −1 : (α, β) → R is the inverse function of Ω.
We are now able to state and prove the following uniqueness result under an Osgood type condition.
then (4.1) has at most one solution on I σ .
Proof. Let x, y be solutions of (4.1) on I σ . Define ψ : I σ → [0, +∞), Ω : (0, +∞) → (0, +∞) as First, note that, given that x, y ∈ AC g (I σ , R n ), we have that each component of x − y is Borel measurable (see Remark 3.19) and, since ψ is the pointwise maximum of Borel measurable maps, we have that ψ is Borel measurable. Now, given that ω is continuous, it follows that ω • ψ is Borel measurable, which guarantees that it is g and g i -measurable, i = 1, 2, . . . , n. Moreover, Remark 3.7 ensures that x − y is bounded, which implies that so is ψ, yielding that ω • ψ is bounded as well. Hence, it follows that ω • ψ is integrable with respect to g and g i , i = 1, 2, . . . , n.
Let K > 0 be an upper bound of ω • ψ. Without loss of generality, we assume that g is continuous, which ensures that g is continuous at t 0 . Then, for each γ ∈ (0, σ), where ε(γ) = Kμ g ([t 0 , t 0 + γ)) + γ > 0. Noting that ε and ω are in the same circumstances as ε and ω in the proof of [15,Theorem 3.9] with g in place of g, we can repeat the same arguments there to see that there exists 0 < R < γ, such that By definition, we have that, for each t ∈ I σ , there is j t ∈ {1, 2, . . . , n}, such that ψ(t) = |x jt (t) − y jt (t)|. Therefore, Theorem 3.3 yields that, for each t ∈ I σ , for all δ ∈ (0, R). Therefore, the assumptions of Lemma 4.7 are satisfied, which guarantees that Applying Ω on both sides of the inequality, we obtain Suppose ψ = 0 on I σ . In that case, there must exist t * ∈ I σ , such that ψ(t * ) > 0. Then, for all δ ∈ (0, R), such that δ < t * − t 0 , , and, by taking the limit as δ → 0 + , we obtain which contradicts (ii). Hence, we must have ψ = 0 on I σ , i.e., x = y on that interval.
Observe that Theorem 4.8 returns [15,Theorem 3.9] in the corresponding setting (namely, when g = (g, g, . . . , g) for some derivator g) as, for any u ∈ (0, +∞) Furthermore, it is possible to obtain the following more general result, which is the analog of the Montel-Tonelli uniqueness result in [15] in the setting of For more information on this integral, we refer the reader to [18].  ( x − y ), g i -a.a. t ∈ I σ , x,y ∈ X, (4.14) then (4.1) has at most one solution on I σ .
Therefore, the result can be proved by reasoning as in the proof of Theorem 4.8 with the appropriate adjustments, i.e., replacing g by g and ε(·) by ε(·) accordingly.

Existence and Uniqueness of Solution
As a final step on the study of (4.1), we combine the uniqueness results in the previous section with some information available in the literature regarding the existence of solution. Specifically, we will use [11,Theorem 4.5], which requires the following definition, a slightly more general version of [6,Definition 4.7].
Definition 4.10. Let g : R → R be a nondecreasing and left-continuous function, J ⊂ R and X ⊂ R m , X = ∅. A function f : J × X → R n is said to be g-Carathéodory if the following properties are satisfied: (i) For each x ∈ X, the map f (·, x) is g-measurable.
(ii) The map f (t, ·) is continuous for g-a.a. t ∈ J.
Naturally, combining the hypotheses of Theorems 4.9 and 4.13, we can obtain an Montel-Osgood-Tonelli type existence and uniqueness result of local solutions of (4.1). This result is a generalization of [15,Theorem 4.3]. To see that this is the case, it is enough to bear in mind expression (4.13).