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

Authors: Z. Lipcsey; J. A. Ugboh; I. M. Esuabana; I. O. Isaac
Journal: Journal of Mathematics
 (2020)

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## Abstract

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. The 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 (φσ0=ξ0) which selects one of the impulsive differential equations by the position of σ0 in a,bν. Solving the selected IVP fully determines the solution on the other scale with a possible delay. (b) The solutions can be continued at each point of α,β×Ω0≕Ω by the conditions in the existence theorem. (c) These changes alter the flow of solutions into a directed tree. This tree however is an in-tree which offers a modelling tool to study among other interactions of generations.


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## Body

## 1. Introduction

The 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 [1–8].The 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. This changes the right side to a measurable function of the time instead of a continuous one.The 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 timet 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.The 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. The existence of solutions of ordinary differential equations with measurable right side has been widely covered by Caratheodory [9]. Therefore, our approach will start from Caratheodory’s existence theorem.The 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 timet and the impulse control time τ, we discussed how existence, uniqueness, and continuation are handled in impulsive differential equations with continuous right side.The 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.


### 1.1. Systems Described by Impulsive Differential Equations

The Bainovian model for the simplest case is as follows. Let the process evolve in a period of timeT (T=α,β⊂R is an open interval). Let Ω0⊂Rn be an open set and Ω≔T×Ω0. Let f:Ω⟶Rn be an at least continuous mapping, which in addition may fulfill local Lipschitz condition in its variable x∈Rn, for each fixed t, ∀t,x∈Ω. Let H⊂Z be an infinite subset of Z (H=N or H=Z will be used). Then, let the time sequence SH=tkk∈H⊂T be increasing without accumulation points in T and tk⟶α,k⟶−∞andtk⟶β,k⟶∞ (equivalently, ∀m,M∈T,m<M,m,M∩SH⊂T, is a finite set). Let g:SH×Rn⟶Rn be continuous and may fulfill Lipschitz condition in its variable x, ∀tk,x∈Ω. Then, the controlling impulsive differential equation is given by(1)x′t=ft,xt,∀t∈T\SH,Δxtk=gtk,xtk,∀tk∈SH,where t,xt∈Ω.The 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:(2)τt≔k−1,iftk−1≤t<tk,∀tk∈SZ.IfH=N, then τt=0∀t<t1.Corollary 1.
The functionτ:R⟶R is a singular ascending function in t which means as an ascending function it is differentiable almost everywhere and(3)dτtdt=0almost everywhere.
The singular ascending functionτ defines a singular measure τ on the Borel sets of R. The domain of the function g is extended to the g˜:Ω⟶Rn from the set SH×Ω0⊂Ω.With measuresτ and g˜, equation (1) can be rewritten in an integral form:(4)xt=x0+∫t0tfs,xsds+g˜s,xsdτ,t0<t∈T,xt0=x0,t0∈T\SH,x0∈Ω0.The 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.Letνλt≔t+τλt=ida,bλt+τλt (τλ represents τ as defined in Corollary 1), which is a strictly ascending function νλ:a,bλ⟶a,bν with bν≔bλ+τλbλ−τλa. As an ascending function νλ has a left- and a right-continuous version,(5)μλ,−λt≔νλt−0,∀t∈a,bλ,μλ,+λt≔νλt+0,∀t∈a,bλ.The mappings defined in (5) give us an increasing function μ^λ:a,bν⟶a,bλ defined as follows:(6)μ^λt≔s,∀t∈a,bν,∃s∈a,bλs.t.t∈μλ,−λs,μλ,+λs,μ^λ⋅μλ,−λt=μ^λ⋅μλ,−λt=ida,bλ.Note thatμ^λ is one-to-one on the set of continuity points (μλ,−λs,μλ,+λs is singleton if μλ,−λs=μλ,+λs). The ascending function ida,bλt is continuous in a,bλ while τ and ν may have a countable set of discontinuity points:(7)Dλ≔tτλt−0≠τλt+0=tνλt−0≠νλt+0.Hence,τλandνλ are continuous in a,bλ\Dλ.


### 1.2. Absolute Continuity and Singularity

Letνλ denote the measure defined by the strictly ascending function νλ. The integral in equation (4) is the sum of integrals with two measures λλandτλ. Both measures λλandτλ are absolute continuous with respect to νλ; therefore, both can be written as an integral of the Radon–Nikodym derivatives [11]. With notations ρλ=dλλ/dνλandρτ=dτλ/dνλ, the following important properties are formulated.LetNτλ≔dτλ/dνλ>0andNλλ≔Nτλ′. Then, λλNτλ=0=τλNλλ and(8)ρλ+ρτ=dλλ+τλ/dνλ=dτλ/dνλ+dλλ/dνλ=1.Moreover, let(9)Nλν≔μ^λ−1Nλλ⊂a,bν,Nτν≔μ^λ−1Nτλ⊂a,bν,Nλν∪Nτν=a,bν,Nλν∩Nτν=∅.These Radon–Nikodym derivatives enable us to rewrite equation (4) using one measure νλ as follows:(10)xt=x0+∫t0tfs,xsdλλdνλ+g˜s,xsdτλdνλdνλ,t∈T,xt0=x0,t0∈T\SH,x0∈Ω0.The details of these assertions are in paper [10].


### 1.3. Measures

The mappingsτλandνλ are ascending not continuous functions, with a common set of discontinuity points Dλ. Therefore, the measures τλandνλ are defined on the semiring Pν,a,bλ,c=s,ts,t∈a,bλ\Dλ,s≤t and the measures τλs,t≔τλt−τλsandνλs,t≔νλt−νλs,∀s,t∈Pν,a,bλ,c can be extended to ℬν,a,bλ,c=σPν,a,bλ,c.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν map the set of discontinuity points Dλ into the set of left-closed right-open intervals:(11)Dλν≔μλ,−t,μλ,+tt∈Dλ,and the set of discontinuity points in a,bν is(12)Dλν≔∪t∈Dλμλ,−t,μλ,+t⊂a,bν.Moreover, the mappingsμλ,−λandμλ,+λ are bijective on the set of continuity points a,bν\Dλν and μλ,−t=μλ,+t,∀t∈a,bλ\Dλ.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν transform s,t∈Pν,a,bλ,c⟶νλs,νλt=μλ,−λs,μλ,−λt=μλ,+λs,μλ,+λt∈Pν,a,bν,cν by Pν,a,bν,cν≔u,vu,v∈a,bν\Dλν.The measureνλν on Pν,a,bν,cν is defined by νλνu,v≔v−u=μλ,−μ^λλv−μλ,−μ^λλu=νλμ^λλu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bλ,c, then μλ,−s,μλ,−t∈Pν,a,bν,cν and νλνμλ,−s,μλ,−t=μλ,−t−μλ,−s=νλs,t. Let the smallest σ-algebra containing the semiring Pλ,a,bν,cν be ℬλ,a,bν,cν≔σPλ,a,bν,cν with the extended measure νλν on it. Then, the following relations hold:(13)∫aμλ,+tf⋅μ^λdνλν=∫atfdνλ,∀a,t∈ℬν,a,bλ,c,∀f∈L1νλ,ℬν,a,bλ,c,∫aσfdνλν=∫aμ^λσf⋅μλ,−dνλ,∀a,σ∈ℬλ,a,bν,cν,∀f∈L1νλν,ℬλ,a,bν,cν.The details of these assertions are in paper [10].


### 1.4. Existence and Continuation of a Solution

Existence, uniqueness, and continuation of solutions are fundamental issues for differential equations of all kinds. These issues, therefore, have been studied by many authors [1, 2, 12–19] just to mention a few. These 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]. The 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.Continuation: by the assumption that the functionf:Ω⟶Rn fulfills the conditions of an existence theorem (Cauchy–Peano’s or Piccard–Lindelöf’s existence theorem [9]), in Ω, any solution that reaches a point s,φs∈Ω,t0<s has a continuation on an interval s,s+δ with a suitable δ>0. The process stops only at a boundary point s,φs∈∂Ω. The impulses, however, change this scenario. If the solution reaches tj,φtj−0∈Ω,tj∈SH, then there is a continuation to φtj+0=φtj−0+gtj,φtj−0. If tj,φtj+0∉Ω, the process stops at tj,φtj−0=tj,φtj∈Ω. Otherwise, if tj,φtj+0∈Ω, then it has a continuation as described above. The jump mapping Jtj,y≔tj,y+gtj,y having a value ∀y∈Ω0 gives further changes. All solutions reaching the impulse point tj continue to φtj+0. If J:SH×Ω0⟶SH×Ω0⊂Ω, then each trajectory is continued beyond the impulse time point. However, the range ℛJ may be a proper subset of SH×Ω0. Then, the solution φ of equation (1) fulfilling the initial value problem s0,φs0∈SH×Ω0∖ℛJ has no history, will have no continuation on the interval s0−δ,s0, and the continuation will exist on s0,s0+δ only. These properties are also mentioned in [1, 14].


#### 1.4.1. Impulsive Delayed Differential Equations

The research on impulsive delay differential equations is very intensive as the cited list of some of the publications [2, 20–37]. The 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. The model of delayed impulsive systems developed by Bainov and his group [38] is based on a discrete set SH⊂R of impulse points with no accumulation points in any bounded interval. In these models, the delayed impact uses these same set SH 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. Leta,b⊃α,β,η≔α+β/2 and let α−h≕γ∈a,b,h>0. Then, let(14)fxt−ϑt,xt≔xt+xt−ϑt,∀t∈α,β,gxt−ϑt,xt≔2xt+xt−ϑt,∀t∈α,β,SH∩α,β=∅,γ∈SH.Let the right continuous solution of the initial value problem of the equation be(15)xt=x0+∫t0tfxs−ϑs,xsds+gxs−ϑs,xsdτ,t∈t0,b,xt0=x0,t0∈a,γ\SH.Assume thatxγ−0=1andxγ+0=−1.Letϑ be continuous ascending function, ϑt<t,∀t∈α,β. Let ut≔t−ϑt⇒ϑt=t−ut.We will now show some simple examples to demonstrate that delay equations may lead to differential equations with measurable right sides:(1)
Letut≔γ−εt−αη−tβ−t∀t∈α,β. This will give uα=uη=uβ=γ; hence, xη−0=1≠−1=xη+0. Therefore, fxt−ϑt,xt 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.(2)
Lettj≔η−β−α/2j,1≤j<∞,tj↗η if j⟶∞. Then,(16)ut≔γ+−1j22η−αt−tj2tj+1−t2,iftj≤t≤tj+1,j∈N.0,otherwise.The functionut=γ,∀t=ti, and ut<γ if ti<t<ti+1 and i is odd and ut>γ if ti<t<ti+1 and i is even. Hence, xti−0=−1andxti+0=1 if i is odd and xti−0=1andxti+0=−1 if i is even.Hence,limsupt↗ηxt−ϑt=1 and liminft↗ηxt−ϑt=−1.The delayed solutionxt−ϑt with this delay has no left limit hence no limit at η∈α,β. Furthermore, there is no limit at tj,∀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. The 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. This means that alternative approach may be necessary to handle such initial value problems.


## 1.1. Systems Described by Impulsive Differential Equations

The Bainovian model for the simplest case is as follows. Let the process evolve in a period of timeT (T=α,β⊂R is an open interval). Let Ω0⊂Rn be an open set and Ω≔T×Ω0. Let f:Ω⟶Rn be an at least continuous mapping, which in addition may fulfill local Lipschitz condition in its variable x∈Rn, for each fixed t, ∀t,x∈Ω. Let H⊂Z be an infinite subset of Z (H=N or H=Z will be used). Then, let the time sequence SH=tkk∈H⊂T be increasing without accumulation points in T and tk⟶α,k⟶−∞andtk⟶β,k⟶∞ (equivalently, ∀m,M∈T,m<M,m,M∩SH⊂T, is a finite set). Let g:SH×Rn⟶Rn be continuous and may fulfill Lipschitz condition in its variable x, ∀tk,x∈Ω. Then, the controlling impulsive differential equation is given by(1)x′t=ft,xt,∀t∈T\SH,Δxtk=gtk,xtk,∀tk∈SH,where t,xt∈Ω.The 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:(2)τt≔k−1,iftk−1≤t<tk,∀tk∈SZ.IfH=N, then τt=0∀t<t1.Corollary 1.
The functionτ:R⟶R is a singular ascending function in t which means as an ascending function it is differentiable almost everywhere and(3)dτtdt=0almost everywhere.
The singular ascending functionτ defines a singular measure τ on the Borel sets of R. The domain of the function g is extended to the g˜:Ω⟶Rn from the set SH×Ω0⊂Ω.With measuresτ and g˜, equation (1) can be rewritten in an integral form:(4)xt=x0+∫t0tfs,xsds+g˜s,xsdτ,t0<t∈T,xt0=x0,t0∈T\SH,x0∈Ω0.The 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.Letνλt≔t+τλt=ida,bλt+τλt (τλ represents τ as defined in Corollary 1), which is a strictly ascending function νλ:a,bλ⟶a,bν with bν≔bλ+τλbλ−τλa. As an ascending function νλ has a left- and a right-continuous version,(5)μλ,−λt≔νλt−0,∀t∈a,bλ,μλ,+λt≔νλt+0,∀t∈a,bλ.The mappings defined in (5) give us an increasing function μ^λ:a,bν⟶a,bλ defined as follows:(6)μ^λt≔s,∀t∈a,bν,∃s∈a,bλs.t.t∈μλ,−λs,μλ,+λs,μ^λ⋅μλ,−λt=μ^λ⋅μλ,−λt=ida,bλ.Note thatμ^λ is one-to-one on the set of continuity points (μλ,−λs,μλ,+λs is singleton if μλ,−λs=μλ,+λs). The ascending function ida,bλt is continuous in a,bλ while τ and ν may have a countable set of discontinuity points:(7)Dλ≔tτλt−0≠τλt+0=tνλt−0≠νλt+0.Hence,τλandνλ are continuous in a,bλ\Dλ.


## 1.2. Absolute Continuity and Singularity

Letνλ denote the measure defined by the strictly ascending function νλ. The integral in equation (4) is the sum of integrals with two measures λλandτλ. Both measures λλandτλ are absolute continuous with respect to νλ; therefore, both can be written as an integral of the Radon–Nikodym derivatives [11]. With notations ρλ=dλλ/dνλandρτ=dτλ/dνλ, the following important properties are formulated.LetNτλ≔dτλ/dνλ>0andNλλ≔Nτλ′. Then, λλNτλ=0=τλNλλ and(8)ρλ+ρτ=dλλ+τλ/dνλ=dτλ/dνλ+dλλ/dνλ=1.Moreover, let(9)Nλν≔μ^λ−1Nλλ⊂a,bν,Nτν≔μ^λ−1Nτλ⊂a,bν,Nλν∪Nτν=a,bν,Nλν∩Nτν=∅.These Radon–Nikodym derivatives enable us to rewrite equation (4) using one measure νλ as follows:(10)xt=x0+∫t0tfs,xsdλλdνλ+g˜s,xsdτλdνλdνλ,t∈T,xt0=x0,t0∈T\SH,x0∈Ω0.The details of these assertions are in paper [10].


## 1.3. Measures

The mappingsτλandνλ are ascending not continuous functions, with a common set of discontinuity points Dλ. Therefore, the measures τλandνλ are defined on the semiring Pν,a,bλ,c=s,ts,t∈a,bλ\Dλ,s≤t and the measures τλs,t≔τλt−τλsandνλs,t≔νλt−νλs,∀s,t∈Pν,a,bλ,c can be extended to ℬν,a,bλ,c=σPν,a,bλ,c.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν map the set of discontinuity points Dλ into the set of left-closed right-open intervals:(11)Dλν≔μλ,−t,μλ,+tt∈Dλ,and the set of discontinuity points in a,bν is(12)Dλν≔∪t∈Dλμλ,−t,μλ,+t⊂a,bν.Moreover, the mappingsμλ,−λandμλ,+λ are bijective on the set of continuity points a,bν\Dλν and μλ,−t=μλ,+t,∀t∈a,bλ\Dλ.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν transform s,t∈Pν,a,bλ,c⟶νλs,νλt=μλ,−λs,μλ,−λt=μλ,+λs,μλ,+λt∈Pν,a,bν,cν by Pν,a,bν,cν≔u,vu,v∈a,bν\Dλν.The measureνλν on Pν,a,bν,cν is defined by νλνu,v≔v−u=μλ,−μ^λλv−μλ,−μ^λλu=νλμ^λλu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bλ,c, then μλ,−s,μλ,−t∈Pν,a,bν,cν and νλνμλ,−s,μλ,−t=μλ,−t−μλ,−s=νλs,t. Let the smallest σ-algebra containing the semiring Pλ,a,bν,cν be ℬλ,a,bν,cν≔σPλ,a,bν,cν with the extended measure νλν on it. Then, the following relations hold:(13)∫aμλ,+tf⋅μ^λdνλν=∫atfdνλ,∀a,t∈ℬν,a,bλ,c,∀f∈L1νλ,ℬν,a,bλ,c,∫aσfdνλν=∫aμ^λσf⋅μλ,−dνλ,∀a,σ∈ℬλ,a,bν,cν,∀f∈L1νλν,ℬλ,a,bν,cν.The details of these assertions are in paper [10].


## 1.4. Existence and Continuation of a Solution

Existence, uniqueness, and continuation of solutions are fundamental issues for differential equations of all kinds. These issues, therefore, have been studied by many authors [1, 2, 12–19] just to mention a few. These 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]. The 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.Continuation: by the assumption that the functionf:Ω⟶Rn fulfills the conditions of an existence theorem (Cauchy–Peano’s or Piccard–Lindelöf’s existence theorem [9]), in Ω, any solution that reaches a point s,φs∈Ω,t0<s has a continuation on an interval s,s+δ with a suitable δ>0. The process stops only at a boundary point s,φs∈∂Ω. The impulses, however, change this scenario. If the solution reaches tj,φtj−0∈Ω,tj∈SH, then there is a continuation to φtj+0=φtj−0+gtj,φtj−0. If tj,φtj+0∉Ω, the process stops at tj,φtj−0=tj,φtj∈Ω. Otherwise, if tj,φtj+0∈Ω, then it has a continuation as described above. The jump mapping Jtj,y≔tj,y+gtj,y having a value ∀y∈Ω0 gives further changes. All solutions reaching the impulse point tj continue to φtj+0. If J:SH×Ω0⟶SH×Ω0⊂Ω, then each trajectory is continued beyond the impulse time point. However, the range ℛJ may be a proper subset of SH×Ω0. Then, the solution φ of equation (1) fulfilling the initial value problem s0,φs0∈SH×Ω0∖ℛJ has no history, will have no continuation on the interval s0−δ,s0, and the continuation will exist on s0,s0+δ only. These properties are also mentioned in [1, 14].


### 1.4.1. Impulsive Delayed Differential Equations

The research on impulsive delay differential equations is very intensive as the cited list of some of the publications [2, 20–37]. The 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. The model of delayed impulsive systems developed by Bainov and his group [38] is based on a discrete set SH⊂R of impulse points with no accumulation points in any bounded interval. In these models, the delayed impact uses these same set SH 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. Leta,b⊃α,β,η≔α+β/2 and let α−h≕γ∈a,b,h>0. Then, let(14)fxt−ϑt,xt≔xt+xt−ϑt,∀t∈α,β,gxt−ϑt,xt≔2xt+xt−ϑt,∀t∈α,β,SH∩α,β=∅,γ∈SH.Let the right continuous solution of the initial value problem of the equation be(15)xt=x0+∫t0tfxs−ϑs,xsds+gxs−ϑs,xsdτ,t∈t0,b,xt0=x0,t0∈a,γ\SH.Assume thatxγ−0=1andxγ+0=−1.Letϑ be continuous ascending function, ϑt<t,∀t∈α,β. Let ut≔t−ϑt⇒ϑt=t−ut.We will now show some simple examples to demonstrate that delay equations may lead to differential equations with measurable right sides:(1)
Letut≔γ−εt−αη−tβ−t∀t∈α,β. This will give uα=uη=uβ=γ; hence, xη−0=1≠−1=xη+0. Therefore, fxt−ϑt,xt 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.(2)
Lettj≔η−β−α/2j,1≤j<∞,tj↗η if j⟶∞. Then,(16)ut≔γ+−1j22η−αt−tj2tj+1−t2,iftj≤t≤tj+1,j∈N.0,otherwise.The functionut=γ,∀t=ti, and ut<γ if ti<t<ti+1 and i is odd and ut>γ if ti<t<ti+1 and i is even. Hence, xti−0=−1andxti+0=1 if i is odd and xti−0=1andxti+0=−1 if i is even.Hence,limsupt↗ηxt−ϑt=1 and liminft↗ηxt−ϑt=−1.The delayed solutionxt−ϑt with this delay has no left limit hence no limit at η∈α,β. Furthermore, there is no limit at tj,∀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. The 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. This means that alternative approach may be necessary to handle such initial value problems.


## 1.4.1. Impulsive Delayed Differential Equations

The research on impulsive delay differential equations is very intensive as the cited list of some of the publications [2, 20–37]. The 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. The model of delayed impulsive systems developed by Bainov and his group [38] is based on a discrete set SH⊂R of impulse points with no accumulation points in any bounded interval. In these models, the delayed impact uses these same set SH 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. Leta,b⊃α,β,η≔α+β/2 and let α−h≕γ∈a,b,h>0. Then, let(14)fxt−ϑt,xt≔xt+xt−ϑt,∀t∈α,β,gxt−ϑt,xt≔2xt+xt−ϑt,∀t∈α,β,SH∩α,β=∅,γ∈SH.Let the right continuous solution of the initial value problem of the equation be(15)xt=x0+∫t0tfxs−ϑs,xsds+gxs−ϑs,xsdτ,t∈t0,b,xt0=x0,t0∈a,γ\SH.Assume thatxγ−0=1andxγ+0=−1.Letϑ be continuous ascending function, ϑt<t,∀t∈α,β. Let ut≔t−ϑt⇒ϑt=t−ut.We will now show some simple examples to demonstrate that delay equations may lead to differential equations with measurable right sides:(1)
Letut≔γ−εt−αη−tβ−t∀t∈α,β. This will give uα=uη=uβ=γ; hence, xη−0=1≠−1=xη+0. Therefore, fxt−ϑt,xt 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.(2)
Lettj≔η−β−α/2j,1≤j<∞,tj↗η if j⟶∞. Then,(16)ut≔γ+−1j22η−αt−tj2tj+1−t2,iftj≤t≤tj+1,j∈N.0,otherwise.The functionut=γ,∀t=ti, and ut<γ if ti<t<ti+1 and i is odd and ut>γ if ti<t<ti+1 and i is even. Hence, xti−0=−1andxti+0=1 if i is odd and xti−0=1andxti+0=−1 if i is even.Hence,limsupt↗ηxt−ϑt=1 and liminft↗ηxt−ϑt=−1.The delayed solutionxt−ϑt with this delay has no left limit hence no limit at η∈α,β. Furthermore, there is no limit at tj,∀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. The 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. This means that alternative approach may be necessary to handle such initial value problems.


## 2. Extended Impulsive Differential Equations and Existence of Their Solution

The 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.


### 2.1. The Extended Impulsive Differential Equations

Let the process evolve in an open time intervalT⊂R and in an open set T×Ω0≕Ω⊂T×Rn. Let f,g:Ω⟶Rn be measurable functions in the time variable t for each fixed spatial value x∈Ω0, and continuous/Lipschitz-continuous functions in the spatial variable x for each fixed t∈T.Letτ: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 NλλandNλτ as follows:(17)dλλ+τλdνλ=dλλdνλ+dτλdνλ=1⇒dλλdνλ=χNλλanddτλdνλ=χNλτ.Putting these into equation (10) and changing g˜ with g, we get the extended impulsive differential equation in t-scale as(18)xt=x0+∫t0tfs,xsχNλλ+gs,xsχNτλdνλ,t∈T,xt0=x0t0∈T\SH,x0∈Ω0.The integral transformations discussed in Section2.3.5 will give a similar result in both ν-scale and τ-scale. We will handle the ν-scale representation first. Let fν≔fs,xsχNλλ⋅μ^λν:Nλν⟶Rn and gν≔gs,xsχNτλ⋅μ^λν:Nτν⟶Rn. Then, let(19)hν≔fνχNλν+gνχNτν:a,bν×Ω0⟶Rnbe the measurable right side of the extended impulsive differential equation in ν-scale.We will use the notationsa,bλ⊂T for t-scale, a,bν for the generated ν-scale, and a,bτ for τ-scale to get the advantages of compact sets.In Section2.2, 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. The basis of our discussion is the approach presented in pg. 43 in [9].


### 2.2. Caratheodory’s Theorem

We present Caratheodory’s existence theorem inRn as it is presented in the cited pages 42-43 for one dimension.We are considering a process on an open setS⊂Ω⊂R×Rn. Let f:S⟶Rn be a function not necessarily continuous.Problem (E): find an intervalI⊂a,b and an absolute continuous function φ:I⊂a,b⟶Rn such that(20)t,φt∈S,φ′t=ft,φt,almost allt∈I.Then, the functionφ:I⟶Rn is a solution of equation (20) in the extended sense.Caratheodory’s existence theorem [9] targets finding a solutions to problem (E) with an initial value t0,ξ∈Ω,φt0=ξ, where the right side is a measurable function of t for each fixed x∈Ω0 on T×Ω0, where ∅≠α,β=T⊂RandΩ0⊂Rn are open sets. Caratheodory’s condition for the existence of the solution is the existence of a local positive integrable dominant m:t0−γ,t0+γ⊂T⟶R+\0andε>0 such that ft,x≤mt,∀t,x∈t0−γ,t0+γ×Bεξ. This condition guarantees that, for any measurable curve φ:t0−γ,t0+γ⟶Bεξ, the measurable function ft,φt,t∈t0−γ,t0+γ is integrable in the intervals t0−γ,t0andt0,t0+γ by ft,φt≤mt,∀t∈t0−γ,t0+γ.Definition 1.
Let a pointt0,ξ∈Ω be selected and let ℛδ,εt0,ξ≔t0−δ,t0+δ×Bεξ⊂Ω,0<δ,ε be a cylinder. Let f:Ω⟶Rn be a measurable function. Then, we will call f locally t-integrable at a point t0,ξ∈Ω if there exists a cylinder ℛδ,εt0,ξ⊂Ω,0<δ,ε and an dominating integrable function (D. I. F.) m:t0−δ,t0+δ⟶R+\0 to f on the cylinder ℛδ,εt0,ξ such that ft,x≤mt∀t,x∈ℛδ,εt0,ξ.Theorem 1.
(Caratheodory). Letf:S⟶Rn be measurable in t for each fixed x, and let it be continuous in x for each fixed t, ∀t,x∈S. Let t0,ξ∈S be a fixed point and let a cylinder ℛδ,εt0,ξ⊂S exist with a dominating integrable function (D.I.F.) m:t0−δ,t0+δ⟶R+\0 to f on the cylinder ℛδ,εt0,ξ. Then, there exists a solution φ of problem (E) in an extended sense in an interval t0−β,t0+β,0<β≤δ, such that t,φt∈ℛδ,εt0,ξ,∀t∈t0−β,t0+β and φt0=ξ.
Caratheodory actually proved the existence on an intervalt0,t0+β interval and used this result to prove the existence on t0−β,t0 by using suitable transformations of symmetry.
Using Caratheodory’s theorem we can prove the existence of solution of the extended impulsive differential equation equation onν-scale with right side (19), precisely.Corollary 2.
Letfν:Nλν×Ω0⟶Rnandgν:Nτν×Ω0⟶Rn; hence, let hν≔fνχNλν+gνχNτν be measurable in σ for each fixed x, and let it be continuous in x for each fixed σ, ∀σ,x∈Ω. Let σ0,ξ0∈Ω be a fixed point and let a cylinder ℛδ,εσ0,ξ0⊂Ω exist with a D.I.F. m:σ0,σ0+δ⟶R+\0 on ℛδ,εσ0,ξ0 to hν. Then, there exists an interval σ0,σ0+β,0<β≤δ for the equation(21)φσ=ξ0+∫σ0σfνv,φvχNλν+gνv,φvχNτνdν=ξ0+∫σ0σhνv,φvdν,such that equation (21) has a solution φ in that interval such that σ,φσ∈ℛδ,εσ0,ξ,∀σ∈σ0,σ0+β and φσ0=ξ0.Corollary 3.
Lethν defined by equation (15) be ℬλ,a,bν,cν-measurable as a function of σ for each fixed x, ∀σ,x∈a,bν×Ω0 in addition to the conditions of Corollary 2. Then, the solution of initial value problem φσ0=ξ0,σ0,ξ0∈Nλν×Ω0 for equation (17) exists on an interval σ0,σ0+β for a suitable β>0 and xλ≔φμλ,+ is a solution of the initial value problem xλt0=ξ0 with t0=μ^λσ0 imposed on the equation:(22)φμλ,+t=ξ0+∫σ0μλ,+thνv,φvdνλν=ξ0+∫t0thνμλ,+s,φμλ,+sdνλ⇒xλt=ξ0+∫t0thλμ^λμλ,+s,xλsdνλ=ξ0+∫t0thλs,xλsdνλ.
This identity follows from equation (13). Details will be discussed later.Remark 1.
Note that the conditiont0∉Dλ used in Corollary 3 is in the Bainovian initial value problem (4). Therefore, the Bainovian case with measurable right side is covered by this simple example.
Note also thathν can be ℬa,bν,ν-measurable which is not covered in this corollary.Remark 2.
With Corollary2, the discussion about general existence theorem has been finished. The rest of this paper will target to include the solutions of equations with ℬa,bν,ν-measurable right sides and the formulation of the conditions for the existence of solutions in terms of the measurable functions fλ and gτ. 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 theν-scale with the help of suitable transformations. We will develop some extensions of the mappings μλ,−λ,μλ,+λ,andμ^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 t0,φt0=t0,ξ0∈Ω exists on an interval t0,t0+δ⊂T if the conditions of one of the existence theorems hold ∀t0,ξ0∈Ω.
Let us consider the impact of this condition on an example (originating from [9]). Let the differential equation be as follows:(23)y′=0,−∞<t<−1,2yt,−1≤t<0,y∈R,0,0≤t<∞.
The differential equation fulfills local Lipschitz condition∀t,y∈R2 on an interval t,t+δt,δt>0.
LetI≔−∞,−1∪0,∞; then, for any t0,y0∈I×R,L=1 is a suitable choice as Lipschitz constant in t0,t0+δt0×Bδt0y0⊂t0,t0+δt0×Bδt0y0¯⊂I×R, with a suitable δt0>0.
Ift0,y0∈−1,0×R and t0,t0+δt0⊂t0,t0+δt0⊂−1,0, then L=max2/tt,y∈t0,t0+δt0×Bδt0y0¯ will serve as the Lipschitz constant. The right side fulfills a local Lipschitz condition in a suitable neighbourhood t0,t0+δt0×Bδt0y0 at any t0,y0∈R2. Hence, no solution trajectory split into two or more trajectories at any point t>t0.
However, the right side does not fulfill Lipschitz condition in any intervala,0,a<0. The solutions of all initial value problems y−1=y1∈R will pass through φ0;−1,y1=0 by the formula φt;−1,y1=y1/−12t2. Therefore, merging of solution trajectories can occur while splitting of trajectories is excluded by having local Lipschitz condition at each point t,y∈R2 in an interval t,t+δt×Bδty. However, note that if φ1t1=y1andφ2t2=y2∈R,t1,y1≠t2,y2 then the two global solutions are different by the initial value problems even if φ1s=φ2s holds at an s>maxt1,t2∈R⇒φ1t=φ2t,∀t≥s. Hence, the global solutions of two different initial value problems are two different trajectories (not necessarily disjoint trajectories).


### 2.3. Timescales and Their Density Functions

This 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, givena,bλ,Nλλ,Nτλ,λλ,andτλ, we obtain a,bν,Nλν,Nτν,andνλν. We now show the reverse order.


#### 2.3.1. Thet-Scale from ν-Scale

We show now thata,bν,Nλν,Nτν,andν determines a,bλ,Nλλ,Nτλ,λλ,andτλ. Since NλνandNτν are measurable sets and a,bν is bounded, the characteristic functions of NλνandNτν are ν-integrable. Let the time scale interval be a,bλ with bλ≔a+∫abνχNλνdν,(24)μ^λs≔a+∫asχNλνdν∈a,bλ,∀s∈a,bν,μλ,−λs≔infμ^λ−1s∈a,bν,∀s∈a,bλ,μλλs≔μλ,+λs≔supμ^λ−1s∈a,bν,∀s∈a,bλ,μ^λ⋅μλ,−λt=μ^λ⋅μλ,−λt=ida,bλ,νλs=a+∫aμλ,+λs1dνν=a+∫aμλ,+λsχNλν+χNτνdνν=a+λλa,s+τλa,s,∀s∈a,bλ,where νλ is right continuous. Since νλ is strictly ascending, Nλλ=νλ−1NλνandNτλ=νλ−1Nτν and τλ is singular with respect to λλ and λλ is singular with respect to τλ by(25)λλNτλ=0=τλNλλ.SH≔Dλ≔tμλ,−t≠μλ,+t,t∈a,bλ is the countable set of discontinuity points in t-scale. From now on, we will use Dλ in the place of SH.Hence, based on equation (22) with hν,xλ≔φμλ,+,andt0≔μ^λσ0 and by equation (17) χNλλdνλ=dλλandχNτλdνλ=dτλ, gives us a t-scale-based impulsive differential equation:(26)xλt=ξ0+∫σ0μλ,+thνμλ,−v,xλvdνλν=ξ0+∫t0thνμλ,+s,xλsdνλ=ξ0+∫t0tfνμλ,+s,xλsdλλ︸absolute continuous+gνμλ,+s,xλsdτλ︸singular,∀t∈a,bλ.It is important to note that theν-scale concepts define a second impulsive system with τ.


#### 2.3.2. Theτ-Scale from ν-Scale

We show now thata,bν,Nλν,Nτν,andν determine a,bτ,Nλτ,Nττ,λτ,andττ. As stated in Section 2.3.1, the characteristic functions of NλνandNτν are ν-integrable. Let the time scale interval be a,bτ with bτ≔a+∫abνχNτνdν,(27)μ^τs≔a+∫asχNτνdν∈a,bτ,∀s∈a,bν,μτ,−s≔infμ^τ−1s∈a,bν,∀s∈a,bτ,μτs≔μτ,+s≔supμ^τ−1s∈a,bν,∀s∈a,bτ,μ^τ⋅μτ,−τt=μ^τ⋅μτ,−τt=ida,bτ,ντs=a+∫aμτ,+τs1dνν=a+∫aμτ,+τsχNλν+χNτνdνν=a+λτa,s+ττa,s,∀s∈a,bτ,where ντ is right continuous.Sinceντ is strictly ascending, hence bijective, Nλτ=ντ−1Nλν,Nττ=ντ−1Nτν, and(28)λτNττ=0=ττNλτ,which means ττ is singular with respect to λτ and λτ is singular with respect to ττ.Dτ≔tμτ,−t≠μτ,+t,t∈a,bτ is the countable set of discontinuity points in τ-scale. Hence, based on equation (22) with hν,xτ≔φμτ,+,andϑ0≔μ^τσ0 and based on equations (17) and (27), χNλτdντ=dλτ,andχNττdντ=dττ, gives us a τ-scale-based impulsive differential equation:(29)xτϑ=ξ0+∫σ0μτ,+ϑhνμτ,−v,xτvdντν=ξ0+∫ϑ0ϑhνμτ,+s,xτsdντ=ξ0+∫ϑ0ϑfνμτ,+s,xτsdλτ︸singular+gνμτ,+s,xτsdττ︸absolut continuous,∀ϑ∈a,bτ.


#### 2.3.3. Interpretation of the Two Representations

We need an interpretation of the two representations (26) and (29) of the Bainovian impulsive systems. The 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. The 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 f-dynamics. Thus, these systems operate in action process/reaction-counter action processes, which is a deep principle in interactions in sciences.Theν-scale representation presents both processes in full details in a time-sharing system. The ν-scale process is presented in equation (30). The ν-scale process is absolute continuous and makes it possible to prove existence theorems using Caratheodory’s techniques.In our presentation, we use the model in equations (1), (4), and (10). The fν-dynamics operates in intervals μλ,−ti,μλ,+ti+1⊂Nλν,∀ti,ti+1∈Dλ while the gν-dynamics operates in intervals μλ,−ti,μλ,+ti⊂Nτν,∀ti∈Dλ. Hence, the ν-scale consists of connected intervals alternating between fν-dynamics and gν-dynamics. The actions of the fν-dynamics and gν-dynamics are as indicated by the column headings in equation (30). The impulse points selected are ti,ti+1,ti+2,andti+3∈Dλ,ti<ti+1<ti+2<ti+3:(30)fν,χNτν=0,τ=const.gν,χNλν=0,t=const.⋯μλ,+λti,μλ,−λti+1⟶μλ,−λti+1,μλ,+λti+1⟶μλ,+λti+1,μλ,−λti+2⟶μλ,−λti+1,μλ,+λti+2⟶μλ,+λti+2,μλ,−λti+3⟶μλ,−λti+2,μλ,+λti+3⋯ti,ti+1,ti+2,ti+3∈Dλ.This scheme of operation follows the rules of timed chess game. The players arefν-dynamics and gν-dynamics. Their clocks are tandτ, respectively. If a solution of equations (1) and (4) is φ:ti,ti+δ⟶Ω with a suitable δ>ti+3−ti, then fν plays on the interval μλ,+λti,μλ,−λti+1 for a period of ti+1−tit-time while gν waiting with stopped τ-time till fν produces φti+1−0. Actions of fν are unknown to gν. At ti+1 the game switches to gν, the t-clock stops, τ-clock operates, and gν performs its job for a period of μλ,+λti+1−μλ,−λti+1 of τ-time. The actions of gν are not known by fν. This continues in this order until the solution exits. Note that the intervals are left-closed right-open by the fact that the new player starts to play at the leftmost point of its domain.Therefore, in thet-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.


#### 2.3.4. 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 initial value problem as presented in the Bainovian model, (1), (4), and (10) requires some clarifications. The Bainovian model discussed in Section 2.3.3 has form (4) or more likely (10) and the initial time t0∈a,bλ\Dλ cannot be a discontinuity point. From the analysis of equation (30) follows that any change between fν⟶gν or gν⟶fν takes place starting from the leftmost point of the domain/interval of the new dynamics. Hence, μλ,+λti,μλ,−λti+1⊂Nλν is in the domain of fν and μλ,−λti+1,μλ,+λti+1⊂Nτν is in the domain of gν in the ti,ti+1 interval, ∀ti,ti+1∈Dλ. Therefore, ∀σ0∈a,bν either σ0∈Nλν⇒μ^λσ0=t0∈Nλλ⊂a,bλ or σ0∈Nτν⇒μ^τσ0=ϑ0∈Nττ⊂a,bτ.Using the left closed right open intervals in equation (30) we obtain a pair of impulsive differential equations such that any initial value problem σ0,ξ∈a,bν×Ω0,φσ0=ξ has a solution on an interval σ0,σ0+δ0⊂Nλν if σ0∈Nλν or has a solution on an interval σ0,σ0+δ0⊂Nτν if σ0∈Nτν. 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 xλ=φ⋅μλ,+andxτ=φ⋅μτ,+, we simply can define the initial value problems as follows:(31)σ0∈Nλν:⇒t0≔μ^λσ0andσ0=μλ,+t0;ξ0,λ≔φμλ,+t0=ξ0ϑ0≔μ^τσ0;ξ0,τ≔φμτ,+ϑ0hence letσ0,τ≔μτ,+ϑ0.Similarly, for the caseσ0∈Nτν,(32)σ0∈Nτν:ϑ0≔μ^τσ0andσ0=μτ,+ϑ0⇒ξ0,τ≔φμτ,+ϑ0=ξ0t0≔μ^λσ0;ξ0,λ≔φμλ,+t0hence letσ0,λ≔μλ,+t0.


#### 2.3.5. Integral Transformations amongt-, τ-, and ν-Scales

The details of the assertions in this section come from paper [10].We summarize the mappings betweent-scale to ν-scale listed in Section 2.3.1 (Table 1).Table 1
The mappings in the first column are strictly ascending left and right continuous versions ofνλ while the mapping in the second column is absolute continuous and ascending.
a,bλ⟶a,bνa,bλ⟷a,bνa,bν⟶a,bλRelation:μλ,−λμλλ≔μλ,+λμ^λμ^λ⋅μλ,−λ=ida,bλμ^λ⋅μλ,+λ=ida,bλThe mappingsτλ,andνλ are not continuous ascending functions, with a common set of discontinuity points Dλ. Therefore, the measures τλandνλ are defined on the semiring Pν,a,bλ,c=s,ts,t∈a,bλ\Dλ,s≤t and the measures τλs,t≔τλt−τλsandνλs,t≔νλt−νλs,∀s,t∈Pν,a,bλ,c can be extended to ℬν,a,bλ,c=σPν,a,bλ,c.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν map the set of discontinuity points Dλ into the set of left-closed right-open intervals:(33)Dλν≔μλ,−t,μλ,+tt∈Dλ,and the set of discontinuity points in a,bν is(34)Dλν≔∪t∈Dλμλ,−t,μλ,+t⊂a,bν.Moreover, the mappingsμλ,−λ,μλ,+λ are bijective on the set of continuity points a,bν\Dλν and μλ,−t=μλ,+t,∀t∈a,bλ\Dλ.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν transform s,t∈Pν,a,bλ,c⟶νλs,νλt=μλ,−λs,μλ,−λt=μλ,+λs,μλ,+λt∈Pν,a,bν,cν by their continuity points with Pν,a,bν,cν≔s,ts,t∈a,bν\Dλν.The measureνλν on Pν,a,bν,cν is defined by νλνu,v≔v−u=μλ,−μ^λλv−μλ,−μ^λλu=νλμ^λλu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bλ,c, then μλ,−s,μλ,−t∈Pν,a,bν,cν and νλνμλ,−s,μλ,−t=μλ,−t−μλ,−s=νλs,t. Let the smallest σ-algebra containing the semiring Pλ,a,bν,cν be ℬλ,a,bν,cν≔σPλ,a,bν,cν with the extended measure νλν on it.From this follows that ifh:a,bν⟶R is ℬλ,a,bν,cν-measurable and νλν-integrable, then(35)∫aμλ,+λthdνλν=∫ath⋅μλ,−λdνλ,∀t∈a,bλ.Conversely, ifh:a,bλ⟶R is ℬλ,a,bν,c-measurable and νλ-integrable, then(36)∫aσh⋅μ^λdνλν=∫aμ^λσhdνλ,∀σ∈a,bν\Dλν.These are some of the main conclusions from paper [10] presented here in a condensed form.We summarize the mappings betweenτ-scale and ν-scale listed in Section 2.3.2 (Table 2):Table 2
The mappings in the first column are strictly ascending left and right continuous versions ofντ while the mapping in the second column is absolute continuous and ascending.
a,bτ⟷a,bνa,bτ⟷a,bνa,bν⟶a,bτRelation:μτ,−τμττ≔μτ,+τμ^τμ^τ⋅μτ,−τ=ida,bτμ^τ⋅μτ,+τ=ida,bτThe mappingsλτandντ are not continuous ascending functions, with a common set of discontinuity points Dτ. Therefore, the measures λτandντ are defined on the semiring Pν,a,bτ,c=s,ts,t∈a,bτ\Dτ,s≤t and the measures λτs,t≔λτt−λτsandντs,t≔ντt−ντs,∀s,t∈Pν,a,bτ,c can be extended to ℬν,a,bτ,c=σPν,a,bτ,c.The mappingsμτ,−τ,μτ,+τ:a,bτ⟶a,bν map the set of discontinuity points Dτ into the set of left-closed right-open intervals:(37)Dτν≔μτ,−t,μτ,+tt∈Dτ,and the set of discontinuity points in a,bν is(38)Dτν≔∪t∈Dτμτ,−t,μτ,+t⊂a,bν.Moreover, the mappingsμτ,−τandμτ,+τ are bijective on the set of continuity points a,bν\Dτν and μτ,−t=μτ,+t,∀t∈a,bτ\Dτ.The mappingsμτ,−τ,μτ,+τ:a,bτ⟶a,bν transform s,t∈Pν,a,bτ,c⟶ντs,ντt=μτ,−τs,μτ,−τt=μτ,+τs,μτ,+τt∈Pν,a,bν,cν by their continuity points with Pν,a,bν,cν≔s,ts,t∈a,bν\Dλν.The measureντν on Pν,a,bν,cν is defined by ντνu,v≔v−u=μτ,−μ^ττv−μτ,−μ^ττu=ντμ^ττu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bτ,c then μτ,−s,μτ,−t∈Pν,a,bν,cν and ντνμτ,−s,μτ,−t=μτ,−t−μτ,−s=ντs,t. Let the smallest σ-algebra containing the semiring Pτ,a,bν,cν be ℬτ,a,bν,cν≔σPτ,a,bν,cν with the extended measure ντν on it.From this follows that ifh:a,bν⟶R is ℬτ,a,bν,cν-measurable and ντν-integrable, then(39)∫aμτ,+τthdντν=∫ath⋅μτ,−τdντ,∀t∈a,bτ.Conversely, ifh:a,bτ⟶R is ℬτ,a,bν,c-measurable and ντ-integrable, then(40)∫aσ0h⋅μ^τdντν=∫aμ^τσ0hdντ,∀σ∈a,bν\Dτν.These are some of the main conclusions of paper [10] presented here in a condensed form.The details about measures and Radon–Nikodym derivatives summarised below are in paper [10].Letℬa,bν,ν denote the Borel sets on a,bν, and let ν be the Lebesgue measure on the σ-algebra ℬa,bν,ν. Then, ℬλ,a,bν,cν⊂ℬa,bν,νandℬτ,a,bν,cν⊂ℬa,bν,ν.Letf:a,bν⟶Rn be a ℬa,bν,ν-measurable ν-integrable function. Let νf,λA≔∫Afdν,∀A∈ℬλ,a,bν,cν and νf,τA≔∫Afdν,∀A∈ℬτ,a,bν,cν be signed measures absolute continuous with respect to the measures νλνandντν, respectively. Then, their Radon–Nikodym derivatives with respect to νλνandντν give(41)f^ν,λ≔dνf,λdνλν⟺∫Afdν=∫Af^ν,λdνλν,∀A∈ℬλ,a,bν,cν,f^ν,λ∈L1νλ,ℬλ,a,bν,cν,(42)f^ν,τ≔dνf,τdντν⟺∫Afdν=∫Af^ν,τdντν,∀A∈ℬτ,a,bν,cν,f^ν,τ∈L1ντν,ℬτ,a,bν,cν.Combining equations (35) and (41) gives(43)∫aμλ,+λtfdν=∫aμλ,+λtf^ν,λdνλν=∫atf^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ.Combining equations (39) and (42) gives(44)∫aμτ,+τϑfdν=∫aμτ,+τϑf^ν,τdντν=∫aϑf^ν,τ⋅μτ,−τdντ,∀ϑ∈a,bτ.It was proved in paper [40] that the Radon–Nikodym derivatives f^ν,λandf^ν,τ fulfill the relations:(45)fs=f^ν,λs,a.e.s∈a,bν\Dλν,fs=f^ν,τs,a.e.s∈a,bν\Dτν,which implies the assertions below as follow ups of the indicated equation (45):(46)∫aμλ,+λtf×χa,bν\Dλνdν=∫aμλ,+λtf^ν,λ×χa,bν\Dλνdνλν=43∫aμλ,+λtf×χa,bν\Dλνdνλν=∫atf×χa,bλ\Dλ⋅μλ,−λdνλ,∀t∈a,bλ,(47)∫aμτ,+τϑf×χa,bν\Dτνdν=∫aμτ,+τϑf^ν,τ×χa,bν\Dτνdντν=43∫aμτ,+τϑf×χa,bν\Dτνdντν=∫aϑf×χa,bτ\Dτ⋅μτ,−τdντ,∀ϑ∈a,bτ.This relation enables us to transform the solution of aν-scale differential equation into solutions of a t-scale/τ-scale impulsive differential equations.The functionf can be written as f=f×χa,bν\Dλν+f×χDλν by a,bν=a,bν\Dλν∪Dλνand∅=a,bν\Dλν∩Dλν.Similarly,f=f×χa,bν\Dτν+f×χDτν by a,bν=a,bν\Dτν∪Dτνand∅=a,bν\Dτν∩Dτν.Therefore, equations (46) and (47) can be rewritten as(48)∫aμλ,+λtfdν=∫aμλ,+λtf×χa,bν\Dλν+f×χDλνdν=∫atf×χa,bλ\Dλ+f×χDλν^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ,∫aμλ,+λtfdν=∫aμτ,+τϑf×χa,bν\Dτν+f×χDτνdν=∫aϑf×χa,bτ\Dτ+f×χDτν^ν,τ,∀ϑ∈a,bτ.The sets inDλν and in Dτν are atoms as described in Lemma 2.9 in [10], in detail. Since an atom in ℬλ,a,bν,cν does not have any proper measurable subset in ℬλ,a,bν,cν but the same set is a nonatomic measurable set in ℬa,bν,ν, f can be integrated on it by ν. Similarly, an atom in ℬτ,a,bν,cν does not have any proper measurable subset in ℬτ,a,bν,cν but the same set is a nonatomic measurable set in ℬa,bν,ν, and f can be integrated on it by ν. Hence, ∀tj∈Dλ the set μλ,−tj,μλ,+tj∈Dλν is an atom. Similarly, ∀τj∈Dτ the set μτ,−τj,μτ,+τj∈Dτν is an atom. Therefore, the Radon–Nikodym derivatives in equation (48) can be written as follows:(49)f×χDλν^ν,λtj≔∫μλ,−tjμλ,+tjfdνμλ,+tj−μλ,−tj,f×χDτν^ν,ττj≔∫μτ,−τjμτ,+τjfdνμτ,+τj−μτ,−τj.


### 2.4. Existence of the Solutions ont-scale and on τ-Scale

In this section, we want to formulate the existence theorem for the pair of impulsive differential equations on thet-scale and τ-scale.First, we assume that Corollary2 is true. The initial value problem in equation (21) has a solution φ:σ0,σ0+β⟶ℛδ,εσ0,ξ,φσ0=ξ.Applying the statements in equations (43) and (44) to equation (21), we get that the solution in t-scale can be xλt=φ⋅μλ,+λt,t∈t0,μ^λσ0+β,t0=μ^λσ0 and in τ-scale it can be xτϑ=φ⋅μτ,+τϑ,ϑ∈ϑ0,μ^τσ0+β,ϑ0=μ^τσ0, which transformed ν-scale solutions. We have to prove that they fulfill the initial value problems with respect to the t-scale and τ-scale versions and that also they fulfill the respective impulsive differential equations.The initial value problemφσ0=ξ on the ν-scale problem fulfills either σ0∈Nλν, which is detailed out in equation (31), or it fulfills σ0∈Nτν, which is detailed out in (32). Hence, the two cases of initial value problems fulfilled by the pair of impulsive differential equations are as follows:Whenσ0∈Nλν, t0≔μ^λσ0andξ0,λ≔φμλ,+t0=ξ and ϑ0≔μ^τσ0andξ0,τ≔φμτ,+ϑ0Whenσ0∈Nτν, ϑ0≔μ^τσ0andξ0,τ≔φμτ,+ϑ0=ξ while t0≔μ^λσ0andξ0,λ≔φμλ,+t0Note that the following equations (31) and (32) for the initial value problems are given in t-scale uniformly t0,ξ0,λ and similarly for τ-scale uniformly ϑ0,ξ0,τ, but the values are obtained differently according to equations (31) and (32) (the difference is whether ξ0,λ=ξ or ξ0,τ=ξ).In these equations, we used thatμλ,+:Nλλ⟶Nλν and μτ,+:Nττ⟶Nτν which are bijective strictly ascending mappings. Hence, the initial conditions are fulfilled.Solution of the pair of impulsive differential equations: let us apply relations (43) and (44) to solution (21) both on t-scale and on τ-scale with hν:(50)φμλ,+λt=ξ0+∫σ0μλ,+λthνv,φvdν=ξ0+∫σ0μλ,+λthν^ν,λvdνλν=ξ0,λ+∫t0thν^ν,λμλ,−λdνλ,∀t∈a,bλ,(51)φμτ,+τϑ=ξ0+∫σ0μτ,+τϑhνv,φvdν=ξ0+∫σ0μτ,+τϑhν^ν,τvdντν=ξ0,τ+∫ϑ0ϑhν^ν,τμτ,−τdντ,∀ϑ∈a,bτ.Let us splithν on ν-scale with Dλν into a component on continuity points hλ,cν≔hν×χa,bν\Dλν and a component on discontinuity points hλ,dν≔hν×χDλν. We can do this with discontinuity points Dτν on ν-scale also. A component on continuity points is hτ,cν≔hν×χa,bν\Dτν, and a component on discontinuity points is hτ,dν≔hν×χDτν:(52)hν=hλ,cν+hλ,dν,hν=hτ,cν+hτ,dν.Let us put the definitionhν≔fνχNλν+gνχNτν into equation (52):(53)hν≔fνχNλν+gνχNτν×χa,bν\Dλν+fνχNλν+gνχNτν×χDλν,hν≔fνχNλν+gνχNτν×χa,bν\Dτν+fνχNλν+gνχNτν×χDτν.ConsideringNλν⊂a,bν\DλνandNτν⊂a,bνDτν, Nλν∩Nτν=∅ and Dλν⊂NτνandDτν⊂Nλν leads to(54)hν≔fνχNλν+gνχNτν×χa,bν\Dλν︸hλ,cν+gν×χDλν︸hλ,dν,hν≔fνχNλν×χa,bν\Dτν+gνχNτν︸hτ,cν+fν×χDτν︸hτ,dν.Let us apply the relations in equations (50) and (51) to hλ,dνandhτ,dν, as expressed in equation (54) (cases of discontinuity):(55)∫σ0μλ,+λthλ,ddν=∫σ0μλ,+λtgν×χDλνv,φvdν=∫σ0μλ,+λtgν×χDλν^ν,λv×χa,bν\Dλνdνλν=∫t0tgν×χDλν^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ,(56)∫σ0μλ,+λthτ,ddν=∫σ0μτ,+τϑfν×χDτνv,φvdν=∫σ0μτ,+τϑfν×χDτνν,τ^v×χa,bν\Dτνdντν=∫ϑ0ϑfν×χDτνν,τ^⋅μτ,−τdντν,∀ϑ∈a,bτ.Let us apply equations (46) and (47) to the components hλ,candhτ,c using equation (54):(57)∫σ0μλ,+λthλ,cνv,φvdν=∫σ0μλ,+λthν^ν,λv×χa,bν\Dλνdνλν=43∫σ0μλ,+λthλ,cνv,φvdνλν=∫t0tfνχNλν⋅μλ,−λ+gνχNτν×χa,bν\Dλν∘μλ,−λdνλ,∀t∈a,bλ,(58)∫σ0μτ,+τϑhτ,cνv,φvdν=∫σ0μτ,+τϑhτ,cν^ν,τv×χa,bν\Dτνdντν=43∫σ0μτ,+τϑhτ,cνv,φvdντν=∫ϑ0ϑfνχNλν×χa,bν\Dτν∘μτ,−τ+gνχNτν∘μτ,−τdντ,∀ϑ∈a,bτ,where equation (45) is obtained from Corollary 2 and Theorem 1 in [40].The Radon–Nikodym derivatives ofhλ,dνandhλ,dν, remains to be determined. The domains of the Radon–Nikodym derivatives are countable unions of pairwise disjoint atoms of the σ-algebras ℬλ,a,bν,cνandℬτ,a,bν,cν, respectively, as defined in equations: Dλν in (11); Dτν in (37). The lists of these atoms are defined: Dλν in (12) and Dτν in (38). This means that the Radon–Nikodym derivatives are fully determined on Dλν if they are determined on the intervals in Dλν, and they are fully determined on Dτν if they are determined on each interval in Dτν.Let impulse time pointstj∈Dλ and τj∈Dτ be selected, and let φ:t0,t0+β⟶Bε0x0 be the solution of equation (21).Based on equation (54), hλ,dνandhτ,dν is replaced by gν×χDλνandfν×χDτν, respectively, in equation (49). This leads to(59)gν×χDλν^ν,λtj,φ≔∫μλ,−tjμλ,+tjgνdνμλ,+tj−μλ,−tj=φμλ,+λtj−φμλ,−λtjμλ,+λtj−μλ,−λtj≕g0^λνtj,φ,∀tj∈Dλ,(60)fν×χDτν^ν,ττj≔∫μτ,−τjμτ,+τjfνdνμτ,+τj−μτ,−τj=φμτ,+τtj−φμτ,−τtjμτ,+τj−μτ,−τj≕f0^τντj,φ,∀τj∈Dτ.LetTa,bν denote all the solution trajectories with domains as subsets of a,bν.With equations (59) and (60), g0^λν is defined on the interval μλ,−λtj,μλ,+λtj,∀tj∈Dλ and f0^λν is defined on the interval μτ,−ττj,μτ,+ττj,∀tj∈Dτ; hence, g0^λν is defined on Dλν and f0^τν is defined on Dτν. Then, we define g^λν&f^τν as follows:(61)g^λνσ,φ≔g0^λνσ,φ,μ^λσ∈Dλ,0,σ∉Dλν,∀σ,φ∈a,bν×Ta,bν,f^τνσ,φ≔f0^τνσ,φ,μ^τσ∈Dτ,0,σ∉Dτν,∀σ,φ∈a,bν×Ta,bν.These jumps in this case are dependent on the solution and the time parameter is based ona,bν. If however, the right sides fulfill Lipschitz condition then the solution is determined by tj,φtj; hence, g^λtj,φ and f^ττj,φ are fully determined by the initial value problems φtj=y,tj,y∈tj×Ω0 and φτj=y,τj,y∈τj×Ω0; hence, the jumps can be written g^λtj,φ⟶g^λtj,φtj and f^ττj,φτj⟶f^ττj,φτj which is Bainov’s formulation. Combining equations (55) and (57)equation and (56) about hλ,cνandhτ,cν with hλ,dνandhτ,dν and equations (59) and (60) on the Radon–Nikodym derivatives results in equations(62)φμλ,+λt=ξ0+∫σ0μλ,+λthλ,cν+hλ,dνv,φvdν=ξ0,λ+∫t0tfνμλ,−λ,φμλ,−λχNλλ+gνμλ,−λ,φμλ,−λχNτλ×χa,bλDλdνλ+∫t0tg^λνμλ,−λ,φdνλ,∀t∈a,bλ,(63)φμτ,+τϑ=ξ0+∫σ0μτ,+τϑhτ,cν+hτ,dνv,φvdν=ξ0,τ+∫ϑ0ϑfνμτ,−τ,φμτ,−τχNλτ×χa,bτ\Dτ+gνμτ,−τ,φμτ,−τχNττdντ+∫ϑ0ϑf^τνμτ,−λ,φdντ,∀ϑ∈a,bτ.Let us put into equations (62) and (63) the expressions xλ≔φμλ,−λandxτ≔φμτ,−τ, and from equation (17) χNλλdνλ=dλ,andχNττdντ=dτ. This leads to(64)xλt=ξ0,λ+∫t0tfνμλ,−λ,xλdλλ︸absolute continuous+∫t0tgνμλ,−λ,xλ×χa,bλ\Dλ︸continuous singular+g^λμλ,−λ,φ︸pure jumping singulardτλ,∀t∈a,bλ,(65)xτϑ=ξ0,τ+∫ϑ0ϑgνμτ,−τ,xτdττ︸absolute continuous+∫ϑ0ϑfνμτ,−τ,xτ×χa,bτ\Dτ︸continuous singular+f^τμτ,−τ,φ︸pure jumping singulardλτ,∀ϑ∈a,bτ.Conclusion 2.
We proved that, from the statement of Corollary2, follows the existence of the solution of the pair of impulsive differential equations. Important to note that the function f is fully known in t-scale and in ν-scale, whereas the function g is fully known in τ-scale and ν-scale. Therefore, we will prove that if fλ is locally t-integrable ∀t,x∈a,bλ×Ω0 and gτ is locally τ-integrable ∀ϑ,x∈a,bτ×Ω0, then the corresponding hν:a,bν⟶Rn is locally ν-integrable ∀σ,x∈a,bν×Ω0 (see the definition of the concept of local t-integrability in Definition 1).Lemma 1.
Leta,b⊂R be a closed bounded interval and let ψ:a,b×Ω0⟶Rn be measurable in its variable t∈a,b for any fixed x∈Ω0, and let it be continuous in its variable x∈Ω0 for each fixed t∈a,b. If ψ is locally integrable at each t0,x0∈a,b×Ω0 then there exists an 0<ε∈R and an integrable dominator mx0:a,b⟶R+\0 such that ψt,x≤mx0t,∀t,x∈a,b×Bεx0⊂a,b×Ω0,∀t0,x0∈a,b×Ω0.Proof.
Letψ be extended to 0 outside a,b⊂R. Let t0,x0∈a,b×Ω0. By the formulation of the lemma ∃ℛδt0,x0,εt0,x0t0,x0andmt0,x0:t0−δt0,x0,t0+δt0,x0⟶R+\0 such that ψt,x≤mt0,x0t,∀t,x∈t0−δt0,x0,t0+δt0,x0×Bεt0,x0x0. Then, a,b⊂∪s∈a,bs−δs,x0,s+δs,x0. Since a,b is compact, a finite subsystem tj,x01≤j≤N can be selected such that a,b⊂∪j=1Ntj−δtj,x0,tj+δtj,x0 covers the interval a,b. Extending mtj,x0 by zero outside tj−δtj,x0,tj+δtj,x0,∀1≤j≤N, we can form a positive integrable dominator mx0t≔maxmtj,x0t1≤j≤N,∀t∈a,b, and can get a positive εx0≔minεtj,x01≤j≤N such that the positive dominator mx0:a,b⟶R+\0 fulfills ψt,x≤mx0t,∀t,x∈a,b×Bεx0x0⊂a,b×Ω0.
Theorem for the existence of solutions of a pair of impulsive differential equations: let us denote the functionsf and g on t-scale by fλ:Nλλ⟶Rnandgλ:Nτλ⟶Rn, on τ-scale by fτ:Nλτ⟶Rnandgτ:Nττ⟶Rn, and on ν-scale by fν:Nλν⟶Rnandgν:Nτν⟶Rn. Let hν≔fλ⋅μ^λχNλν+gτ⋅μ^τχNτν:a,bν⟶Rn.Lemma 2.
The following statements are equivalent:Statement 1: the mappingfλ is locally t-integrable ∀t,x∈a,bλ×Ω0 and the mapping gτ is locally τ-integrable ∀ϑ,x∈a,bτ×Ω0Statement 2: the mappinghν is locally ν-integrable ∀σ,x∈a,bν×Ω0Proof.
Statement 1⇒ Statement 2:(1)
fλ is l  Statement 2: the mapping hν is locally ν-integrable ∀σ,x∈a,bν×Ω0 and locally t-integrable ∀t0,x0∈a,bλ×Ω0⇒∃0<ελ,x0∈Randmλ,x0:a,bλ⟶R+\0 such that fλt,x≤mλ,x0t,∀t,x∈a,bλ×Bελ,x0x0⊂a,bλ×Ω0∀x0∈Ω0 by Lemma 1.(2)
gτ is locally τ-integrable ∀ϑ0,x0∈a,bτ×Ω0⇒∃0<ετ,x0∈Randmτ,x0:a,bτ⟶R+\0 such that gτϑ,x≤mτ,x0ϑ,∀ϑ,x∈a,bτ×Bετ,x0x0⊂a,bτ×Ω0,∀x0∈Ω0 by Lemma 1.(3)
By point 1 ina,bλ, let x0∈Ω0; then, ∃0<ελ,x0∈R such that fλt,x≤mλ,x0t,∀t,x∈a,bλ×Bελ,x0x0⊂a,bλ×Ω0⇒fλμ^λσ,x≤mλ,x0μ^λσ,∀σ,x∈a,bν.×Bελ,x0x0⊂a,bν×Ω0. Hence, with fνσ,x≔fλμ^λσ,x×χNλνσ,fνσ,x≤mλ,x0μ^λσ×χNλνσ,∀σ,x∈Nλν×Bελ,x0x0⊂a,bν×Ω0.(4)
By point 1 ina,bτ, let x0∈Ω0; then, ∃0<ετ,x0∈R such that gτϑ,x≤mτ,x0ϑ,∀ϑ,x∈a,bτ×Bετ,x0x0⊂a,bτ×Ω0⇒gτμ^τσ,x≤mτ,x0μ^τσ,∀σ,x.∈a,bν×Bετ,x0x0⊂a,bν×Ω0. Hence, with gνσ,x≔gτμ^τσ,x×χNτνσ,gνσ,x≤mτ,x0μ^τσ×χNτνσ,∀σ,x∈Nλν×Bετ,x0x0⊂a,bν×Ω0.(5)
By point 3,fνσ,x≤mλ,x0μ^λσ×χNλνσ in Nλν×Bελ,x0x0, and by point 4, gνσ,x≤mτ,x0μ^τσ×χNτνσ hold in Nλν×Bετ,x0x0. Then, with mν,x0σ≔maxmλ,x0μ^λσ×χNλνσ,mτ,x0μ^τσ×χNτνσ,∀σ∈a,bν and with εν,x0≔minελ,x0,ετ,x0, we obtain that hνσ,x=fνσ,x+gνσ,x≤mν,x0σ,∀σ,x∈a,bν×Bεν,x0x0.
Statement 2⇒ Statement 1: assume that ∃0<εν,x0∈Randmν,x0:a,bν⟶R+\0∀x0∈Ω0 such that hνσ,x≤mν,x0σ∀σ,x∈a,bν×Bεν,x0x0, where hν≔fνχNλν+gνχNτν:a,bν×Ω0⟶Rn.(1)
hνσ,x≤mν,x0σ⇒‖hνμλ,−t,x≤mν,x0μλ,−t holds ∀t,x∈a,bλ×Bεν,x0x0. Multiplying both sides by the characteristic function of Nλλ, we obtain that(66)hνμλ,−t,xχNλλ=fνχNλν+gνχNτνμλ,−t,xχNλλ=fνμλ,−t,xχNλλt+gνμλ,−t,xχNτλtχNλλt=fνμλ,−t,xχNλλt≤mν,x0μλ,−t∀t,x∈a,bλ×Bεν,x0x0,which proves the case for thet-scale. The case for τ-scale is word for word the same as the case of t-scale and is left to the reader.Conclusion 3.
With this we proved that the initial value problem prescribed for a pair of impulsive differential equations has a solution if Caratheodory’s condition holds forfλ:Nλλ⟶Rn and for gτ:Nττ⟶Rn. This condition is equivalent to the condition of Corollary 2. We state a final version of the existence theorem.Condition 1.
C1: letfλ:t,x∈Nλλ×Ω0⟶fλt,x∈Rn and gτ:ϑ,x∈Nττ×Ω0⟶gτϑ,x∈Rn be measurable functions of t and ϑ, for each fixed x, respectively, and let they be continuous in x for each fixed t and ϑ, respectively, ∀t,x∈Nλλ×Ω0and∀ϑ,x∈Nττ×Ω0.C2: letfνσ,x≔fλμ^λσ,xandgνσ,x≔gτμ^τσ,x∀σ,x∈a,bν×Ω0. Then, by condition C1, hνσ,x≔fνσ,xχNλνσ+gνσ,xχNτνσ is measurable in σ for each fixed x, and it is continuous in x for each fixed σ, ∀σ,x∈a,bν×Ω0.C3: letfλ and gτ be locally t- and τ-integrable on their respective domains or equivalently let hν be locally ν-integrable on a,bν×Ω0.Theorem 2.
Let the mappingsfλ,gτ,andhν fulfill Condition 1 C1, C2, and C3. Let σ0,ξ0∈a,bν×Ω0 be a fixed point and let a cylinder ℛδ,εσ0,ξ0⊂a,bν×Ω0 exist with a D.I.F. m:σ0,σ0+δ⟶R+\0 on ℛδ,εσ0,ξ0 to hν. Then, there exists an interval σ0,σ0+β,0<β≤δ such that equation (21) has a solution φ in that interval such that σ,φσ∈ℛδ,εσ0,ξ,∀σ∈σ0,σ0+β and φσ0=ξ0.Moreover, lett0=μ^λσ0∈a,bλ and ϑ0=μ^τσ0∈a,bλ. Let xλ≔φ⋅μλ,+λ:t0,μ^λσ0+β⟶Bεξ0 and let xτ≔φ⋅μτ,+τ:ϑ0,μ^τσ0+β⟶Bεξ0. Let xλt0≔φμλ,+λt0≕ξ0,λ and let xτϑ0≔φμτ,+τϑ0≕ξ0,τ. Then, the following equations hold:(67)xλt=ξ0,λ+∫t0tfλs,xλsdλλ+∫t0tgτμ^τμλ,−λs,xλs×χa,bλ\Dλ+g^νμλ,−λsdτλ,∀t∈t0,μ^λσ0+β,xτϑ=ξ0,τ+∫ϑ0ϑgτη,xτηdττ+∫ϑ0ϑfλμ^λμτ,−τη,xτη×χa,bτ\Dτ+f^νμτ,−τηdλτ,∀ϑ∈ϑ0,μ^λσ0+β.Proof.
In the theorem,fνσ,x=fλμ^λσ,x and gνσ,x=gτμ^τσ,x by 2.1, C2. Then, fνμλ,−λt,x=fλμ^λμλ,−λt,x=fλida,bλt,x=fλt,x and gνμτ,−τϑ,x=gτμ^τμτ,−τϑ,x=gτida,bτt,x=gτϑ,x, where we applied the identities listed in equations (24) and (27).Conclusion 4.
The 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 f-dynamics 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. The relationship is an action/reaction interaction between the two processes. This 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. This will make the flow of solutions to be a tree structure instead of a connected flow as in ordinary differential equations. The leaves of the tree consist of trajectories coming from discontinuity points without history. This tree is directed, with orientation from the leaves to the root which is called in-tree or antiarborescence [41]. This gives wide range of modelling facilities by enabling one to model and study mixing new generations in addition to studying flows of solutions.


## 2.1. The Extended Impulsive Differential Equations

Let the process evolve in an open time intervalT⊂R and in an open set T×Ω0≕Ω⊂T×Rn. Let f,g:Ω⟶Rn be measurable functions in the time variable t for each fixed spatial value x∈Ω0, and continuous/Lipschitz-continuous functions in the spatial variable x for each fixed t∈T.Letτ: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 NλλandNλτ as follows:(17)dλλ+τλdνλ=dλλdνλ+dτλdνλ=1⇒dλλdνλ=χNλλanddτλdνλ=χNλτ.Putting these into equation (10) and changing g˜ with g, we get the extended impulsive differential equation in t-scale as(18)xt=x0+∫t0tfs,xsχNλλ+gs,xsχNτλdνλ,t∈T,xt0=x0t0∈T\SH,x0∈Ω0.The integral transformations discussed in Section2.3.5 will give a similar result in both ν-scale and τ-scale. We will handle the ν-scale representation first. Let fν≔fs,xsχNλλ⋅μ^λν:Nλν⟶Rn and gν≔gs,xsχNτλ⋅μ^λν:Nτν⟶Rn. Then, let(19)hν≔fνχNλν+gνχNτν:a,bν×Ω0⟶Rnbe the measurable right side of the extended impulsive differential equation in ν-scale.We will use the notationsa,bλ⊂T for t-scale, a,bν for the generated ν-scale, and a,bτ for τ-scale to get the advantages of compact sets.In Section2.2, 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. The basis of our discussion is the approach presented in pg. 43 in [9].


## 2.2. Caratheodory’s Theorem

We present Caratheodory’s existence theorem inRn as it is presented in the cited pages 42-43 for one dimension.We are considering a process on an open setS⊂Ω⊂R×Rn. Let f:S⟶Rn be a function not necessarily continuous.Problem (E): find an intervalI⊂a,b and an absolute continuous function φ:I⊂a,b⟶Rn such that(20)t,φt∈S,φ′t=ft,φt,almost allt∈I.Then, the functionφ:I⟶Rn is a solution of equation (20) in the extended sense.Caratheodory’s existence theorem [9] targets finding a solutions to problem (E) with an initial value t0,ξ∈Ω,φt0=ξ, where the right side is a measurable function of t for each fixed x∈Ω0 on T×Ω0, where ∅≠α,β=T⊂RandΩ0⊂Rn are open sets. Caratheodory’s condition for the existence of the solution is the existence of a local positive integrable dominant m:t0−γ,t0+γ⊂T⟶R+\0andε>0 such that ft,x≤mt,∀t,x∈t0−γ,t0+γ×Bεξ. This condition guarantees that, for any measurable curve φ:t0−γ,t0+γ⟶Bεξ, the measurable function ft,φt,t∈t0−γ,t0+γ is integrable in the intervals t0−γ,t0andt0,t0+γ by ft,φt≤mt,∀t∈t0−γ,t0+γ.Definition 1.
Let a pointt0,ξ∈Ω be selected and let ℛδ,εt0,ξ≔t0−δ,t0+δ×Bεξ⊂Ω,0<δ,ε be a cylinder. Let f:Ω⟶Rn be a measurable function. Then, we will call f locally t-integrable at a point t0,ξ∈Ω if there exists a cylinder ℛδ,εt0,ξ⊂Ω,0<δ,ε and an dominating integrable function (D. I. F.) m:t0−δ,t0+δ⟶R+\0 to f on the cylinder ℛδ,εt0,ξ such that ft,x≤mt∀t,x∈ℛδ,εt0,ξ.Theorem 1.
(Caratheodory). Letf:S⟶Rn be measurable in t for each fixed x, and let it be continuous in x for each fixed t, ∀t,x∈S. Let t0,ξ∈S be a fixed point and let a cylinder ℛδ,εt0,ξ⊂S exist with a dominating integrable function (D.I.F.) m:t0−δ,t0+δ⟶R+\0 to f on the cylinder ℛδ,εt0,ξ. Then, there exists a solution φ of problem (E) in an extended sense in an interval t0−β,t0+β,0<β≤δ, such that t,φt∈ℛδ,εt0,ξ,∀t∈t0−β,t0+β and φt0=ξ.
Caratheodory actually proved the existence on an intervalt0,t0+β interval and used this result to prove the existence on t0−β,t0 by using suitable transformations of symmetry.
Using Caratheodory’s theorem we can prove the existence of solution of the extended impulsive differential equation equation onν-scale with right side (19), precisely.Corollary 2.
Letfν:Nλν×Ω0⟶Rnandgν:Nτν×Ω0⟶Rn; hence, let hν≔fνχNλν+gνχNτν be measurable in σ for each fixed x, and let it be continuous in x for each fixed σ, ∀σ,x∈Ω. Let σ0,ξ0∈Ω be a fixed point and let a cylinder ℛδ,εσ0,ξ0⊂Ω exist with a D.I.F. m:σ0,σ0+δ⟶R+\0 on ℛδ,εσ0,ξ0 to hν. Then, there exists an interval σ0,σ0+β,0<β≤δ for the equation(21)φσ=ξ0+∫σ0σfνv,φvχNλν+gνv,φvχNτνdν=ξ0+∫σ0σhνv,φvdν,such that equation (21) has a solution φ in that interval such that σ,φσ∈ℛδ,εσ0,ξ,∀σ∈σ0,σ0+β and φσ0=ξ0.Corollary 3.
Lethν defined by equation (15) be ℬλ,a,bν,cν-measurable as a function of σ for each fixed x, ∀σ,x∈a,bν×Ω0 in addition to the conditions of Corollary 2. Then, the solution of initial value problem φσ0=ξ0,σ0,ξ0∈Nλν×Ω0 for equation (17) exists on an interval σ0,σ0+β for a suitable β>0 and xλ≔φμλ,+ is a solution of the initial value problem xλt0=ξ0 with t0=μ^λσ0 imposed on the equation:(22)φμλ,+t=ξ0+∫σ0μλ,+thνv,φvdνλν=ξ0+∫t0thνμλ,+s,φμλ,+sdνλ⇒xλt=ξ0+∫t0thλμ^λμλ,+s,xλsdνλ=ξ0+∫t0thλs,xλsdνλ.
This identity follows from equation (13). Details will be discussed later.Remark 1.
Note that the conditiont0∉Dλ used in Corollary 3 is in the Bainovian initial value problem (4). Therefore, the Bainovian case with measurable right side is covered by this simple example.
Note also thathν can be ℬa,bν,ν-measurable which is not covered in this corollary.Remark 2.
With Corollary2, the discussion about general existence theorem has been finished. The rest of this paper will target to include the solutions of equations with ℬa,bν,ν-measurable right sides and the formulation of the conditions for the existence of solutions in terms of the measurable functions fλ and gτ. 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 theν-scale with the help of suitable transformations. We will develop some extensions of the mappings μλ,−λ,μλ,+λ,andμ^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 t0,φt0=t0,ξ0∈Ω exists on an interval t0,t0+δ⊂T if the conditions of one of the existence theorems hold ∀t0,ξ0∈Ω.
Let us consider the impact of this condition on an example (originating from [9]). Let the differential equation be as follows:(23)y′=0,−∞<t<−1,2yt,−1≤t<0,y∈R,0,0≤t<∞.
The differential equation fulfills local Lipschitz condition∀t,y∈R2 on an interval t,t+δt,δt>0.
LetI≔−∞,−1∪0,∞; then, for any t0,y0∈I×R,L=1 is a suitable choice as Lipschitz constant in t0,t0+δt0×Bδt0y0⊂t0,t0+δt0×Bδt0y0¯⊂I×R, with a suitable δt0>0.
Ift0,y0∈−1,0×R and t0,t0+δt0⊂t0,t0+δt0⊂−1,0, then L=max2/tt,y∈t0,t0+δt0×Bδt0y0¯ will serve as the Lipschitz constant. The right side fulfills a local Lipschitz condition in a suitable neighbourhood t0,t0+δt0×Bδt0y0 at any t0,y0∈R2. Hence, no solution trajectory split into two or more trajectories at any point t>t0.
However, the right side does not fulfill Lipschitz condition in any intervala,0,a<0. The solutions of all initial value problems y−1=y1∈R will pass through φ0;−1,y1=0 by the formula φt;−1,y1=y1/−12t2. Therefore, merging of solution trajectories can occur while splitting of trajectories is excluded by having local Lipschitz condition at each point t,y∈R2 in an interval t,t+δt×Bδty. However, note that if φ1t1=y1andφ2t2=y2∈R,t1,y1≠t2,y2 then the two global solutions are different by the initial value problems even if φ1s=φ2s holds at an s>maxt1,t2∈R⇒φ1t=φ2t,∀t≥s. Hence, the global solutions of two different initial value problems are two different trajectories (not necessarily disjoint trajectories).


## 2.3. Timescales and Their Density Functions

This 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, givena,bλ,Nλλ,Nτλ,λλ,andτλ, we obtain a,bν,Nλν,Nτν,andνλν. We now show the reverse order.


### 2.3.1. Thet-Scale from ν-Scale

We show now thata,bν,Nλν,Nτν,andν determines a,bλ,Nλλ,Nτλ,λλ,andτλ. Since NλνandNτν are measurable sets and a,bν is bounded, the characteristic functions of NλνandNτν are ν-integrable. Let the time scale interval be a,bλ with bλ≔a+∫abνχNλνdν,(24)μ^λs≔a+∫asχNλνdν∈a,bλ,∀s∈a,bν,μλ,−λs≔infμ^λ−1s∈a,bν,∀s∈a,bλ,μλλs≔μλ,+λs≔supμ^λ−1s∈a,bν,∀s∈a,bλ,μ^λ⋅μλ,−λt=μ^λ⋅μλ,−λt=ida,bλ,νλs=a+∫aμλ,+λs1dνν=a+∫aμλ,+λsχNλν+χNτνdνν=a+λλa,s+τλa,s,∀s∈a,bλ,where νλ is right continuous. Since νλ is strictly ascending, Nλλ=νλ−1NλνandNτλ=νλ−1Nτν and τλ is singular with respect to λλ and λλ is singular with respect to τλ by(25)λλNτλ=0=τλNλλ.SH≔Dλ≔tμλ,−t≠μλ,+t,t∈a,bλ is the countable set of discontinuity points in t-scale. From now on, we will use Dλ in the place of SH.Hence, based on equation (22) with hν,xλ≔φμλ,+,andt0≔μ^λσ0 and by equation (17) χNλλdνλ=dλλandχNτλdνλ=dτλ, gives us a t-scale-based impulsive differential equation:(26)xλt=ξ0+∫σ0μλ,+thνμλ,−v,xλvdνλν=ξ0+∫t0thνμλ,+s,xλsdνλ=ξ0+∫t0tfνμλ,+s,xλsdλλ︸absolute continuous+gνμλ,+s,xλsdτλ︸singular,∀t∈a,bλ.It is important to note that theν-scale concepts define a second impulsive system with τ.


### 2.3.2. Theτ-Scale from ν-Scale

We show now thata,bν,Nλν,Nτν,andν determine a,bτ,Nλτ,Nττ,λτ,andττ. As stated in Section 2.3.1, the characteristic functions of NλνandNτν are ν-integrable. Let the time scale interval be a,bτ with bτ≔a+∫abνχNτνdν,(27)μ^τs≔a+∫asχNτνdν∈a,bτ,∀s∈a,bν,μτ,−s≔infμ^τ−1s∈a,bν,∀s∈a,bτ,μτs≔μτ,+s≔supμ^τ−1s∈a,bν,∀s∈a,bτ,μ^τ⋅μτ,−τt=μ^τ⋅μτ,−τt=ida,bτ,ντs=a+∫aμτ,+τs1dνν=a+∫aμτ,+τsχNλν+χNτνdνν=a+λτa,s+ττa,s,∀s∈a,bτ,where ντ is right continuous.Sinceντ is strictly ascending, hence bijective, Nλτ=ντ−1Nλν,Nττ=ντ−1Nτν, and(28)λτNττ=0=ττNλτ,which means ττ is singular with respect to λτ and λτ is singular with respect to ττ.Dτ≔tμτ,−t≠μτ,+t,t∈a,bτ is the countable set of discontinuity points in τ-scale. Hence, based on equation (22) with hν,xτ≔φμτ,+,andϑ0≔μ^τσ0 and based on equations (17) and (27), χNλτdντ=dλτ,andχNττdντ=dττ, gives us a τ-scale-based impulsive differential equation:(29)xτϑ=ξ0+∫σ0μτ,+ϑhνμτ,−v,xτvdντν=ξ0+∫ϑ0ϑhνμτ,+s,xτsdντ=ξ0+∫ϑ0ϑfνμτ,+s,xτsdλτ︸singular+gνμτ,+s,xτsdττ︸absolut continuous,∀ϑ∈a,bτ.


### 2.3.3. Interpretation of the Two Representations

We need an interpretation of the two representations (26) and (29) of the Bainovian impulsive systems. The 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. The 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 f-dynamics. Thus, these systems operate in action process/reaction-counter action processes, which is a deep principle in interactions in sciences.Theν-scale representation presents both processes in full details in a time-sharing system. The ν-scale process is presented in equation (30). The ν-scale process is absolute continuous and makes it possible to prove existence theorems using Caratheodory’s techniques.In our presentation, we use the model in equations (1), (4), and (10). The fν-dynamics operates in intervals μλ,−ti,μλ,+ti+1⊂Nλν,∀ti,ti+1∈Dλ while the gν-dynamics operates in intervals μλ,−ti,μλ,+ti⊂Nτν,∀ti∈Dλ. Hence, the ν-scale consists of connected intervals alternating between fν-dynamics and gν-dynamics. The actions of the fν-dynamics and gν-dynamics are as indicated by the column headings in equation (30). The impulse points selected are ti,ti+1,ti+2,andti+3∈Dλ,ti<ti+1<ti+2<ti+3:(30)fν,χNτν=0,τ=const.gν,χNλν=0,t=const.⋯μλ,+λti,μλ,−λti+1⟶μλ,−λti+1,μλ,+λti+1⟶μλ,+λti+1,μλ,−λti+2⟶μλ,−λti+1,μλ,+λti+2⟶μλ,+λti+2,μλ,−λti+3⟶μλ,−λti+2,μλ,+λti+3⋯ti,ti+1,ti+2,ti+3∈Dλ.This scheme of operation follows the rules of timed chess game. The players arefν-dynamics and gν-dynamics. Their clocks are tandτ, respectively. If a solution of equations (1) and (4) is φ:ti,ti+δ⟶Ω with a suitable δ>ti+3−ti, then fν plays on the interval μλ,+λti,μλ,−λti+1 for a period of ti+1−tit-time while gν waiting with stopped τ-time till fν produces φti+1−0. Actions of fν are unknown to gν. At ti+1 the game switches to gν, the t-clock stops, τ-clock operates, and gν performs its job for a period of μλ,+λti+1−μλ,−λti+1 of τ-time. The actions of gν are not known by fν. This continues in this order until the solution exits. Note that the intervals are left-closed right-open by the fact that the new player starts to play at the leftmost point of its domain.Therefore, in thet-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.


### 2.3.4. 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 initial value problem as presented in the Bainovian model, (1), (4), and (10) requires some clarifications. The Bainovian model discussed in Section 2.3.3 has form (4) or more likely (10) and the initial time t0∈a,bλ\Dλ cannot be a discontinuity point. From the analysis of equation (30) follows that any change between fν⟶gν or gν⟶fν takes place starting from the leftmost point of the domain/interval of the new dynamics. Hence, μλ,+λti,μλ,−λti+1⊂Nλν is in the domain of fν and μλ,−λti+1,μλ,+λti+1⊂Nτν is in the domain of gν in the ti,ti+1 interval, ∀ti,ti+1∈Dλ. Therefore, ∀σ0∈a,bν either σ0∈Nλν⇒μ^λσ0=t0∈Nλλ⊂a,bλ or σ0∈Nτν⇒μ^τσ0=ϑ0∈Nττ⊂a,bτ.Using the left closed right open intervals in equation (30) we obtain a pair of impulsive differential equations such that any initial value problem σ0,ξ∈a,bν×Ω0,φσ0=ξ has a solution on an interval σ0,σ0+δ0⊂Nλν if σ0∈Nλν or has a solution on an interval σ0,σ0+δ0⊂Nτν if σ0∈Nτν. 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 xλ=φ⋅μλ,+andxτ=φ⋅μτ,+, we simply can define the initial value problems as follows:(31)σ0∈Nλν:⇒t0≔μ^λσ0andσ0=μλ,+t0;ξ0,λ≔φμλ,+t0=ξ0ϑ0≔μ^τσ0;ξ0,τ≔φμτ,+ϑ0hence letσ0,τ≔μτ,+ϑ0.Similarly, for the caseσ0∈Nτν,(32)σ0∈Nτν:ϑ0≔μ^τσ0andσ0=μτ,+ϑ0⇒ξ0,τ≔φμτ,+ϑ0=ξ0t0≔μ^λσ0;ξ0,λ≔φμλ,+t0hence letσ0,λ≔μλ,+t0.


### 2.3.5. Integral Transformations amongt-, τ-, and ν-Scales

The details of the assertions in this section come from paper [10].We summarize the mappings betweent-scale to ν-scale listed in Section 2.3.1 (Table 1).Table 1
The mappings in the first column are strictly ascending left and right continuous versions ofνλ while the mapping in the second column is absolute continuous and ascending.
a,bλ⟶a,bνa,bλ⟷a,bνa,bν⟶a,bλRelation:μλ,−λμλλ≔μλ,+λμ^λμ^λ⋅μλ,−λ=ida,bλμ^λ⋅μλ,+λ=ida,bλThe mappingsτλ,andνλ are not continuous ascending functions, with a common set of discontinuity points Dλ. Therefore, the measures τλandνλ are defined on the semiring Pν,a,bλ,c=s,ts,t∈a,bλ\Dλ,s≤t and the measures τλs,t≔τλt−τλsandνλs,t≔νλt−νλs,∀s,t∈Pν,a,bλ,c can be extended to ℬν,a,bλ,c=σPν,a,bλ,c.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν map the set of discontinuity points Dλ into the set of left-closed right-open intervals:(33)Dλν≔μλ,−t,μλ,+tt∈Dλ,and the set of discontinuity points in a,bν is(34)Dλν≔∪t∈Dλμλ,−t,μλ,+t⊂a,bν.Moreover, the mappingsμλ,−λ,μλ,+λ are bijective on the set of continuity points a,bν\Dλν and μλ,−t=μλ,+t,∀t∈a,bλ\Dλ.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν transform s,t∈Pν,a,bλ,c⟶νλs,νλt=μλ,−λs,μλ,−λt=μλ,+λs,μλ,+λt∈Pν,a,bν,cν by their continuity points with Pν,a,bν,cν≔s,ts,t∈a,bν\Dλν.The measureνλν on Pν,a,bν,cν is defined by νλνu,v≔v−u=μλ,−μ^λλv−μλ,−μ^λλu=νλμ^λλu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bλ,c, then μλ,−s,μλ,−t∈Pν,a,bν,cν and νλνμλ,−s,μλ,−t=μλ,−t−μλ,−s=νλs,t. Let the smallest σ-algebra containing the semiring Pλ,a,bν,cν be ℬλ,a,bν,cν≔σPλ,a,bν,cν with the extended measure νλν on it.From this follows that ifh:a,bν⟶R is ℬλ,a,bν,cν-measurable and νλν-integrable, then(35)∫aμλ,+λthdνλν=∫ath⋅μλ,−λdνλ,∀t∈a,bλ.Conversely, ifh:a,bλ⟶R is ℬλ,a,bν,c-measurable and νλ-integrable, then(36)∫aσh⋅μ^λdνλν=∫aμ^λσhdνλ,∀σ∈a,bν\Dλν.These are some of the main conclusions from paper [10] presented here in a condensed form.We summarize the mappings betweenτ-scale and ν-scale listed in Section 2.3.2 (Table 2):Table 2
The mappings in the first column are strictly ascending left and right continuous versions ofντ while the mapping in the second column is absolute continuous and ascending.
a,bτ⟷a,bνa,bτ⟷a,bνa,bν⟶a,bτRelation:μτ,−τμττ≔μτ,+τμ^τμ^τ⋅μτ,−τ=ida,bτμ^τ⋅μτ,+τ=ida,bτThe mappingsλτandντ are not continuous ascending functions, with a common set of discontinuity points Dτ. Therefore, the measures λτandντ are defined on the semiring Pν,a,bτ,c=s,ts,t∈a,bτ\Dτ,s≤t and the measures λτs,t≔λτt−λτsandντs,t≔ντt−ντs,∀s,t∈Pν,a,bτ,c can be extended to ℬν,a,bτ,c=σPν,a,bτ,c.The mappingsμτ,−τ,μτ,+τ:a,bτ⟶a,bν map the set of discontinuity points Dτ into the set of left-closed right-open intervals:(37)Dτν≔μτ,−t,μτ,+tt∈Dτ,and the set of discontinuity points in a,bν is(38)Dτν≔∪t∈Dτμτ,−t,μτ,+t⊂a,bν.Moreover, the mappingsμτ,−τandμτ,+τ are bijective on the set of continuity points a,bν\Dτν and μτ,−t=μτ,+t,∀t∈a,bτ\Dτ.The mappingsμτ,−τ,μτ,+τ:a,bτ⟶a,bν transform s,t∈Pν,a,bτ,c⟶ντs,ντt=μτ,−τs,μτ,−τt=μτ,+τs,μτ,+τt∈Pν,a,bν,cν by their continuity points with Pν,a,bν,cν≔s,ts,t∈a,bν\Dλν.The measureντν on Pν,a,bν,cν is defined by ντνu,v≔v−u=μτ,−μ^ττv−μτ,−μ^ττu=ντμ^ττu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bτ,c then μτ,−s,μτ,−t∈Pν,a,bν,cν and ντνμτ,−s,μτ,−t=μτ,−t−μτ,−s=ντs,t. Let the smallest σ-algebra containing the semiring Pτ,a,bν,cν be ℬτ,a,bν,cν≔σPτ,a,bν,cν with the extended measure ντν on it.From this follows that ifh:a,bν⟶R is ℬτ,a,bν,cν-measurable and ντν-integrable, then(39)∫aμτ,+τthdντν=∫ath⋅μτ,−τdντ,∀t∈a,bτ.Conversely, ifh:a,bτ⟶R is ℬτ,a,bν,c-measurable and ντ-integrable, then(40)∫aσ0h⋅μ^τdντν=∫aμ^τσ0hdντ,∀σ∈a,bν\Dτν.These are some of the main conclusions of paper [10] presented here in a condensed form.The details about measures and Radon–Nikodym derivatives summarised below are in paper [10].Letℬa,bν,ν denote the Borel sets on a,bν, and let ν be the Lebesgue measure on the σ-algebra ℬa,bν,ν. Then, ℬλ,a,bν,cν⊂ℬa,bν,νandℬτ,a,bν,cν⊂ℬa,bν,ν.Letf:a,bν⟶Rn be a ℬa,bν,ν-measurable ν-integrable function. Let νf,λA≔∫Afdν,∀A∈ℬλ,a,bν,cν and νf,τA≔∫Afdν,∀A∈ℬτ,a,bν,cν be signed measures absolute continuous with respect to the measures νλνandντν, respectively. Then, their Radon–Nikodym derivatives with respect to νλνandντν give(41)f^ν,λ≔dνf,λdνλν⟺∫Afdν=∫Af^ν,λdνλν,∀A∈ℬλ,a,bν,cν,f^ν,λ∈L1νλ,ℬλ,a,bν,cν,(42)f^ν,τ≔dνf,τdντν⟺∫Afdν=∫Af^ν,τdντν,∀A∈ℬτ,a,bν,cν,f^ν,τ∈L1ντν,ℬτ,a,bν,cν.Combining equations (35) and (41) gives(43)∫aμλ,+λtfdν=∫aμλ,+λtf^ν,λdνλν=∫atf^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ.Combining equations (39) and (42) gives(44)∫aμτ,+τϑfdν=∫aμτ,+τϑf^ν,τdντν=∫aϑf^ν,τ⋅μτ,−τdντ,∀ϑ∈a,bτ.It was proved in paper [40] that the Radon–Nikodym derivatives f^ν,λandf^ν,τ fulfill the relations:(45)fs=f^ν,λs,a.e.s∈a,bν\Dλν,fs=f^ν,τs,a.e.s∈a,bν\Dτν,which implies the assertions below as follow ups of the indicated equation (45):(46)∫aμλ,+λtf×χa,bν\Dλνdν=∫aμλ,+λtf^ν,λ×χa,bν\Dλνdνλν=43∫aμλ,+λtf×χa,bν\Dλνdνλν=∫atf×χa,bλ\Dλ⋅μλ,−λdνλ,∀t∈a,bλ,(47)∫aμτ,+τϑf×χa,bν\Dτνdν=∫aμτ,+τϑf^ν,τ×χa,bν\Dτνdντν=43∫aμτ,+τϑf×χa,bν\Dτνdντν=∫aϑf×χa,bτ\Dτ⋅μτ,−τdντ,∀ϑ∈a,bτ.This relation enables us to transform the solution of aν-scale differential equation into solutions of a t-scale/τ-scale impulsive differential equations.The functionf can be written as f=f×χa,bν\Dλν+f×χDλν by a,bν=a,bν\Dλν∪Dλνand∅=a,bν\Dλν∩Dλν.Similarly,f=f×χa,bν\Dτν+f×χDτν by a,bν=a,bν\Dτν∪Dτνand∅=a,bν\Dτν∩Dτν.Therefore, equations (46) and (47) can be rewritten as(48)∫aμλ,+λtfdν=∫aμλ,+λtf×χa,bν\Dλν+f×χDλνdν=∫atf×χa,bλ\Dλ+f×χDλν^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ,∫aμλ,+λtfdν=∫aμτ,+τϑf×χa,bν\Dτν+f×χDτνdν=∫aϑf×χa,bτ\Dτ+f×χDτν^ν,τ,∀ϑ∈a,bτ.The sets inDλν and in Dτν are atoms as described in Lemma 2.9 in [10], in detail. Since an atom in ℬλ,a,bν,cν does not have any proper measurable subset in ℬλ,a,bν,cν but the same set is a nonatomic measurable set in ℬa,bν,ν, f can be integrated on it by ν. Similarly, an atom in ℬτ,a,bν,cν does not have any proper measurable subset in ℬτ,a,bν,cν but the same set is a nonatomic measurable set in ℬa,bν,ν, and f can be integrated on it by ν. Hence, ∀tj∈Dλ the set μλ,−tj,μλ,+tj∈Dλν is an atom. Similarly, ∀τj∈Dτ the set μτ,−τj,μτ,+τj∈Dτν is an atom. Therefore, the Radon–Nikodym derivatives in equation (48) can be written as follows:(49)f×χDλν^ν,λtj≔∫μλ,−tjμλ,+tjfdνμλ,+tj−μλ,−tj,f×χDτν^ν,ττj≔∫μτ,−τjμτ,+τjfdνμτ,+τj−μτ,−τj.


## 2.3.1. Thet-Scale from ν-Scale

We show now thata,bν,Nλν,Nτν,andν determines a,bλ,Nλλ,Nτλ,λλ,andτλ. Since NλνandNτν are measurable sets and a,bν is bounded, the characteristic functions of NλνandNτν are ν-integrable. Let the time scale interval be a,bλ with bλ≔a+∫abνχNλνdν,(24)μ^λs≔a+∫asχNλνdν∈a,bλ,∀s∈a,bν,μλ,−λs≔infμ^λ−1s∈a,bν,∀s∈a,bλ,μλλs≔μλ,+λs≔supμ^λ−1s∈a,bν,∀s∈a,bλ,μ^λ⋅μλ,−λt=μ^λ⋅μλ,−λt=ida,bλ,νλs=a+∫aμλ,+λs1dνν=a+∫aμλ,+λsχNλν+χNτνdνν=a+λλa,s+τλa,s,∀s∈a,bλ,where νλ is right continuous. Since νλ is strictly ascending, Nλλ=νλ−1NλνandNτλ=νλ−1Nτν and τλ is singular with respect to λλ and λλ is singular with respect to τλ by(25)λλNτλ=0=τλNλλ.SH≔Dλ≔tμλ,−t≠μλ,+t,t∈a,bλ is the countable set of discontinuity points in t-scale. From now on, we will use Dλ in the place of SH.Hence, based on equation (22) with hν,xλ≔φμλ,+,andt0≔μ^λσ0 and by equation (17) χNλλdνλ=dλλandχNτλdνλ=dτλ, gives us a t-scale-based impulsive differential equation:(26)xλt=ξ0+∫σ0μλ,+thνμλ,−v,xλvdνλν=ξ0+∫t0thνμλ,+s,xλsdνλ=ξ0+∫t0tfνμλ,+s,xλsdλλ︸absolute continuous+gνμλ,+s,xλsdτλ︸singular,∀t∈a,bλ.It is important to note that theν-scale concepts define a second impulsive system with τ.


## 2.3.2. Theτ-Scale from ν-Scale

We show now thata,bν,Nλν,Nτν,andν determine a,bτ,Nλτ,Nττ,λτ,andττ. As stated in Section 2.3.1, the characteristic functions of NλνandNτν are ν-integrable. Let the time scale interval be a,bτ with bτ≔a+∫abνχNτνdν,(27)μ^τs≔a+∫asχNτνdν∈a,bτ,∀s∈a,bν,μτ,−s≔infμ^τ−1s∈a,bν,∀s∈a,bτ,μτs≔μτ,+s≔supμ^τ−1s∈a,bν,∀s∈a,bτ,μ^τ⋅μτ,−τt=μ^τ⋅μτ,−τt=ida,bτ,ντs=a+∫aμτ,+τs1dνν=a+∫aμτ,+τsχNλν+χNτνdνν=a+λτa,s+ττa,s,∀s∈a,bτ,where ντ is right continuous.Sinceντ is strictly ascending, hence bijective, Nλτ=ντ−1Nλν,Nττ=ντ−1Nτν, and(28)λτNττ=0=ττNλτ,which means ττ is singular with respect to λτ and λτ is singular with respect to ττ.Dτ≔tμτ,−t≠μτ,+t,t∈a,bτ is the countable set of discontinuity points in τ-scale. Hence, based on equation (22) with hν,xτ≔φμτ,+,andϑ0≔μ^τσ0 and based on equations (17) and (27), χNλτdντ=dλτ,andχNττdντ=dττ, gives us a τ-scale-based impulsive differential equation:(29)xτϑ=ξ0+∫σ0μτ,+ϑhνμτ,−v,xτvdντν=ξ0+∫ϑ0ϑhνμτ,+s,xτsdντ=ξ0+∫ϑ0ϑfνμτ,+s,xτsdλτ︸singular+gνμτ,+s,xτsdττ︸absolut continuous,∀ϑ∈a,bτ.


## 2.3.3. Interpretation of the Two Representations

We need an interpretation of the two representations (26) and (29) of the Bainovian impulsive systems. The 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. The 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 f-dynamics. Thus, these systems operate in action process/reaction-counter action processes, which is a deep principle in interactions in sciences.Theν-scale representation presents both processes in full details in a time-sharing system. The ν-scale process is presented in equation (30). The ν-scale process is absolute continuous and makes it possible to prove existence theorems using Caratheodory’s techniques.In our presentation, we use the model in equations (1), (4), and (10). The fν-dynamics operates in intervals μλ,−ti,μλ,+ti+1⊂Nλν,∀ti,ti+1∈Dλ while the gν-dynamics operates in intervals μλ,−ti,μλ,+ti⊂Nτν,∀ti∈Dλ. Hence, the ν-scale consists of connected intervals alternating between fν-dynamics and gν-dynamics. The actions of the fν-dynamics and gν-dynamics are as indicated by the column headings in equation (30). The impulse points selected are ti,ti+1,ti+2,andti+3∈Dλ,ti<ti+1<ti+2<ti+3:(30)fν,χNτν=0,τ=const.gν,χNλν=0,t=const.⋯μλ,+λti,μλ,−λti+1⟶μλ,−λti+1,μλ,+λti+1⟶μλ,+λti+1,μλ,−λti+2⟶μλ,−λti+1,μλ,+λti+2⟶μλ,+λti+2,μλ,−λti+3⟶μλ,−λti+2,μλ,+λti+3⋯ti,ti+1,ti+2,ti+3∈Dλ.This scheme of operation follows the rules of timed chess game. The players arefν-dynamics and gν-dynamics. Their clocks are tandτ, respectively. If a solution of equations (1) and (4) is φ:ti,ti+δ⟶Ω with a suitable δ>ti+3−ti, then fν plays on the interval μλ,+λti,μλ,−λti+1 for a period of ti+1−tit-time while gν waiting with stopped τ-time till fν produces φti+1−0. Actions of fν are unknown to gν. At ti+1 the game switches to gν, the t-clock stops, τ-clock operates, and gν performs its job for a period of μλ,+λti+1−μλ,−λti+1 of τ-time. The actions of gν are not known by fν. This continues in this order until the solution exits. Note that the intervals are left-closed right-open by the fact that the new player starts to play at the leftmost point of its domain.Therefore, in thet-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.


## 2.3.4. 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 initial value problem as presented in the Bainovian model, (1), (4), and (10) requires some clarifications. The Bainovian model discussed in Section 2.3.3 has form (4) or more likely (10) and the initial time t0∈a,bλ\Dλ cannot be a discontinuity point. From the analysis of equation (30) follows that any change between fν⟶gν or gν⟶fν takes place starting from the leftmost point of the domain/interval of the new dynamics. Hence, μλ,+λti,μλ,−λti+1⊂Nλν is in the domain of fν and μλ,−λti+1,μλ,+λti+1⊂Nτν is in the domain of gν in the ti,ti+1 interval, ∀ti,ti+1∈Dλ. Therefore, ∀σ0∈a,bν either σ0∈Nλν⇒μ^λσ0=t0∈Nλλ⊂a,bλ or σ0∈Nτν⇒μ^τσ0=ϑ0∈Nττ⊂a,bτ.Using the left closed right open intervals in equation (30) we obtain a pair of impulsive differential equations such that any initial value problem σ0,ξ∈a,bν×Ω0,φσ0=ξ has a solution on an interval σ0,σ0+δ0⊂Nλν if σ0∈Nλν or has a solution on an interval σ0,σ0+δ0⊂Nτν if σ0∈Nτν. 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 xλ=φ⋅μλ,+andxτ=φ⋅μτ,+, we simply can define the initial value problems as follows:(31)σ0∈Nλν:⇒t0≔μ^λσ0andσ0=μλ,+t0;ξ0,λ≔φμλ,+t0=ξ0ϑ0≔μ^τσ0;ξ0,τ≔φμτ,+ϑ0hence letσ0,τ≔μτ,+ϑ0.Similarly, for the caseσ0∈Nτν,(32)σ0∈Nτν:ϑ0≔μ^τσ0andσ0=μτ,+ϑ0⇒ξ0,τ≔φμτ,+ϑ0=ξ0t0≔μ^λσ0;ξ0,λ≔φμλ,+t0hence letσ0,λ≔μλ,+t0.


## 2.3.5. Integral Transformations amongt-, τ-, and ν-Scales

The details of the assertions in this section come from paper [10].We summarize the mappings betweent-scale to ν-scale listed in Section 2.3.1 (Table 1).Table 1
The mappings in the first column are strictly ascending left and right continuous versions ofνλ while the mapping in the second column is absolute continuous and ascending.
a,bλ⟶a,bνa,bλ⟷a,bνa,bν⟶a,bλRelation:μλ,−λμλλ≔μλ,+λμ^λμ^λ⋅μλ,−λ=ida,bλμ^λ⋅μλ,+λ=ida,bλThe mappingsτλ,andνλ are not continuous ascending functions, with a common set of discontinuity points Dλ. Therefore, the measures τλandνλ are defined on the semiring Pν,a,bλ,c=s,ts,t∈a,bλ\Dλ,s≤t and the measures τλs,t≔τλt−τλsandνλs,t≔νλt−νλs,∀s,t∈Pν,a,bλ,c can be extended to ℬν,a,bλ,c=σPν,a,bλ,c.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν map the set of discontinuity points Dλ into the set of left-closed right-open intervals:(33)Dλν≔μλ,−t,μλ,+tt∈Dλ,and the set of discontinuity points in a,bν is(34)Dλν≔∪t∈Dλμλ,−t,μλ,+t⊂a,bν.Moreover, the mappingsμλ,−λ,μλ,+λ are bijective on the set of continuity points a,bν\Dλν and μλ,−t=μλ,+t,∀t∈a,bλ\Dλ.The mappingsμλ,−λ,μλ,+λ:a,bλ⟶a,bν transform s,t∈Pν,a,bλ,c⟶νλs,νλt=μλ,−λs,μλ,−λt=μλ,+λs,μλ,+λt∈Pν,a,bν,cν by their continuity points with Pν,a,bν,cν≔s,ts,t∈a,bν\Dλν.The measureνλν on Pν,a,bν,cν is defined by νλνu,v≔v−u=μλ,−μ^λλv−μλ,−μ^λλu=νλμ^λλu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bλ,c, then μλ,−s,μλ,−t∈Pν,a,bν,cν and νλνμλ,−s,μλ,−t=μλ,−t−μλ,−s=νλs,t. Let the smallest σ-algebra containing the semiring Pλ,a,bν,cν be ℬλ,a,bν,cν≔σPλ,a,bν,cν with the extended measure νλν on it.From this follows that ifh:a,bν⟶R is ℬλ,a,bν,cν-measurable and νλν-integrable, then(35)∫aμλ,+λthdνλν=∫ath⋅μλ,−λdνλ,∀t∈a,bλ.Conversely, ifh:a,bλ⟶R is ℬλ,a,bν,c-measurable and νλ-integrable, then(36)∫aσh⋅μ^λdνλν=∫aμ^λσhdνλ,∀σ∈a,bν\Dλν.These are some of the main conclusions from paper [10] presented here in a condensed form.We summarize the mappings betweenτ-scale and ν-scale listed in Section 2.3.2 (Table 2):Table 2
The mappings in the first column are strictly ascending left and right continuous versions ofντ while the mapping in the second column is absolute continuous and ascending.
a,bτ⟷a,bνa,bτ⟷a,bνa,bν⟶a,bτRelation:μτ,−τμττ≔μτ,+τμ^τμ^τ⋅μτ,−τ=ida,bτμ^τ⋅μτ,+τ=ida,bτThe mappingsλτandντ are not continuous ascending functions, with a common set of discontinuity points Dτ. Therefore, the measures λτandντ are defined on the semiring Pν,a,bτ,c=s,ts,t∈a,bτ\Dτ,s≤t and the measures λτs,t≔λτt−λτsandντs,t≔ντt−ντs,∀s,t∈Pν,a,bτ,c can be extended to ℬν,a,bτ,c=σPν,a,bτ,c.The mappingsμτ,−τ,μτ,+τ:a,bτ⟶a,bν map the set of discontinuity points Dτ into the set of left-closed right-open intervals:(37)Dτν≔μτ,−t,μτ,+tt∈Dτ,and the set of discontinuity points in a,bν is(38)Dτν≔∪t∈Dτμτ,−t,μτ,+t⊂a,bν.Moreover, the mappingsμτ,−τandμτ,+τ are bijective on the set of continuity points a,bν\Dτν and μτ,−t=μτ,+t,∀t∈a,bτ\Dτ.The mappingsμτ,−τ,μτ,+τ:a,bτ⟶a,bν transform s,t∈Pν,a,bτ,c⟶ντs,ντt=μτ,−τs,μτ,−τt=μτ,+τs,μτ,+τt∈Pν,a,bν,cν by their continuity points with Pν,a,bν,cν≔s,ts,t∈a,bν\Dλν.The measureντν on Pν,a,bν,cν is defined by ντνu,v≔v−u=μτ,−μ^ττv−μτ,−μ^ττu=ντμ^ττu,v∀u,v∈Pν,a,bν,cν. Also, if s,t∈Pν,a,bτ,c then μτ,−s,μτ,−t∈Pν,a,bν,cν and ντνμτ,−s,μτ,−t=μτ,−t−μτ,−s=ντs,t. Let the smallest σ-algebra containing the semiring Pτ,a,bν,cν be ℬτ,a,bν,cν≔σPτ,a,bν,cν with the extended measure ντν on it.From this follows that ifh:a,bν⟶R is ℬτ,a,bν,cν-measurable and ντν-integrable, then(39)∫aμτ,+τthdντν=∫ath⋅μτ,−τdντ,∀t∈a,bτ.Conversely, ifh:a,bτ⟶R is ℬτ,a,bν,c-measurable and ντ-integrable, then(40)∫aσ0h⋅μ^τdντν=∫aμ^τσ0hdντ,∀σ∈a,bν\Dτν.These are some of the main conclusions of paper [10] presented here in a condensed form.The details about measures and Radon–Nikodym derivatives summarised below are in paper [10].Letℬa,bν,ν denote the Borel sets on a,bν, and let ν be the Lebesgue measure on the σ-algebra ℬa,bν,ν. Then, ℬλ,a,bν,cν⊂ℬa,bν,νandℬτ,a,bν,cν⊂ℬa,bν,ν.Letf:a,bν⟶Rn be a ℬa,bν,ν-measurable ν-integrable function. Let νf,λA≔∫Afdν,∀A∈ℬλ,a,bν,cν and νf,τA≔∫Afdν,∀A∈ℬτ,a,bν,cν be signed measures absolute continuous with respect to the measures νλνandντν, respectively. Then, their Radon–Nikodym derivatives with respect to νλνandντν give(41)f^ν,λ≔dνf,λdνλν⟺∫Afdν=∫Af^ν,λdνλν,∀A∈ℬλ,a,bν,cν,f^ν,λ∈L1νλ,ℬλ,a,bν,cν,(42)f^ν,τ≔dνf,τdντν⟺∫Afdν=∫Af^ν,τdντν,∀A∈ℬτ,a,bν,cν,f^ν,τ∈L1ντν,ℬτ,a,bν,cν.Combining equations (35) and (41) gives(43)∫aμλ,+λtfdν=∫aμλ,+λtf^ν,λdνλν=∫atf^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ.Combining equations (39) and (42) gives(44)∫aμτ,+τϑfdν=∫aμτ,+τϑf^ν,τdντν=∫aϑf^ν,τ⋅μτ,−τdντ,∀ϑ∈a,bτ.It was proved in paper [40] that the Radon–Nikodym derivatives f^ν,λandf^ν,τ fulfill the relations:(45)fs=f^ν,λs,a.e.s∈a,bν\Dλν,fs=f^ν,τs,a.e.s∈a,bν\Dτν,which implies the assertions below as follow ups of the indicated equation (45):(46)∫aμλ,+λtf×χa,bν\Dλνdν=∫aμλ,+λtf^ν,λ×χa,bν\Dλνdνλν=43∫aμλ,+λtf×χa,bν\Dλνdνλν=∫atf×χa,bλ\Dλ⋅μλ,−λdνλ,∀t∈a,bλ,(47)∫aμτ,+τϑf×χa,bν\Dτνdν=∫aμτ,+τϑf^ν,τ×χa,bν\Dτνdντν=43∫aμτ,+τϑf×χa,bν\Dτνdντν=∫aϑf×χa,bτ\Dτ⋅μτ,−τdντ,∀ϑ∈a,bτ.This relation enables us to transform the solution of aν-scale differential equation into solutions of a t-scale/τ-scale impulsive differential equations.The functionf can be written as f=f×χa,bν\Dλν+f×χDλν by a,bν=a,bν\Dλν∪Dλνand∅=a,bν\Dλν∩Dλν.Similarly,f=f×χa,bν\Dτν+f×χDτν by a,bν=a,bν\Dτν∪Dτνand∅=a,bν\Dτν∩Dτν.Therefore, equations (46) and (47) can be rewritten as(48)∫aμλ,+λtfdν=∫aμλ,+λtf×χa,bν\Dλν+f×χDλνdν=∫atf×χa,bλ\Dλ+f×χDλν^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ,∫aμλ,+λtfdν=∫aμτ,+τϑf×χa,bν\Dτν+f×χDτνdν=∫aϑf×χa,bτ\Dτ+f×χDτν^ν,τ,∀ϑ∈a,bτ.The sets inDλν and in Dτν are atoms as described in Lemma 2.9 in [10], in detail. Since an atom in ℬλ,a,bν,cν does not have any proper measurable subset in ℬλ,a,bν,cν but the same set is a nonatomic measurable set in ℬa,bν,ν, f can be integrated on it by ν. Similarly, an atom in ℬτ,a,bν,cν does not have any proper measurable subset in ℬτ,a,bν,cν but the same set is a nonatomic measurable set in ℬa,bν,ν, and f can be integrated on it by ν. Hence, ∀tj∈Dλ the set μλ,−tj,μλ,+tj∈Dλν is an atom. Similarly, ∀τj∈Dτ the set μτ,−τj,μτ,+τj∈Dτν is an atom. Therefore, the Radon–Nikodym derivatives in equation (48) can be written as follows:(49)f×χDλν^ν,λtj≔∫μλ,−tjμλ,+tjfdνμλ,+tj−μλ,−tj,f×χDτν^ν,ττj≔∫μτ,−τjμτ,+τjfdνμτ,+τj−μτ,−τj.


## 2.4. Existence of the Solutions ont-scale and on τ-Scale

In this section, we want to formulate the existence theorem for the pair of impulsive differential equations on thet-scale and τ-scale.First, we assume that Corollary2 is true. The initial value problem in equation (21) has a solution φ:σ0,σ0+β⟶ℛδ,εσ0,ξ,φσ0=ξ.Applying the statements in equations (43) and (44) to equation (21), we get that the solution in t-scale can be xλt=φ⋅μλ,+λt,t∈t0,μ^λσ0+β,t0=μ^λσ0 and in τ-scale it can be xτϑ=φ⋅μτ,+τϑ,ϑ∈ϑ0,μ^τσ0+β,ϑ0=μ^τσ0, which transformed ν-scale solutions. We have to prove that they fulfill the initial value problems with respect to the t-scale and τ-scale versions and that also they fulfill the respective impulsive differential equations.The initial value problemφσ0=ξ on the ν-scale problem fulfills either σ0∈Nλν, which is detailed out in equation (31), or it fulfills σ0∈Nτν, which is detailed out in (32). Hence, the two cases of initial value problems fulfilled by the pair of impulsive differential equations are as follows:Whenσ0∈Nλν, t0≔μ^λσ0andξ0,λ≔φμλ,+t0=ξ and ϑ0≔μ^τσ0andξ0,τ≔φμτ,+ϑ0Whenσ0∈Nτν, ϑ0≔μ^τσ0andξ0,τ≔φμτ,+ϑ0=ξ while t0≔μ^λσ0andξ0,λ≔φμλ,+t0Note that the following equations (31) and (32) for the initial value problems are given in t-scale uniformly t0,ξ0,λ and similarly for τ-scale uniformly ϑ0,ξ0,τ, but the values are obtained differently according to equations (31) and (32) (the difference is whether ξ0,λ=ξ or ξ0,τ=ξ).In these equations, we used thatμλ,+:Nλλ⟶Nλν and μτ,+:Nττ⟶Nτν which are bijective strictly ascending mappings. Hence, the initial conditions are fulfilled.Solution of the pair of impulsive differential equations: let us apply relations (43) and (44) to solution (21) both on t-scale and on τ-scale with hν:(50)φμλ,+λt=ξ0+∫σ0μλ,+λthνv,φvdν=ξ0+∫σ0μλ,+λthν^ν,λvdνλν=ξ0,λ+∫t0thν^ν,λμλ,−λdνλ,∀t∈a,bλ,(51)φμτ,+τϑ=ξ0+∫σ0μτ,+τϑhνv,φvdν=ξ0+∫σ0μτ,+τϑhν^ν,τvdντν=ξ0,τ+∫ϑ0ϑhν^ν,τμτ,−τdντ,∀ϑ∈a,bτ.Let us splithν on ν-scale with Dλν into a component on continuity points hλ,cν≔hν×χa,bν\Dλν and a component on discontinuity points hλ,dν≔hν×χDλν. We can do this with discontinuity points Dτν on ν-scale also. A component on continuity points is hτ,cν≔hν×χa,bν\Dτν, and a component on discontinuity points is hτ,dν≔hν×χDτν:(52)hν=hλ,cν+hλ,dν,hν=hτ,cν+hτ,dν.Let us put the definitionhν≔fνχNλν+gνχNτν into equation (52):(53)hν≔fνχNλν+gνχNτν×χa,bν\Dλν+fνχNλν+gνχNτν×χDλν,hν≔fνχNλν+gνχNτν×χa,bν\Dτν+fνχNλν+gνχNτν×χDτν.ConsideringNλν⊂a,bν\DλνandNτν⊂a,bνDτν, Nλν∩Nτν=∅ and Dλν⊂NτνandDτν⊂Nλν leads to(54)hν≔fνχNλν+gνχNτν×χa,bν\Dλν︸hλ,cν+gν×χDλν︸hλ,dν,hν≔fνχNλν×χa,bν\Dτν+gνχNτν︸hτ,cν+fν×χDτν︸hτ,dν.Let us apply the relations in equations (50) and (51) to hλ,dνandhτ,dν, as expressed in equation (54) (cases of discontinuity):(55)∫σ0μλ,+λthλ,ddν=∫σ0μλ,+λtgν×χDλνv,φvdν=∫σ0μλ,+λtgν×χDλν^ν,λv×χa,bν\Dλνdνλν=∫t0tgν×χDλν^ν,λ⋅μλ,−λdνλ,∀t∈a,bλ,(56)∫σ0μλ,+λthτ,ddν=∫σ0μτ,+τϑfν×χDτνv,φvdν=∫σ0μτ,+τϑfν×χDτνν,τ^v×χa,bν\Dτνdντν=∫ϑ0ϑfν×χDτνν,τ^⋅μτ,−τdντν,∀ϑ∈a,bτ.Let us apply equations (46) and (47) to the components hλ,candhτ,c using equation (54):(57)∫σ0μλ,+λthλ,cνv,φvdν=∫σ0μλ,+λthν^ν,λv×χa,bν\Dλνdνλν=43∫σ0μλ,+λthλ,cνv,φvdνλν=∫t0tfνχNλν⋅μλ,−λ+gνχNτν×χa,bν\Dλν∘μλ,−λdνλ,∀t∈a,bλ,(58)∫σ0μτ,+τϑhτ,cνv,φvdν=∫σ0μτ,+τϑhτ,cν^ν,τv×χa,bν\Dτνdντν=43∫σ0μτ,+τϑhτ,cνv,φvdντν=∫ϑ0ϑfνχNλν×χa,bν\Dτν∘μτ,−τ+gνχNτν∘μτ,−τdντ,∀ϑ∈a,bτ,where equation (45) is obtained from Corollary 2 and Theorem 1 in [40].The Radon–Nikodym derivatives ofhλ,dνandhλ,dν, remains to be determined. The domains of the Radon–Nikodym derivatives are countable unions of pairwise disjoint atoms of the σ-algebras ℬλ,a,bν,cνandℬτ,a,bν,cν, respectively, as defined in equations: Dλν in (11); Dτν in (37). The lists of these atoms are defined: Dλν in (12) and Dτν in (38). This means that the Radon–Nikodym derivatives are fully determined on Dλν if they are determined on the intervals in Dλν, and they are fully determined on Dτν if they are determined on each interval in Dτν.Let impulse time pointstj∈Dλ and τj∈Dτ be selected, and let φ:t0,t0+β⟶Bε0x0 be the solution of equation (21).Based on equation (54), hλ,dνandhτ,dν is replaced by gν×χDλνandfν×χDτν, respectively, in equation (49). This leads to(59)gν×χDλν^ν,λtj,φ≔∫μλ,−tjμλ,+tjgνdνμλ,+tj−μλ,−tj=φμλ,+λtj−φμλ,−λtjμλ,+λtj−μλ,−λtj≕g0^λνtj,φ,∀tj∈Dλ,(60)fν×χDτν^ν,ττj≔∫μτ,−τjμτ,+τjfνdνμτ,+τj−μτ,−τj=φμτ,+τtj−φμτ,−τtjμτ,+τj−μτ,−τj≕f0^τντj,φ,∀τj∈Dτ.LetTa,bν denote all the solution trajectories with domains as subsets of a,bν.With equations (59) and (60), g0^λν is defined on the interval μλ,−λtj,μλ,+λtj,∀tj∈Dλ and f0^λν is defined on the interval μτ,−ττj,μτ,+ττj,∀tj∈Dτ; hence, g0^λν is defined on Dλν and f0^τν is defined on Dτν. Then, we define g^λν&f^τν as follows:(61)g^λνσ,φ≔g0^λνσ,φ,μ^λσ∈Dλ,0,σ∉Dλν,∀σ,φ∈a,bν×Ta,bν,f^τνσ,φ≔f0^τνσ,φ,μ^τσ∈Dτ,0,σ∉Dτν,∀σ,φ∈a,bν×Ta,bν.These jumps in this case are dependent on the solution and the time parameter is based ona,bν. If however, the right sides fulfill Lipschitz condition then the solution is determined by tj,φtj; hence, g^λtj,φ and f^ττj,φ are fully determined by the initial value problems φtj=y,tj,y∈tj×Ω0 and φτj=y,τj,y∈τj×Ω0; hence, the jumps can be written g^λtj,φ⟶g^λtj,φtj and f^ττj,φτj⟶f^ττj,φτj which is Bainov’s formulation. Combining equations (55) and (57)equation and (56) about hλ,cνandhτ,cν with hλ,dνandhτ,dν and equations (59) and (60) on the Radon–Nikodym derivatives results in equations(62)φμλ,+λt=ξ0+∫σ0μλ,+λthλ,cν+hλ,dνv,φvdν=ξ0,λ+∫t0tfνμλ,−λ,φμλ,−λχNλλ+gνμλ,−λ,φμλ,−λχNτλ×χa,bλDλdνλ+∫t0tg^λνμλ,−λ,φdνλ,∀t∈a,bλ,(63)φμτ,+τϑ=ξ0+∫σ0μτ,+τϑhτ,cν+hτ,dνv,φvdν=ξ0,τ+∫ϑ0ϑfνμτ,−τ,φμτ,−τχNλτ×χa,bτ\Dτ+gνμτ,−τ,φμτ,−τχNττdντ+∫ϑ0ϑf^τνμτ,−λ,φdντ,∀ϑ∈a,bτ.Let us put into equations (62) and (63) the expressions xλ≔φμλ,−λandxτ≔φμτ,−τ, and from equation (17) χNλλdνλ=dλ,andχNττdντ=dτ. This leads to(64)xλt=ξ0,λ+∫t0tfνμλ,−λ,xλdλλ︸absolute continuous+∫t0tgνμλ,−λ,xλ×χa,bλ\Dλ︸continuous singular+g^λμλ,−λ,φ︸pure jumping singulardτλ,∀t∈a,bλ,(65)xτϑ=ξ0,τ+∫ϑ0ϑgνμτ,−τ,xτdττ︸absolute continuous+∫ϑ0ϑfνμτ,−τ,xτ×χa,bτ\Dτ︸continuous singular+f^τμτ,−τ,φ︸pure jumping singulardλτ,∀ϑ∈a,bτ.Conclusion 2.
We proved that, from the statement of Corollary2, follows the existence of the solution of the pair of impulsive differential equations. Important to note that the function f is fully known in t-scale and in ν-scale, whereas the function g is fully known in τ-scale and ν-scale. Therefore, we will prove that if fλ is locally t-integrable ∀t,x∈a,bλ×Ω0 and gτ is locally τ-integrable ∀ϑ,x∈a,bτ×Ω0, then the corresponding hν:a,bν⟶Rn is locally ν-integrable ∀σ,x∈a,bν×Ω0 (see the definition of the concept of local t-integrability in Definition 1).Lemma 1.
Leta,b⊂R be a closed bounded interval and let ψ:a,b×Ω0⟶Rn be measurable in its variable t∈a,b for any fixed x∈Ω0, and let it be continuous in its variable x∈Ω0 for each fixed t∈a,b. If ψ is locally integrable at each t0,x0∈a,b×Ω0 then there exists an 0<ε∈R and an integrable dominator mx0:a,b⟶R+\0 such that ψt,x≤mx0t,∀t,x∈a,b×Bεx0⊂a,b×Ω0,∀t0,x0∈a,b×Ω0.Proof.
Letψ be extended to 0 outside a,b⊂R. Let t0,x0∈a,b×Ω0. By the formulation of the lemma ∃ℛδt0,x0,εt0,x0t0,x0andmt0,x0:t0−δt0,x0,t0+δt0,x0⟶R+\0 such that ψt,x≤mt0,x0t,∀t,x∈t0−δt0,x0,t0+δt0,x0×Bεt0,x0x0. Then, a,b⊂∪s∈a,bs−δs,x0,s+δs,x0. Since a,b is compact, a finite subsystem tj,x01≤j≤N can be selected such that a,b⊂∪j=1Ntj−δtj,x0,tj+δtj,x0 covers the interval a,b. Extending mtj,x0 by zero outside tj−δtj,x0,tj+δtj,x0,∀1≤j≤N, we can form a positive integrable dominator mx0t≔maxmtj,x0t1≤j≤N,∀t∈a,b, and can get a positive εx0≔minεtj,x01≤j≤N such that the positive dominator mx0:a,b⟶R+\0 fulfills ψt,x≤mx0t,∀t,x∈a,b×Bεx0x0⊂a,b×Ω0.
Theorem for the existence of solutions of a pair of impulsive differential equations: let us denote the functionsf and g on t-scale by fλ:Nλλ⟶Rnandgλ:Nτλ⟶Rn, on τ-scale by fτ:Nλτ⟶Rnandgτ:Nττ⟶Rn, and on ν-scale by fν:Nλν⟶Rnandgν:Nτν⟶Rn. Let hν≔fλ⋅μ^λχNλν+gτ⋅μ^τχNτν:a,bν⟶Rn.Lemma 2.
The following statements are equivalent:Statement 1: the mappingfλ is locally t-integrable ∀t,x∈a,bλ×Ω0 and the mapping gτ is locally τ-integrable ∀ϑ,x∈a,bτ×Ω0Statement 2: the mappinghν is locally ν-integrable ∀σ,x∈a,bν×Ω0Proof.
Statement 1⇒ Statement 2:(1)
fλ is l  Statement 2: the mapping hν is locally ν-integrable ∀σ,x∈a,bν×Ω0 and locally t-integrable ∀t0,x0∈a,bλ×Ω0⇒∃0<ελ,x0∈Randmλ,x0:a,bλ⟶R+\0 such that fλt,x≤mλ,x0t,∀t,x∈a,bλ×Bελ,x0x0⊂a,bλ×Ω0∀x0∈Ω0 by Lemma 1.(2)
gτ is locally τ-integrable ∀ϑ0,x0∈a,bτ×Ω0⇒∃0<ετ,x0∈Randmτ,x0:a,bτ⟶R+\0 such that gτϑ,x≤mτ,x0ϑ,∀ϑ,x∈a,bτ×Bετ,x0x0⊂a,bτ×Ω0,∀x0∈Ω0 by Lemma 1.(3)
By point 1 ina,bλ, let x0∈Ω0; then, ∃0<ελ,x0∈R such that fλt,x≤mλ,x0t,∀t,x∈a,bλ×Bελ,x0x0⊂a,bλ×Ω0⇒fλμ^λσ,x≤mλ,x0μ^λσ,∀σ,x∈a,bν.×Bελ,x0x0⊂a,bν×Ω0. Hence, with fνσ,x≔fλμ^λσ,x×χNλνσ,fνσ,x≤mλ,x0μ^λσ×χNλνσ,∀σ,x∈Nλν×Bελ,x0x0⊂a,bν×Ω0.(4)
By point 1 ina,bτ, let x0∈Ω0; then, ∃0<ετ,x0∈R such that gτϑ,x≤mτ,x0ϑ,∀ϑ,x∈a,bτ×Bετ,x0x0⊂a,bτ×Ω0⇒gτμ^τσ,x≤mτ,x0μ^τσ,∀σ,x.∈a,bν×Bετ,x0x0⊂a,bν×Ω0. Hence, with gνσ,x≔gτμ^τσ,x×χNτνσ,gνσ,x≤mτ,x0μ^τσ×χNτνσ,∀σ,x∈Nλν×Bετ,x0x0⊂a,bν×Ω0.(5)
By point 3,fνσ,x≤mλ,x0μ^λσ×χNλνσ in Nλν×Bελ,x0x0, and by point 4, gνσ,x≤mτ,x0μ^τσ×χNτνσ hold in Nλν×Bετ,x0x0. Then, with mν,x0σ≔maxmλ,x0μ^λσ×χNλνσ,mτ,x0μ^τσ×χNτνσ,∀σ∈a,bν and with εν,x0≔minελ,x0,ετ,x0, we obtain that hνσ,x=fνσ,x+gνσ,x≤mν,x0σ,∀σ,x∈a,bν×Bεν,x0x0.
Statement 2⇒ Statement 1: assume that ∃0<εν,x0∈Randmν,x0:a,bν⟶R+\0∀x0∈Ω0 such that hνσ,x≤mν,x0σ∀σ,x∈a,bν×Bεν,x0x0, where hν≔fνχNλν+gνχNτν:a,bν×Ω0⟶Rn.(1)
hνσ,x≤mν,x0σ⇒‖hνμλ,−t,x≤mν,x0μλ,−t holds ∀t,x∈a,bλ×Bεν,x0x0. Multiplying both sides by the characteristic function of Nλλ, we obtain that(66)hνμλ,−t,xχNλλ=fνχNλν+gνχNτνμλ,−t,xχNλλ=fνμλ,−t,xχNλλt+gνμλ,−t,xχNτλtχNλλt=fνμλ,−t,xχNλλt≤mν,x0μλ,−t∀t,x∈a,bλ×Bεν,x0x0,which proves the case for thet-scale. The case for τ-scale is word for word the same as the case of t-scale and is left to the reader.Conclusion 3.
With this we proved that the initial value problem prescribed for a pair of impulsive differential equations has a solution if Caratheodory’s condition holds forfλ:Nλλ⟶Rn and for gτ:Nττ⟶Rn. This condition is equivalent to the condition of Corollary 2. We state a final version of the existence theorem.Condition 1.
C1: letfλ:t,x∈Nλλ×Ω0⟶fλt,x∈Rn and gτ:ϑ,x∈Nττ×Ω0⟶gτϑ,x∈Rn be measurable functions of t and ϑ, for each fixed x, respectively, and let they be continuous in x for each fixed t and ϑ, respectively, ∀t,x∈Nλλ×Ω0and∀ϑ,x∈Nττ×Ω0.C2: letfνσ,x≔fλμ^λσ,xandgνσ,x≔gτμ^τσ,x∀σ,x∈a,bν×Ω0. Then, by condition C1, hνσ,x≔fνσ,xχNλνσ+gνσ,xχNτνσ is measurable in σ for each fixed x, and it is continuous in x for each fixed σ, ∀σ,x∈a,bν×Ω0.C3: letfλ and gτ be locally t- and τ-integrable on their respective domains or equivalently let hν be locally ν-integrable on a,bν×Ω0.Theorem 2.
Let the mappingsfλ,gτ,andhν fulfill Condition 1 C1, C2, and C3. Let σ0,ξ0∈a,bν×Ω0 be a fixed point and let a cylinder ℛδ,εσ0,ξ0⊂a,bν×Ω0 exist with a D.I.F. m:σ0,σ0+δ⟶R+\0 on ℛδ,εσ0,ξ0 to hν. Then, there exists an interval σ0,σ0+β,0<β≤δ such that equation (21) has a solution φ in that interval such that σ,φσ∈ℛδ,εσ0,ξ,∀σ∈σ0,σ0+β and φσ0=ξ0.Moreover, lett0=μ^λσ0∈a,bλ and ϑ0=μ^τσ0∈a,bλ. Let xλ≔φ⋅μλ,+λ:t0,μ^λσ0+β⟶Bεξ0 and let xτ≔φ⋅μτ,+τ:ϑ0,μ^τσ0+β⟶Bεξ0. Let xλt0≔φμλ,+λt0≕ξ0,λ and let xτϑ0≔φμτ,+τϑ0≕ξ0,τ. Then, the following equations hold:(67)xλt=ξ0,λ+∫t0tfλs,xλsdλλ+∫t0tgτμ^τμλ,−λs,xλs×χa,bλ\Dλ+g^νμλ,−λsdτλ,∀t∈t0,μ^λσ0+β,xτϑ=ξ0,τ+∫ϑ0ϑgτη,xτηdττ+∫ϑ0ϑfλμ^λμτ,−τη,xτη×χa,bτ\Dτ+f^νμτ,−τηdλτ,∀ϑ∈ϑ0,μ^λσ0+β.Proof.
In the theorem,fνσ,x=fλμ^λσ,x and gνσ,x=gτμ^τσ,x by 2.1, C2. Then, fνμλ,−λt,x=fλμ^λμλ,−λt,x=fλida,bλt,x=fλt,x and gνμτ,−τϑ,x=gτμ^τμτ,−τϑ,x=gτida,bτt,x=gτϑ,x, where we applied the identities listed in equations (24) and (27).Conclusion 4.
The 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 f-dynamics 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. The relationship is an action/reaction interaction between the two processes. This 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. This will make the flow of solutions to be a tree structure instead of a connected flow as in ordinary differential equations. The leaves of the tree consist of trajectories coming from discontinuity points without history. This tree is directed, with orientation from the leaves to the root which is called in-tree or antiarborescence [41]. This gives wide range of modelling facilities by enabling one to model and study mixing new generations in addition to studying flows of solutions.



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