Applications of Stieltjes Derivatives to Periodic Boundary Value Inclusions

In the present paper, we are interested in studying first-order Stieltjes differential inclusions with periodic boundary conditions. Relying on recent results obtained by the authors in the single-valued case, the existence of regulated solutions is obtained via the multivalued Bohnenblust–Karlin fixed-point theorem and a result concerning the dependence on the data of the solution set is provided.


Introduction
Allowing the study in a unique framework of many classical problems: ordinary differential or difference equations (in the case of an absolutely continuous measure-with respect to the Lebesgue measure-respectively of a discrete measure), impulsive differential problems (for a sum of Lebesgue measure with a discrete one), dynamic equations on time scales (see [1]) and generalized differential equations (e.g., [2,3]), it is clear why the theory of differential equations driven by measures has seen a significant growth (e.g., [1,4]).
Using a natural notion of Stieltjes derivative with respect to a non-decreasing function (c.f. [5], see also [6] or [7,8] for applications), measure-driven differential equations can be expressed, in an equivalent form, as a Stieltjes differential equation.
Based on the results obtained in [4] for measure-driven differential equations with periodic boundary conditions, in the present paper we focus on nonlinear differential inclusions of the form: u g (t) + b(t)u(t) ∈ F(t, u(t)), t ∈ [0, T] u(0) = u(T) (1) where u g denotes the Stieltjes derivative of the state u with respect to a left-continuous non-decreasing function g : [0, T] → R. This form is preferred since in many real-world problems the linear, respectively the nonlinear term has different practical meanings.
In the particular case of the identical function g, periodic differential problems have been widely considered in the literature; to mention only a few works, we refer to [16][17][18] for the single-valued setting and to [19,20] (without impulses) or [21,22] (allowing impulses) in the set-valued framework.
It is not difficult to check that: for some regulated function χ : [0, T] → R.
In the whole paper, g : [0, T] → R will be a non-decreasing left-continuous function and µ g the Stieltjes measure defined by g. Without any loss of generality, suppose g(0) = 0. We deal with the Kurzweil-Stieltjes integral; we recall below the basic facts concerning this integral. [2,3,27,28] or [29]) One says that f : [0, T] → R d is Kurzweil-Stieltjes integrable (or KS-integrable) with respect to g : [0, T] → R if there is T 0 f (s)dg(s) ∈ R d with the property that for every ε > 0, one can find δ ε : [0, T] → R + satisfying
In general, the Lebesgue-Stieltjes integrability with respect to g (i.e., the abstract Lebesgue integrability with respect to the Stieltjes measure µ g ) yields the Kurzweil-Stieltjes integrability with respect to g. When g is left-continuous and non-decreasing, by ( [28], Theorem 6.11.3) (or ( [27], (Ref [29], Proposition 2.3.16) asserts that the KS-primitive F : Consequently, if g is continuous at some point, then F is also continuous.
To recall more properties of the primitive, we need a notion of (Stieltjes) derivative of a function with respect to another function, given in [5] (see also [33]).

Definition 2.
Let g : [0, T] → R be non-decreasing and left-continuous. The derivative of f : [0, T] → R d with respect to g (or the g-derivative) at the point t ∈ [0, T] is if g is continuous at t, if the limit exists.
The g-derivative has found interesting applications in solving real-world problems where periods of time where no activity occurs and instants with abrupt changes are both present, such as [7] or [8].
Define the following set: namely the collection of atoms of µ g ; remark that if t ∈ D g , then There is a set where Definition 2 has no meaning, more precisely, It is convenient, when working with the g-derivative, to also disregard the points of the set where C g = n∈N (u n , v n ) is a pairwise disjoint decomposition of C g (such a writing is possible due to the fact that C g is open in the usual topology of the real line, see [5]).
To warrant this, take into account that µ g (C g ) = µ g (N g ) = 0 [5] and, when studying differential equations, the equation has to be satisfied µ g -almost everywhere.
As our aim is to study a differential inclusion, we end this section with basic notions of set-valued analysis (the reader is referred to [34,35] or [36]).
Let P bc (R d ) be the space of all non-empty bounded, closed and convex subsets of R d endowed with the Hausdorff-Pompeiu distance where the (Pompeiu-) excess of the set A ∈ P bc (R d ) over A ∈ P bc (R d ) is given by Let X, Y be Banach spaces and let F : X → P (Y) be a multimapping. F is said to be upper semicontinuous at u 0 ∈ X if for each ε > 0 there is δ ε,u 0 > 0 such that whenever u − u 0 < δ ε,u 0 , B being the closed unit ball of Y.
Moreover, F has closed graph if for all (u n ) n∈N ⊂ X, we have v ∈ F(u).

Preliminary Result-Existence Theory for the Single-Valued Problem
In this section, relying on the theory in [4], we present an existence result for the linear Stieltjes differential equation with periodic boundary conditions where g : [0, T] → R is non-decreasing and left-continuous and b : [0, T] → R is a µ g -measurable function satisfying the non-resonance condition: and u(0) = u(T).
Let us remark that when b ∈ L 1 (µ g ), the following condition is fulfilled: Indeed, if D g is countable, we note its elements by {t n } n∈N and we get (4) comes from the Limit Comparison Criterion for the convergence of numerical series. If D g is finite, then (4) is trivially fulfilled.
It turns out (see [4]) that for some positive constant δ, To solve the problem (2), the sign of 1 − b(t)µ g ({t}) has to be taken into account.
Applying Theorem 1, the following existence result can be proved: is a solution of problem (2).
Proof. Obviously, the LS-integrability ofb with respect to g follows from condition (4) and the LS-integrability of b.
One can see that for all t ∈ [0, T], We notice that α is constant on a neighborhood of t, so, by the product differentiation rule (see [5], Proposition 2.2), When calculating the g-derivative of the exponential function, we used a chain rule ([5], Theorem 2.3) together with Theorem 1, namely: The equality u g (t) = −b(t)u(t) + f (t) at the points in D g can be proved exactly as in ( [4], Theorem 17).

Remark 2.
If we impose the LS-integrability with respect to g of f , then the LS-integrability (therefore, the KS-integrability) of

Remark 3.
The reciprocal assertion of Theorem 2 is also valid (see [4], Theorem 19). Specifically, if b,g, f are as postulated in Theorem 2 and u : [0, T] → R d is a solution of (2), then then α(t) = 1 for every t ∈ [0, T], therefore the formulas and the computations become much simpler.

Existence of Solutions
We aim to obtain the existence of solutions for the set-valued periodic boundary value problem (1): The notion of solution adapted from the single-valued case (Definition 3) reads as follows.
We shall apply the following fixed-point theorem for multivalued operators. • there exists a function φ LS-integrable with respect to g such that Then the Stieltjes differential inclusion (1) has solutions. Moreover, the solution set of (1) is C -bounded.
Proof. Let X g be the subspace of G([0, T], R d ) consisting of the functions being continuous on [0, T] \ D g .
Condition (4) together with the LS-integrability with respect to g of b imply thatb has the same feature.
Following Remark 4, we note by By condition (4), for every t ∈ [0, T], so we shall denote by . and the operator A : M → P (M) given, for each u ∈ M, by withg as in Theorem 2 and A is well defined: for each u ∈ X g , S F(·,u(·)) is non-empty and whenever u ∈ X g , i.e., u is regulated and continuous on [0, T] \ D g , each element of Au has the same feature. Indeed, we note that α is constant in a neighborhood of t ∈ [0, T] \ D g , and writing each element of Au as in (6) Indeed, fix t ∈ [0, T]. Then every v ∈ Au is given (by the definition of the operator A) by some selection f of F(·, u(·)) and we can see, by Remark 4, that v(t) ≤ max 1, 1 Let us next check that the operator has closed, convex values. Let u ∈ M. Obviously, S F(·,u(·)) is convex (recall that F has convex values), therefore, Au is convex as well.
To prove that it is closed, take (v n ) n∈N ⊂ Au uniformly convergent to v ∈ M; specifically, for each n ∈ N, one can find f n ∈ S F(·,u(·)) such that v n (t) = 1 and v n → v uniformly.
One can see that so there exists a subsequence ( f n k ) k∈N weakly L 1 (µ g , R d ) convergent to a function f ∈ L 1 (µ g , R d ) (Dunford-Pettis Theorem). In a classical way (Mazur's theorem and properties of norm-convergent sequences in L 1 (µ g , R d )), a sequence of convex combinations tends pointwise µ g -a.e. to f , whence f (·) ∈ S F(·,u(·)) .
By a dominated convergence result (see [28], Theorem 6.8.7) applied for the components of ( f n k ) k∈N , f , one deduces that v n k (t) = 1 thus Au is closed.
We will prove that A satisfies the hypotheses of Theorem 3.
We check that A(M) is relatively compact, using Lemma 1.

Remark 1 yields now that the set A(M) is equiregulated.
The pointwise boundedness is immediate, therefore Lemma 1 implies that {Au : u ∈ M} is relatively compact.
Next, let us prove that A is upper semicontinuous. As A(M) is relatively compact, it suffices to verify that A has closed graph (see [36], Proposition 2.23).
Let (u n ) n∈N ⊂ M converge uniformly to u ∈ M and (v n ) n∈N ⊂ M converge uniformly to v ∈ M be such that v n ∈ Au n for all n ∈ N.
One can find, for every n ∈ N, f n ∈ S F(·,u n (·)) such that v n (t) = 1 As before, f n (t) ≤ φ(t), ∀n ∈ N, t ∈ [0, T], so there is a subsequence ( f n k ) k∈N convergent in the weak-L 1 (µ g , R d ) topology to a function f ∈ L 1 (µ g , R d ). It follows that a sequence of convex combinations of { f n k : k ∈ N} tends pointwise (µ g -a.e.) to f . On the other hand, F is upper semicontinuous with respect to the second value also with closed values , thus it has closed graph with respect to the second value (see [36], Proposition 2.17). Combining these two facts, we may easily check that f ∈ S F(·,u(·)) .
By a dominated convergence result (see [28], Theorem 6.8.7) applied for the components of ( f n k ) k∈N and f , one deduces that the corresponding sequence of convex combinations of (v n k ) k converges to 1 and consequently v ∈ Au. Finally, Bohnenblust-Karlin fixed-point theorem yields that the operator has fixed points, which are solutions to problem (1) by Theorem 2.

Dependence on the Data
Let us now study in which manner the solution set of problem (1) depends on the data. For this purpose, we are forced to drop the dependence on the state of the right-hand side. To be more precise, if we consider functions b 1 , b 2 as in Theorem 4 and multifunctions F 1 , F 2 : [0, T] → P bc (R d ) such that the considered problem has solutions, we are interested in finding the relation between the solution set S 1 of and the solution set S 2 of The perturbation of b shall be measured through while the perturbation of F through Correspondingly, one can measure the distance between the C -bounded sets S 1 , S 2 of regulated functions in the following ways: where the Pompeiu-excess of the set S 1 over the set S 2 is defined by where the excess of S 1 over S 2 is Let us note that Let F 1 , F 2 : [0, T] → P bc (R d ) satisfy the following hypotheses: there exists a function φ LS-integrable with respect to g such that Then there exist positive constants C i , i = 1, 6 such that for every u 1 ∈ S 1 , one can find u 2 ∈ S 2 satisfying, for all t ∈ [0, T], Proof. Let u 1 ∈ S 1 . Then there exists a selection f 1 of F 1 which is LS-integrable with respect to g such that (with the obvious convention t 1 0 = 0 and t 1 k+1 = T), By ( [34], Corollary 8.2.13) we can choose f 2 as the µ g -measurable selection of F 2 satisfying so, by the very definition of the Pompeiu-Hausdorff distance, Consider now the function u 2 : [0, T] → R d given by , i f t ∈ D g andg 2 (t, s) = 1 α 2 (T)e T 0b 2 (r)dg(r) − 1 Obviously, u 2 ∈ S 2 by Theorem 2. Let us see that it satisfies the requested inequality for some well-chosen constants C i , i = 1, ..., 6.
First, we may write Using the remark that |α 1 (t)| = 1 and also |α 2 (t)| = 1 for every t ∈ [0, T], we obtain (please note that 1 Let us evaluate the first term of the sum (7): and by Remark 2, where δ 1 , δ 2 are the corresponding positive constants in Remark 2 for b 1 , b 2 respectively. Since the condition (4) is verified, |b 2 (s)µ g ({s})| is bounded, say by m 2 . Then We can also see, by the choice of f 2 that We are now evaluating the second term of the sum (7): As in Remark 4, we denote by and so, It follows that We are now evaluating the differenceg 1 (t, s) −g 2 (t, s). It can be seen that In the first case (0 ≤ s ≤ t ≤ T), Similarly, in the second case (0 ≤ t < s ≤ T) it can be proved that Denoting bỹ we may say that for every s, t ∈ [0, T], We use next Remark 5(ii) and the fact that from (4), any t ∈ D g satisfies It is immediate that for each t ∈ [0, T], Finally, exploiting (8), (9), (10) we obtain that for all t ∈ [0, T],  Proof. Under the additional hypothesis on b 1 and b 2 , it can be seen that α 1 (t) = α 2 (t) = 1 on the whole interval and so, Theorem 5 yields that for every u 1 ∈ S 1 one can find u 2 ∈ S 2 such that for all t ∈ [0, T], T 0 |b 1 (s) − b 2 (s)|φ(s)dg(s) (i) By taking the supremum in (11) over t ∈ [0, T], By the definition of the Pompeiu-excess, it follows that and, by interchanging the roles of S 1 and S 2 , one obtains the announced estimation.
(ii) If φ is bounded, the inequality (11) implies that By integrating it with respect to g on [0, T] we get whence e L 1 (S 1 , S 2 ) ≤ C 1 + C 4 sup t∈[0,T] φ(t) g(T) b 1 − b 2 L 1 + C 5 g(T)D L 1 (F 1 , F 2 ). and the inequality comes from interchanging the roles of S 1 and S 2 .
Author Contributions: Both authors have equally contributed to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.