Existence Theorem for Impulsive Differential Equations with Measurable Right Side for Handling Delay Problems

Due to noncontinuous solution, impulsive differential equations with delay may have a measurable right side and not a continuous one. In order to support handling impulsive differential equations with delay like in other chapters of differential equations, we formulated and proved existence and uniqueness theorems for impulsive differential equations with measurable right sides following Caratheodory’s techniques. +e new setup had an impact on the formulation of initial value problems (IVP), the continuation of solutions, and the structure of the system of trajectories. (a) We have two impulsive differential equations to solve with one IVP (φ(σ0) � ξ0) which selects one of the impulsive differential equations by the position of σ0 in [a, b]]. Solving the selected IVP fully determines the solution on the other scale with a possible delay. (b)+e solutions can be continued at each point of (α, β) ×Ω0≕Ω by the conditions in the existence theorem. (c)+ese changes alter the flow of solutions into a directed tree.+is tree however is an in-tree which offers a modelling tool to study among other interactions of generations.


Introduction
e innovation of the theory of impulsive systems is manifested in the fact that the time development of the state of such a system forms a mapping of bounded variation instead of continuously differentiable solution of a differential equation [1][2][3][4][5][6][7][8].
e motivation for this research came from two observations, which arose in the theory and applications of impulsive differential equations.
One is the effect of the discontinuity of the right side of the impulsive differential equation, originating from the discontinuity of solutions of delayed equations. A discontinuity of the first order of the solution at a time point may create a set of discontinuity points on the right side (dynamics) of the equation in later time points. is changes the right side to a measurable function of the time instead of a continuous one. e second issue comes from the representation of a function of bounded variation in terms of two integral forms. In the "usual" representation, the absolute continuous part is a function of time t and the singular part is a function of the singular timer (impulse timer in impulsive differential equations) τ, while in the other representation, the function of τ is the absolute continuous part and the function of t is singular. In impulsive differential equations, the first form is in use. e purpose of this paper, therefore, is to analyse and formulate the concept of initial value problem suitable to initialize the obtained pair of impulsive differential equations having measurable right sides and to give conditions for the existence, uniqueness, and continuation of solutions. e existence of solutions of ordinary differential equations with measurable right side has been widely covered by Caratheodory [9]. erefore, our approach will start from Caratheodory's existence theorem.
e analysis and proof are presented in the following steps.
After giving a brief summary of processes described by impulsive differential equation-Bainovian model and introduction of the system time t and the impulse control time τ, we discussed how existence, uniqueness, and continuation are handled in impulsive differential equations with continuous right side. e analysis of problems arising from delayed equations and their handling leads us to the necessity of the formulation of the extended concept of impulsive differential equations and analysis of the difference between the Bainovian and extended models.

Systems Described by Impulsive Differential Equations.
e Bainovian model for the simplest case is as follows. Let the process evolve in a period of time T (T � (α, β) ⊂ R is an open interval). Let Ω 0 ⊂ R n be an open set and Ω ≔ T × Ω 0 . Let f: Ω ⟶ R n be an at least continuous mapping, which in addition may fulfill local Lipschitz condition in its variable x ∈ R n , for each fixed t, ∀(t, x) ∈ Ω. Let H ⊂ Z be an infinite subset of Z (H � N or H � Z will be used). en, let the time sequence S H � t k k∈H ⊂ T be increasing without accumulation points in T and t k ⟶ α, k ⟶ − ∞ and t k ⟶ β, k ⟶ ∞ (equivalently, ∀m, M ∈ T, m < M, [m, M]∩S H ⊂ T, is a finite set). Let g: S H × R n ⟶ R n be continuous and may fulfill Lipschitz condition in its variable x, ∀(t k , x) ∈ Ω. en, the controlling impulsive differential equation is given by where (t, x(t)) ∈ Ω. e impulsive differential equation (1) can be rewritten as an integral equation. We define an ascending step function τ: R ⟶ Z with unit jumps at the impulse points: If H � N, then τ(t) � 0∀t < t 1 .
Corollary 1. e function τ: R ⟶ R is a singular ascending function in t which means as an ascending function it is differentiable almost everywhere and The singular ascending function τ defines a singular measure τ on the Borel sets of R. e domain of the function g is extended to the g: Ω ⟶ R n from the set S H × Ω 0 ⊂ Ω.
With measures τ and g, equation (1) can be rewritten in an integral form: e technical details of these facts are discussed in [10]. Equation (4) has two measures; therefore, it does not look like an integral equation of an (impulsive) differential equation. We will change the parametrization of this equation.
Let ] λ (t) ≔ t + τ λ (t) � id [a,b λ ] (t) + τ λ (t) (τ λ represents τ as defined in Corollary 1), which is a strictly ascending function ] λ : As an ascending function ] λ has a left-and a rightcontinuous version, e mappings defined in (5) give us an increasing function μ λ : [a, b ] ] ⟶ [a, b λ ] defined as follows: Note that μ λ is one-to-one on the set of continuity points and ] may have a countable set of discontinuity points: Hence, τ λ and ] λ are continuous in [a, b λ ]\D λ .

Absolute Continuity and Singularity.
Let ] λ denote the measure defined by the strictly ascending function ] λ . e integral in equation (4) is the sum of integrals with two measures λ λ and τ λ . Both measures λ λ and τ λ are absolute continuous with respect to ] λ ; therefore, both can be written as an integral of the Radon-Nikodym derivatives [11]. With notations ρ λ � dλ λ /d] λ and ρ τ � dτ λ /d] λ , the following important properties are formulated. Let ese Radon-Nikodym derivatives enable us to rewrite equation (4) using one measure ] λ as follows: Journal of Mathematics e details of these assertions are in paper [10].

1.3.
Measures. e mappings τ λ and ] λ are ascending not continuous functions, with a common set of discontinuity points D λ . erefore, the measures τ λ and ] λ are defined on the ] map the set of discontinuity points D λ into the set of left-closed rightopen intervals: and the set of discontinuity points in Moreover, the mappings μ λ λ,− and μ λ λ,+ are bijective on the set of continuity points [a, b ] ]\D ] λ and μ λ, [a,b ] ],c ) with the extended measure ] ] λ on it. en, the following relations hold: e details of these assertions are in paper [10].

Existence and Continuation of a Solution.
Existence, uniqueness, and continuation of solutions are fundamental issues for differential equations of all kinds. ese issues, therefore, have been studied by many authors [1,2,[12][13][14][15][16][17][18][19] just to mention a few. ese articles consider initial value problems and boundary value problems for impulsive differential equations with an at least continuous right side f in Ω or in addition to the continuity f fulfills Lipschitz condition in its spatial variables; therefore, the analysis is based on Cauchy-Peano's or Piccard-Lindelöf's existence theorems [9]. e sources of discontinuities are arranged so that any closed bounded interval contains finite number of discontinuity points of the first type. We give a summary of these systems by pointing out the major differences in properties compared with the ordinary differential equations. If the initial value problem is not prescribed at a discontinuity time point then Cauchy-Peano's or Piccard-Lindelöf's existence theorems [9] provide solutions extendible in line with the rules of ordinary differential equations.

Impulsive Delayed Differential Equations.
e research on impulsive delay differential equations is very intensive as the cited list of some of the publications [2,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. e right side of the equations is still continuous or may fulfill Lipschitz conditions and sustains the finiteness of discontinuity points in closed bounded time intervals. e model of delayed impulsive systems developed by Bainov and his group [38] is based on a discrete set S H ⊂ R of impulse points with no accumulation points in any bounded Journal of Mathematics 3 interval. In these models, the delayed impact uses these same set S H of impulse points which regulate the occurrence of impulses at impulse time points which maybe a costly assumption. In some other approaches, different ways are used to meet the condition of local finiteness of the set of discontinuity points. Hence, to guarantee the local finiteness of discontinuity points of the right side is increasingly difficult in delayed systems, it is worth to see the effect of delay on the right side as presented in the following examples. Important examples: let the right side of the impulsive differential equation be defined as follows. Let Let the right continuous solution of the initial value problem of the equation be Assume that We will now show some simple examples to demonstrate that delay equations may lead to differential equations with measurable right sides: ), x(t)) has both left and right limits which are not the same. Hence, f is measurable and not a continuous function of t in [a, b] and ϑ is ascending with suitable selection of ε.
with this delay has no left limit hence no limit at η ∈ (α, β). Furthermore, there is no limit at t j , ∀1 ≤ j < ∞. Hence, f with delayed arguments is measurable and not continuous function of t in [a, b].
(3) Continuous descending delay leads to bijective mapping of the impulse points; hence, in this case, there are no accumulation points of the images of impact points but the statement about measurable right side remains valid.
Conclusion 1. Examples 1 and 3 can be handled with the help of the existence theorems such as Cauchy-Peano's or Piccard-Lindelöf's [9] since the discontinuity points have no accumulation points. e second example, however, requires limit theorems and additional reasoning. If Example 2 is combined with the construction of Cantor's triadic set [39], then we get a set of discontinuity points of continuum cardinality.
is means that alternative approach may be necessary to handle such initial value problems.

Extended Impulsive Differential Equations
and Existence of Their Solution e extended impulsive differential equations mean changing some basic assumptions used in Bainov's model as described in equation (1) or in rewritten form in equations (3) and (10). Major changes include the time control of the impulses may have infinite discontinuities but has to be of bounded variation on every closed bounded interval, and the system dynamics is measurable as a function of the time and not necessarily continuous. Let τ: T ⟶ R + be a singular ascending function of the time parameter t as the singular "impulse timer". It is important to see that τ may have a countably infinite set of jump points, where the total lengths of these jumps must be bounded on any closed bounded interval. Using equation (8), we can rewrite the Radon-Nikodym derivatives in terms of characteristic functions of the sets N λ λ and N τ λ as follows: Putting these into equation (10) and changing g with g, we get the extended impulsive differential equation in t-scale as e integral transformations discussed in Section 2.3.5 will give a similar result in both ]-scale and τ-scale. We will handle the ]-scale representation first.
be the measurable right side of the extended impulsive differential equation in ]-scale.
We will use the notations for the generated ]-scale, and [a, b τ ] for τ-scale to get the advantages of compact sets.
In Section 2.2, we will discuss the main results of this paper which is formulation of the extension of Caratheodory's existence theorem for the extended impulsive differential equations with measurable right side. e basis of our discussion is the approach presented in pg. 43 in [9].

Caratheodory's eorem.
We present Caratheodory's existence theorem in R n as it is presented in the cited pages 42-43 for one dimension.
We are considering a process on an open set S ⊂ Ω ⊂ R × R n . Let f: S ⟶ R n be a function not necessarily continuous.
Problem (E): find an interval I ⊂ [a, b] and an absolute continuous function φ: en, the function φ: I ⟶ R n is a solution of equation (20) in the extended sense.
Caratheodory actually proved the existence on an interval [t 0 , t 0 + β) interval and used this result to prove the existence on (t 0 − β, t 0 ] by using suitable transformations of symmetry. Using Caratheodory's theorem we can prove the existence of solution of the extended impulsive differential equation equation on ]-scale with right side (19), precisely.
Remark 1. Note that the condition t 0 ∉ D λ used in Corollary 3 is in the Bainovian initial value problem (4). erefore, the Bainovian case with measurable right side is covered by this simple example.
Note also that h ] can be B ([a, b ] ], ])-measurable which is not covered in this corollary.

Remark 2.
With Corollary 2, the discussion about general existence theorem has been finished. e rest of this paper will target to include the solutions of equations with B([a, b ] ], ])-measurable right sides and the formulation of the conditions for the existence of solutions in terms of the measurable functions f λ and g τ . Some issues will have to be clarified about the initial value problems.
As shown in the example, solutions of the impulsive differential equations are obtained from the absolute continuous solutions on the ]-scale with the help of suitable transformations. We will develop some extensions of the mappings μ λ λ,− , μ λ λ,+ , and μ λ . Uniqueness: the solution of an initial value problem is unique if the right side of the differential equation fulfills local Lipschitz condition [9]. Although there are other conditions for uniqueness, we will demonstrate our presentation on this condition. We concluded in Section 1.4 about continuation of solutions that, for impulsive differential equations, the solution of an initial value problem (t 0 , φ(t 0 )) � (t 0 , ξ 0 ) ∈ Ω exists on an interval [t 0 , t 0 + δ) ⊂ T if the conditions of one of the existence theorems hold ∀(t 0 , ξ 0 ) ∈ Ω.
Let us consider the impact of this condition on an example (originating from [9]). Let the differential equation be as follows: e differential equation fulfills local Lipschitz condition ∀(t, y) ∈ R 2 on an interval [t, t + δ t ), δ t > 0.
erefore, merging of solution trajectories can occur while splitting of trajectories is excluded by having local Lipschitz condition at each point (t, y) ∈ R 2 in an interval [t, t+ δ t ) × B δ t (y). However, note that if φ 1 (t 1 ) � y 1 and φ 2 (t 2 ) � y 2 ∈ R, (t 1 , y 1 ) ≠ (t 2 , y 2 ) then the two global solutions are different by the initial value problems even if φ 1 (s) � φ 2 (s) holds at an s > max t 1 , t 2 ∈ R⇒φ 1 (t) � φ 2 (t), ∀t ≥ s. Hence, the global solutions of two different initial value problems are two different trajectories (not necessarily disjoint trajectories). 6 Journal of Mathematics

Timescales and eir Density Functions.
is section summarises the concepts which will serve as the basis of most of our coming discussion and were developed in [10] and partly in [40]. We use the notations and concepts formulated in Section 1.1 and our starting point will be the integral equation (4).
We showed that, given [a, b λ ], N λ λ , N λ τ , λ λ , and τ λ , we obtain [a, b ] where ] λ is right continuous. Since ] λ is strictly ascending, and τ λ is singular with respect to λ λ and λ λ is singular with respect to τ λ by is the countable set of discontinuity points in t-scale. From now on, we will use D λ in the place of S H .

Journal of Mathematics
where ] τ is right continuous. Since ] τ is strictly ascending, hence bijective, which means τ τ is singular with respect to λ τ and λ τ is singular with respect to τ τ .

Interpretation of the Two Representations.
We need an interpretation of the two representations (26) and (29) of the Bainovian impulsive systems. e impulsive process described by equations (1) and (4) is composed from a process with f-dynamics and a process with g-dynamics. In t-scale equation (26), the fully described process with f-dynamics (absolute continuous component) is exposed to impulses generated by the (singular) impulse generator with g-dynamics.
e second τ-scale representation equation (29) gives a full description of the (absolute continuous) impulse generator with g-dynamics while being exposed to the (singular) counter impacts caused by the process with fdynamics.
us, these systems operate in action process/ reaction-counter action processes, which is a deep principle in interactions in sciences. e ]-scale representation presents both processes in full details in a time-sharing system. e ]-scale process is presented in equation (30). e ]-scale process is absolute continuous and makes it possible to prove existence theorems using Caratheodory's techniques.
e f ] -dynamics operates in intervals τ , ∀t i ∈ D λ . Hence, the ]-scale consists of connected intervals alternating between f ] -dynamics and g ] -dynamics. e actions of the f ] -dynamics and g ] -dynamics are as indicated by the column headings in equation (30). e impulse points selected are t i , t i+1 , t i+2 , and t i+3 ∈ D λ , t i < t i+1 < t i+2 < t i+3 : is scheme of operation follows the rules of timed chess game. e players are f ] -dynamics and g ] -dynamics. eir clocks are t and τ, respectively. If a solution of equations (1) and (4) is φ: ) for a period of t i+1 − t i t-time while g ] waiting with stopped τ-time till f ] produces φ(t i+1 − 0). Actions of f ] are unknown to g ] . At t i+1 the game switches to g ] , the t-clock stops, τ-clock operates, and g ] performs its job for a period of μ λ λ, erefore, in the t-scale process all g-actions are hidden and each g-interval appears in the form of jump. In the case of τ-scale representation, all f-intervals appear as jumps. Hence, the two equations are completely symmetric and form a pair of impulsive differential equations.

Initial Value Problem for a Pair of Impulsive Differential Equations.
From the interpretation of a pair of impulsive differential equation follows that the concept of  (30) we obtain a pair of impulsive differential equations such that any initial value problem

Using the left closed right open intervals in equation
τ . Finally, an initial value problem will give initial value problems on the t-scale and on the τ-scale as follows. Since the solution is x λ � φ · μ λ,+ and x τ � φ · μ τ,+ , we simply can define the initial value problems as follows: Similarly, for the case σ 0 ∈ N ] τ ,

Integral Transformations among t-, τ-, and ]-Scales.
e details of the assertions in this section come from paper [10].
We summarize the mappings between t-scale to ]-scale listed in Section 2.3.1 (Table 1) and the set of discontinuity points in Moreover, the mappings μ λ λ,− , μ λ λ,+ are bijective on the set of continuity points [a, b ] ]\D ] λ and μ λ,− (t) � μ λ, ese are some of the main conclusions from paper [10] presented here in a condensed form.
We summarize the mappings between τ-scale and ]-scale listed in Section 2.3.2 (Table 2): e mappings λ τ and ] τ are not continuous ascending functions, with a common set of discontinuity points D τ . erefore, the measures λ τ and ] τ are defined on the sem- and the set of discontinuity points in [a, b ] ] is Moreover, the mappings μ τ τ,− and μ τ τ,+ are bijective on the set of continuity points [a, b ] ]\D ] τ and μ τ,− (t) � μ τ,   [a,b ] ],c -measurable and ] τ -integrable, then ese are some of the main conclusions of paper [10] presented here in a condensed form. e details about measures and Radon-Nikodym derivatives summarised below are in paper [10].

Conclusion 4.
e Bainovian impulsive processes describe the movements of a process (f-dynamics) under the impulses of another process (g-dynamics). In this paper, we developed a technique to split such a system into two impulsive processes. One is the Bainovian process moving with fdynamics under the impulses of the process with g-dynamics, while the other is the process with g-dynamics under the impacts of f-impulses on it. e relationship is an action/reaction interaction between the two processes. is gives a pair of impulsive differential equation with the relationship between them analysed. We established an existence theorem for impulsive differential equations with right side being a measurable function of time which facilitates the analysis of delayed impulsive differential equations. On the contrary, trajectories may be connected together by impulse effects. is will make the flow of solutions to be a tree structure instead of a connected flow as in ordinary differential equations. e leaves of the tree consist of trajectories coming from discontinuity points without history. is tree is directed, with orientation from the leaves to the root which is called in-tree or antiarborescence [41].
is gives wide range of modelling facilities by enabling one to model and study mixing new generations in addition to studying flows of solutions.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.