Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations

. In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.


Introduction
The important role played by the theory of initial value impulsive differential problems in describing the evolution of many processes in the real life is well-known [1,15,27]. The most encountered framework in literature is that of impulsive equations with impulses occurring at fixed times [1,5].
The more general setting of state-dependent time discrete perturbations is (despite its wide applicability, e.g. [6,12,24]) far less studied, due to its complexity -see [2,4,10] or [25] and the references therein. To give just an idea, fixed point results are not applicable since the continuity of Nemytskii operator cannot be checked, while the control of the number and position of the state-dependent impulse moments requires strong specific assumptions.
At the same time, the theory of differential equations with Stieltjes derivative -see [19] (called Stieltjes differential equations, e.g. [11,17]), which has been shown to be generally equivalent to the theory of measure differential equations (see [8,9,21]) covers a wide variety Email: bisatco@usm.ro of real life processes. For instance, it allows the occurrence of stationary intervals (where the derivator g is constant) coupled with moments with abrupt changes in the state (where g has discontinuities).
We have in mind the possibility to allow both behaviours: stationary intervals coupled with pre-established moments with abrupt changes and also with state-dependent time impulses.
We thus focus on Stieltjes first-order Cauchy differential problem with impulses depending on the state where g : [0, 1] → R is a left-continuous nondecreasing function which induces the Stieltjes measure µ g , B ⊂ R is a closed set containing x 0 , f : [0, 1] × B → R is the function describing the rate of change of the unknown function, while I i : R → R, i = 1, . . . , k give the jumps at the points where the barriers γ i : R → [0, 1], i = 1, . . . , k are reached.
By A, A i x and A x (x being a real valued function on [0, 1]) one denotes the sets A = the set of points of discontinuity of g, A i x = {t ∈ [0, 1] : t = γ i (x(t))} for every i = 1, . . . , k, respectively To avoid ambiguity at the common points of A i x and A j x (with i ̸ = j), respectively of A i x and A, we impose the conditions H4).iii), respectively H4).iv) below.
Using of the Stieltjes derivative x ′ g with respect to a left-continuous nondecreasing map g enables the presence of dead times (intervals where the process is stationary -corresponding to intervals where g is constant) as well as of fixed-time discrete perturbations (at the discontinuities of g).
In the particular case where g(t) = t for every t ∈ [0, 1], the existence of solutions for this problem has been provided e.g. in [2,10,13] or [25]. However, even in this specific case, basic properties of the set of solutions are difficult to be proved (we refer to [13] or [33] for a detailed discussion).
The very wide framework of Stieltjes differential problems (which already covers many classical cases, such as ordinary differential and difference equations, impulsive equations, time-scales dynamic equations) with state-dependent discontinuities is studied here for the first time, as far as the author knows.
More precisely, we first present an existence result inspired by [22] (available for measure differential equations without allowing state dependent discontinuities, in particular for impulsive problems with fixed time impulse moments) by taking the advantage of the method used in [10] for state-dependent impulsive equations with g(t) = t.
Finally, we prove, using a nice result for measure differential problems without variable time impulses in [22], that a least and a greatest solution can be found. Note that, by a different method and different hypotheses, the existence of extremal solutions has been obtained in [13] when g(t) = t under assumptions involving that each barrier is hit only once.

Notions and preliminary facts
A function u : [0, 1] → R is said to be regulated if for every t ∈ [0, 1) there exists the limit u(t+) and for every s ∈ (0, 1] there exists the limit u(s−). The set of discontinuity points of a regulated function is at most countable and the bounded variation or continuous functions are, without any doubt, regulated. The space G([0, 1], R) of regulated functions u : [0, 1] → R is a Banach space with respect to the sup-norm. By G − ([0, 1], R) we denote its subspace consisting in left-continuous functions.
Given a left-continuous nondecreasing function g : [0, 1] → R, the measurability with respect to (in short, w.r.t.) the σ-algebra defined by g will be called g-measurability, µ g denotes the Stieltjes measure generated by g and the Lebesgue-Stieltjes (shortly, LS-) integrability w.r.t. g means the abstract Lebesgue integrability w.r.t. the Stieltjes measure µ g . It is well known that if f is LS-integrable w.r.t. g, the primitive · 0 f (s)dg(s) = [0,·) f (s)dg(s) is a g-absolutely continuous function in the following sense (see [31], [11] or [19]): a function u : We shall also use the theory of Kurzweil-Stieltjes integral (we refer the reader to [14,23,30], see also [28,29]) motivated by the fact that it is easy to handle (by integral sums), it fits well with the setting of regulated functions (i.e. it covers the situation where both the integrand and the integrator possess discontinuities) and, moreover, it can integrate functions that are not absolutely integrable.
Below are listed the basic properties of KS-integrals.
Therefore, h is left-continuous, respectively right-continuous at the points where g has the same property.
Note that the Lebesgue-Stieltjes integrability of a function f implies the Kurzweil-Stieltjes integrability and in the framework of a left-continuous nondecreasing function g, as a consequence of [23, Theorem 6.11.3] (see also [26,Theorem 8 In order to recall more properties of the primitive, we need the notion of (Stieltjes) derivative of a function with respect to another function, given in [19] (see also [31]).
if the limit exists.
The g-derivative has found meaningful applications in solving real-world problems where periods of time where no activity occurs and instants with abrupt changes are both involved, such as [11], [18] or [20].
Remark that if t is a discontinuity point of g, then There is a set where Definition 2.3 does not work, more precisely, but we must take into account that µ g (C g ) = 0 [19] and, when studying differential equations, the equation has to be satisfied µ g -almost everywhere. The connection between Stieltjes integrals and the Stieltjes derivative is given by Fundamental Theorems of Calculus [19, Theorems 5.4, 6.2, 6.5].
For Lebesgue-Stieltjes integrals, it is contained in [19,Theorem 5.4], we give the entire statement below. Theorem 2.4. Let g : [0, 1] → R be a nondecreasing left-continuous function. Then f : [0, 1] → R is g-absolutely continuous if and only if it is g-differentiable µ g -a.e., f ′ g is Lebesgue-Stieltjes integrable w.r.t g and

Main results
We are concerned with the Stieltjes initial value differential problem with state-dependent discontinuities and A x is the union of these A i x .

Existence result
it is a solution of the impulsive integral equation ii) (e.g. [10]) We say that a function x • it is g-absolutely continuous and x is left-continuous, it has finite right limit and Consider B ⊂ R a compact set. We shall impose the following hypotheses on f : Remark 3.2. Using [22, Lemma 3.1], from the preceding assumptions it follows that for each The assumptions on the barriers γ i : R → [0, 1] (known as transversality assumptions) and on the jumps I i : R → R, i = 1, . . . , k are described below:

H4)
i) The maps γ i , i = 1, . . . , k are strictly monotone and continuous; H5) There is a positive integer M such that each integral solution of (1.1) on any subinterval of [0, 1] hits the barriers at at most M points.
We make the convention that, whenever a solution hits the intersection of two barriers, the moment is counted only once.

Remark 3.3. The last part of Condition H4) means that for every
and some i ∈ {1, . . . , k}, Condition H5) is presented in a very general form, but we stress that it is ensured by the hypotheses imposed in other works on state-dependent impulsive differential problems when g(t) = t.
For instance, in [10] it is assumed that the distance between any two consecutive points where a solution hits the barriers is bigger than some constant, see (3.4) in Theorem 3.1. Also, in [2] there are a fixed number of barriers which are hit at most once by any solution, while in [25] there is only one barrier hit exactly once by any solution, see [25,Lemma 5.1].
By combining the hypotheses imposed for integral measure driven equations in [22] with the method used in the framework of state dependent impulsive equations in [10], we can prove an existence result for the state dependent impulsive Stieltjes differential problem (1.1): Theorem 3.4. Let f : [0, 1] × B → R satisfy the hypotheses H1)-H3) and the barriers and jumps satisfy H4), H5). Suppose that where Then the problem (1.1) admits integral solutions on [0, 1].
Let us next look at the measure-driven Cauchy problem It is not difficult to see that as for each x in this set, as before, We can thus apply [22,Theorem 3.2] and one can continue the process and we claim that it will be finished after less than M + 1 steps (otherwise, hypothesis H5) would be contradicted).
Under stronger assumptions on f and keeping the hypothesis on the barriers, one can obtain the existence of g-Carathéodory solutions for the impulsive measure differential problem (1.1).
Proof. We follow the same lines as in the previous result. Consider first the measure-driven Cauchy problem By the Peano existence result [11,Theorem 7.5], one can find a g-Carathéodory solution x 1 on [0, 1].
Define then r i,1 : [0, 1] → R as before and we can fall into one of the following situations: • if r i,1 (t) ̸ = 0 for all i = 1, . . . , k and t ∈ (0, 1] \ A, then x 1 is a g-Carathéodory solution of (1.1) on [0, 1]; • if r i,1 (t) = 0 for some i ∈ {1, . . . , k} and t ∈ (0, 1] \ A, then let t 1 ∈ (0, 1] \ A be chosen such that r i 1 ,1 (t 1 ) = 0 for some i 1 and r i, Consider then the measure-driven Cauchy problem We can again apply [11,Theorem 7.5] in order to get a g-Carathéodory solution on [t 1 , 1] and so on. Remark 3.6. We could have obtained the previous result by applying Theorem 3.4 and remarking that the assumptions H1 ′ ), H2 ′ ) together with the Fundamental Theorem of Calculus imply that any integral solution of our problem is a g-Carathéodory solution.

Existence of extremal solutions
Using the existence of extremal solutions for measure differential equations ( [22,Theorem 4.4]), we get the existence of extremal solutions for measure differential equations with statedependent impulses.
We need several additional assumptions.
H6) One of the following sets of conditions holds:
Let us next look at the problem: for each i ∈ {1, . . . , k}.
The hypothesis on B implies that, by [22,Theorem 4.4], we can find, for each of these problems, a least solution on [t 2 , 1], denoted by y 3 and one can continue the process, which will be finished after less than M + 1 steps (otherwise, hypothesis H5) would be contradicted).
Let us see that the solution constructed in this way, namely y − , is a least solution of (1.1) on [0, 1]. Suppose that H6).a) is satisfied (the case b) can be analyzed in a similar way).
Let x be an arbitrary solution of (1.1) on [0, 1]. We first show that ii) If there are points in (A x \ A) ∩ [0, t 1 ), let us focus on the first one since their number is finite and for all such points the discussion can be led in the same way; let τ 1 ∈ (A x \ A) ∩ [0, t 1 ) be the first point where x hits some barrier γ i 0 . Then the following situations can be encountered: If in (3.3) one has equality, then the proof on [0, t 1 ] has to be repeated in order to get the assertion on [t 1 , t 2 ]. If strict inequality holds, then some modifications are necessary but we take the same steps as on [0, t 1 ] in order to prove that y − (t) ≤ x(t) for every t ∈ [t 1 , t 2 ]. Thus: We could be in the following situations: i).a) [t 1 , t] ∩ A = ∅, in which case x and y − are continuous on (t 1 , t], y − (t 1 +) < x(t 1 +) is valid and so there is a point t in this interval where the two trajectories intersect; then the solution defined by would contradict the definition of y − on (t 1 , t 2 ] as being the least solution of the measure integral equation. .a), the fact that y − is a least solution of the measure integral equation would be disobeyed.
So, by H7), y − ( t k +) ≤ x( t k +) and, as y − (t) > x(t), as in i).a), the fact that y − is the least solution of the measure integral equation on [t 1 , t 2 ) is contradicted.
• this is a countable set { t i , i ∈ N} accumulating towards t < t; then, as before, at each such point y − ( t i ) ≤ x( t i ) which, taking into account the left continuity of x, y − , imply and thus y − (t+) ≤ x(t+).
Again it would follow that y − is not a least solution on (t, t].
• this is a countable set { t i , i ∈ N} accumulating towards t, in which case, as before, , let us focus only on the first one where x hits some barrier γ i 0 . Then the following situations can be encountered: ii.a) none of the discontinuity points of g lies in between t 1 and τ 1 ; in this case, let us put together (3.2) and the fact that y − (τ 1 ) ≤ x(τ 1 ) = γ −1 i 0 (τ 1 ) (since otherwise, together with (3.3) and the continuity of y − and x it would be contradicted, as before, the choice of y − as the least solution of the measure integral equation on [t 1 , t 2 )). By the continuity of γ −1 i 0 and y − on (t 1 , τ 1 ), it would then follow that y − hits the barrier γ i 0 on (t 1 , τ 1 ], contradiction with the choice of t 2 . ii.b) if there are discontinuity points of g lying in between t 1 and τ 1 , we can fall again into one of the three cases below: • this is a finite subset of A, { t i , i = 1, . . . , k}; then for each i, x( t i ), y − ( t i ) > γ −1 i 0 ( t i ) since otherwise the graphs of x, y − would hit the barrier γ i 0 on (t 1 , τ 1 ) and this is not possible. nights, while during the days the evaporation speed is maximum at middays) states that the evolution of x can be described by the Stieltjes differential problem Note that in [11] the map f is supposed to be linear in x, but the nonlinear framework is realistic as well. The nondecreasing left-continuous function g can be chosen conveniently [11, page 20], for instance if we want to refill the tank every morning with an amount of water depending on to the level before refilling, then one may set g(t) = t 0 max (sin(πs), 0) ds + max{k ∈ N : 2k ≤ t} and f (2k, x(2k)) = ∆ + x(2k) = λ k x(2k), λ k > 0 (the intervals [2k, 2k + 1), k ∈ N correspond to day times and, obviously, the intervals [2k + 1, 2k + 2), k ∈ N to night times).
In other words, the moments 2k, k ∈ N are fixed-time impulsive moments with ∆ + g(2k) = 1, ∀k and so far, the problem can be solved through the theory of Stieltjes differential equations.
Suppose now that we want to add an amount of water (equal to I(x(t))) whenever a state-dependent condition is satisfied, such as x(t) = β(t), where β is a decreasing function measuring the water level in a huge second tank where the level water decreases due to evaporation, without adding or removing any quantity and without stationary intervals.
In this case, the theory in [11] cannot be applied due to the occurrence of state-dependent impulses. At the same time, nor the studies developed for state-dependent impulsive problems ([10], [2] or [25]) apply since the involved derivative is the Stieltjes derivative (not the usual derivative).