ExistenceTheoremforImpulsiveDifferentialEquationswith...

Research ArticleExistence Theorem for Impulsive Differential Equations withMeasurable Right Side for Handling Delay Problems

Z Lipcsey1 J A Ugboh1 I M Esuabana 1 and I O Isaac2

1Department of Mathematics University of Calabar Calabar Nigeria2Department of Mathematics Akwa Ibom State University Mkpat-Enin Nigeria

Correspondence should be addressed to I M Esuabana esuabanaunicaledung

Received 7 August 2019 Revised 12 May 2020 Accepted 18 May 2020 Published 26 June 2020

Academic Editor Nan-Jing Huang

Copyright copy 2020 Z Lipcsey et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to noncontinuous solution impulsive differential equations with delay may have a measurable right side and not a con-tinuous one In order to support handling impulsive differential equations with delay like in other chapters of differentialequations we formulated and proved existence and uniqueness theorems for impulsive differential equations with measurableright sides following Caratheodoryrsquos techniques +e new setup had an impact on the formulation of initial value problems (IVP)the continuation of solutions and the structure of the system of trajectories (a) We have two impulsive differential equations tosolve with one IVP (φ(σ0) ξ0) which selects one of the impulsive differential equations by the position of σ0 in [a b]] Solving theselected IVP fully determines the solution on the other scale with a possible delay (b)+e solutions can be continued at each pointof (α β) timesΩ0≕Ω by the conditions in the existence theorem (c)+ese changes alter the flow of solutions into a directed tree+istree however is an in-tree which offers a modelling tool to study among other interactions of generations

1 Introduction

+e innovation of the theory of impulsive systems ismanifested in the fact that the time development of the stateof such a system forms a mapping of bounded variationinstead of continuously differentiable solution of a differ-ential equation [1ndash8]

+e motivation for this research came from two ob-servations which arose in the theory and applications ofimpulsive differential equations

One is the effect of the discontinuity of the right side ofthe impulsive differential equation originating from thediscontinuity of solutions of delayed equations A discon-tinuity of the first order of the solution at a time point maycreate a set of discontinuity points on the right side (dy-namics) of the equation in later time points+is changes theright side to a measurable function of the time instead ofa continuous one

+e second issue comes from the representation ofa function of bounded variation in terms of two integralforms In the ldquousualrdquo representation the absolute

continuous part is a function of time t and the singular partis a function of the singular timer (impulse timer in im-pulsive differential equations) τ while in the other repre-sentation the function of τ is the absolute continuous partand the function of t is singular In impulsive differentialequations the first form is in use

+e purpose of this paper therefore is to analyse andformulate the concept of initial value problem suitable toinitialize the obtained pair of impulsive differential equa-tions having measurable right sides and to give conditionsfor the existence uniqueness and continuation of solutions+e existence of solutions of ordinary differential equationswith measurable right side has been widely covered byCaratheodory [9] +erefore our approach will start fromCaratheodoryrsquos existence theorem

+e analysis and proof are presented in the followingsteps

After giving a brief summary of processes described byimpulsive differential equation-Bainovian model and in-troduction of the system time t and the impulse control timeτ we discussed how existence uniqueness and continuation

HindawiJournal of MathematicsVolume 2020 Article ID 7089313 17 pageshttpsdoiorg10115520207089313

are handled in impulsive differential equations with con-tinuous right side

+e analysis of problems arising from delayed equationsand their handling leads us to the necessity of the formu-lation of the extended concept of impulsive differentialequations and analysis of the difference between the Bai-novian and extended models

11 Systems Described by Impulsive Differential Equations+e Bainovian model for the simplest case is as follows Letthe process evolve in a period of time T (T (α β) sub R is anopen interval) Let Ω0 sub Rn be an open set and Ω ≔ T timesΩ0Let f Ω⟶ Rn be an at least continuous mapping whichin additionmay fulfill local Lipschitz condition in its variablex isin Rn for each fixed t forall(t x) isin Ω Let H sub Z be an infinitesubset ofZ (H N orH Zwill be used)+en let the timesequence SH tk1113864 1113865kisinH sub T be increasing without accumu-lation points in T and tk⟶ α k⟶ minus infinand tk⟶ β k⟶infin (equivalently forallm M isin T mltM [m M]capSH sub T is a finite set) Let g SH times Rn⟶ Rn becontinuous and may fulfill Lipschitz condition in its variablex forall(tk x) isin Ω +en the controlling impulsive differentialequation is given by

xprime(t) f(t x(t)) forallt isin TSH

Δx tk( 1113857 g tk x tk( 1113857( 1113857 foralltk isin SH

⎧⎨

⎩ (1)

where (t x(t)) isin Ω+e impulsive differential equation (1) can be re-

written as an integral equation We define an ascendingstep function τ R⟶ Z with unit jumps at the impulsepoints

τ(t) ≔ k minus 1 if tkminus 1 le tlt tkforalltk isin SZ (2)

If H N then τ(t) 0foralltlt t1

Corollary 1 e function τ R⟶ R is a singular ascendingfunction in t which means as an ascending function it isdifferentiable almost everywhere and

dτ(t)

dt 0 almost everywhere (3)

The singular ascending function τ defines a singularmeasure τ on the Borel sets of R +e domain of the functiong is extended to the 1113957g Ω⟶ Rn from the set SH timesΩ0 sub Ω

With measures τ and 1113957g equation (1) can be rewritten inan integral form

x(t) x0 + 1113946t

t0

(f(s x(s))ds + 1113957g(s x(s))dτ) t0 lt t isin T

x t0( 1113857 x0 t0 isin TSH x0 isin Ω0(4)

+e technical details of these facts are discussed in [10]

Equation (4) has two measures therefore it does notlook like an integral equation of an (impulsive) differentialequation We will change the parametrization of thisequation

Let ]λ(t) ≔ t + τλ(t) id[abλ](t) + τλ(t) (τλ represents τas defined in Corollary 1) which is a strictly ascendingfunction ]λ [a bλ]⟶ [a b]] with b] ≔ bλ + τλ(bλ)minus

τλ(a) As an ascending function ]λ has a left- and a right-continuous version

μλλminus (t) ≔ ]λ(t minus 0) forallt isin a bλ1113858 1113859

μλλ+(t) ≔ ]λ(t + 0) forallt isin a bλ1113858 1113859(5)

+e mappings defined in (5) give us an increasingfunction 1113954μλ [a b]]⟶ [a bλ] defined as follows

1113954μλ(t) ≔ s forallt isin a b]1113858 1113859 exists isin a bλ1113858 1113859st t isin μλλminus (s) μλλ+(s)1113960 1113961

1113954μλ middot μλλminus (t) 1113954μλ middot μλλminus (t) id abλ[ ]

(6)

Note that 1113954μλ is one-to-one on the set of continuity points([μλλminus (s) μλλ+(s)] is singleton if μλλminus (s) μλλ+(s) ) +e as-cending function id[abλ](t) is continuous in [a bλ] while τand ] may have a countable set of discontinuity points

Dλ ≔ t | τλ(t minus 0)ne τλ(t + 0)1113966 1113967 t | ]λ(t minus 0)ne ]λ(t + 0)1113966 1113967

(7)

Hence τλ and ]λ are continuous in [a bλ]Dλ

12 Absolute Continuity and Singularity Let ]λ denote themeasure defined by the strictly ascending function ]λ +eintegral in equation (4) is the sum of integrals with twomeasures λλ and τλ Both measures λλ and τλ are absolutecontinuous with respect to ]λ therefore both can be writtenas an integral of the RadonndashNikodym derivatives [11] Withnotations ρλ dλλd]λ and ρτ dτλd]λ the following im-portant properties are formulated

Let Nλτ ≔ (dτλd]λ)gt 01113864 1113865 andNλ

λ ≔ (Nλτ)prime +en

λλ(Nλτ) 0 τλ(Nλ

λ) and

ρλ + ρτ d λλ + τλ1113872 1113873d]λ dτλd]λ + dλλd]λ 1 (8)

Moreover let

N]λ ≔ 1113954μminus 1

λ Nλλ1113872 1113873 sub a b]1113858 1113859

N]τ ≔ 1113954μminus 1

λ Nλτ1113872 1113873 sub a b]1113858 1113859

N]λcupN

]τ a b]1113858 1113859

N]λcapN

]τ empty

(9)

+ese RadonndashNikodym derivatives enable us to rewriteequation (4) using one measure ]λ as follows

2 Journal of Mathematics

x(t) x0 + 1113938t

t0f(s x(s))

dλλ

d]λ+ 1113957g(s x(s))

dτλ

d]λ1113888 1113889d]λ t isin T

x t0( 1113857 x0 t0 isin TSH x0 isin Ω0

(10)

+e details of these assertions are in paper [10]

13 Measures +e mappings τλ and ]λ are ascending notcontinuous functions with a common set of discontinuitypointsDλ+erefore themeasures τλ and ]λ are defined on thesemiring P][abλ]c [s t) | s t isin [a bλ]Dλ sle t1113864 1113865 and themeasures τλ([s t)) ≔ τλ(t) minus τλ(s) and ]λ([s t)) ≔ ]λ(t) minus

]λ(s) forall[s t) isin P][abλ]c can be extended to B][abλ]c

σ(P][abλ]c)+e mappings μλλminus μλλ+ [a bλ]⟶ [a b]] map the set

of discontinuity points Dλ into the set of left-closed right-open intervals

D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (11)

and the set of discontinuity points in [a b]] is

D]λ ≔ cup

tisinDλ

μλminus (t) μλ+(t)1113960 1113873 sub a b]1113858 1113859 (12)

Moreover the mappings μλλminus and μλλ+ are bijective on the

set of continuity points [a b]]D]λ and μλminus (t)

μλ+(t) forallt isin [a bλ]Dλ+e mappings μλλminus μλλ+ [a bλ]⟶ [a b]] transform

[s t) isin P][abλ]c⟶ []λ(s) ]λ(t)) [μλλminus (s) μλλminus (t))

[μλλ+(s) μλλ+(t)) isin P]][ab]]c

by P]][ab]]c≔ [u v) | u v isin

[a b]]D]λ

+e measure ]]λ on P]][ab]]c

is defined by ]]λ ([u v)) ≔v minus u μλminus (1113954μλλ(v)) minus μλminus (1113954μλλ(u)) ]λ(1113954μλλ([u v)))forall[u v) isinP]][ab]]c

Also if [s t) isin P][abλ]c then [μλminus (s) μλminus (t)) isinP]][ab]]c

and ]]λ([μλminus (s) μλminus (t))) μλminus (t) minus μλminus (s)

]λ([s t)) Let the smallest σ-algebra containing the semiringP]λ[ab]]c

be B]λ[ab]]c≔ σ(P]

λ[ab]]c) with the extended

measure ]]λ on it +en the following relations hold

1113938μλ+(t)

af middot 1113954μλd]]λ 1113938

t

afd]λ forall[a t) isinB] abλ[ ]cforallf isin L1 ]λB] abλ[ ]c1113874 1113875

1113938σa

fd]]λ 11139381113954μλ(σ)

af middot μλminus d]λ forall[a σ) isinB]

λ ab][ ]cforallf isin L1 ]]λB

]λ ab][ ]c

1113874 1113875

(13)


14 Existence and Continuation of a Solution Existenceuniqueness and continuation of solutions are fundamentalissues for differential equations of all kinds +ese issuestherefore have been studied by many authors [1 2 12ndash19]just to mention a few +ese articles consider initial valueproblems and boundary value problems for impulsive dif-ferential equations with an at least continuous right side f inΩ or in addition to the continuity f fulfills Lipschitz con-dition in its spatial variables therefore the analysis is based onCauchyndashPeanorsquos or PiccardndashLindelofrsquos existence theorems[9] +e sources of discontinuities are arranged so that anyclosed bounded interval contains finite number of disconti-nuity points of the first type We give a summary of thesesystems by pointing out the major differences in propertiescompared with the ordinary differential equations If theinitial value problem is not prescribed at a discontinuity timepoint then CauchyndashPeanorsquos or PiccardndashLindelofrsquos existencetheorems [9] provide solutions extendible in line with therules of ordinary differential equations

Continuation by the assumption that the functionf Ω⟶ Rn fulfills the conditions of an existence theorem(CauchyndashPeanorsquos or PiccardndashLindelofrsquos existence theorem[9]) in Ω any solution that reaches a point (sφ(s)) isinΩ t0 lt s has a continuation on an interval [s s + δ) witha suitable δ gt 0 +e process stops only at a boundary point

(sφ(s)) isinzΩ +e impulses however change this scenarioIf the solution reaches (tjφ(tj minus 0) isin Ω tj isin SH then thereis a continuation to φ(tj + 0) φ(tj minus 0)+ g(tjφ(tj minus 0))If (tjφ(tj + 0)) notin Ω the process stops at(tjφ(tj minus 0)) (tjφ(tj)) isin Ω Otherwise if (tjφ(tj+

0))) isin Ω then it has a continuation as described above +ejump mapping J(tj y) ≔ (tj y + g(tj y)) having a valueforally isin Ω0 gives further changes All solutions reaching theimpulse point tj continue to φ(tj + 0) If J SHtimes

Ω0⟶ SH timesΩ0 sub Ω then each trajectory is continuedbeyond the impulse time point However the range R(J)

may be a proper subset of SH timesΩ0 +en the solution φ ofequation (1) fulfilling the initial value problem(s0φ(s0)) isin (SH times (Ω0)∖R(J)) has no history will haveno continuation on the interval (s0 minus δ s0] and the con-tinuation will exist on [s0 s0 + δ) only +ese properties arealso mentioned in [1 14]

141 Impulsive Delayed Differential Equations +e re-search on impulsive delay differential equations is veryintensive as the cited list of some of the publications[2 20ndash37] +e right side of the equations is still continuousor may fulfill Lipschitz conditions and sustains the finitenessof discontinuity points in closed bounded time intervals+emodel of delayed impulsive systems developed by Bainovand his group [38] is based on a discrete set SH sub R ofimpulse points with no accumulation points in any bounded

Journal of Mathematics 3

interval In these models the delayed impact uses these sameset SH of impulse points which regulate the occurrence ofimpulses at impulse time points which maybe a costly as-sumption In some other approaches different ways are usedto meet the condition of local finiteness of the set of dis-continuity points Hence to guarantee the local finiteness ofdiscontinuity points of the right side is increasingly difficultin delayed systems it is worth to see the effect of delay on theright side as presented in the following examples

Important examples let the right side of the impulsivedifferential equation be defined as follows Let

[a b]I(α β) η ≔ (α + β)2 and let α minus h≕ c isin (a b] hgt 0+en let

f(x(t minus ϑ(t)) x(t)) ≔ x(t) + x(t minus ϑ(t)) forallt isin (α β)

g((x(t minus ϑ(t)) x(t)) ≔ 2x(t) + x(t minus ϑ(t)) forallt isin (α β)

SHcap(α β) empty c isin SH

(14)

Let the right continuous solution of the initial valueproblem of the equation be

x(t) x0 + 1113938t

t0(f(x(s minus ϑ(s)) x(s))ds + g(x(s minus ϑ(s)) x(s))dτ) t isin t0 b1113858 1113859

x t0( 1113857 x0 t0 isin (a c)SH(15)

Assume that x(c minus 0) 1 andx(c + 0) minus 1Let ϑ be continuous ascending function ϑ(t)lt tforallt isin

(α β) Let u(t) ≔ t minus ϑ(t)rArrϑ(t) t minus u(t)We will now show some simple examples to demonstrate

that delay equations may lead to differential equations withmeasurable right sides

(1) Let u(t) ≔ c minus ε(t minus α)(η minus t)(β minus t)forallt isin (α β) +iswill give u(α) u(η) u(β) c hence x(η minus 0)

1ne minus 1 x(η + 0) +erefore f(x(t minus ϑ(t)) x(t))

has both left and right limits which are not the sameHence f is measurable and not a continuousfunction of t in [a b] and ϑ is ascending with suitableselection of ε

(2) Let tj ≔ η minus ((β minus α)2j) 1le jltinfin tjη if j⟶infin +en

u(t) ≔c +

(minus 1)j

22(η minus α)t minus tj1113872 1113873

2tj+1 minus t1113872 1113873

2 if tj le tle tj+1 j isin N

0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(16)

+e function u(t) cforallt ti and u(t)lt c ifti lt tlt ti+1 and i is odd and u(t)gt c if ti lt tlt ti+1 and i

is even Hence x(ti minus 0) minus 1 and x(ti + 0) 1 if i isodd and x(ti minus 0) 1 andx(ti + 0) minus 1 if i is evenHence lim suptηx(t minus ϑ(t)) 1 and lim inf tηx(tminus

ϑ(t)) minus 1+e delayed solution x(t minus ϑ(t)) with this delay hasno left limit hence no limit at η isin (α β) Further-more there is no limit at tjforall1le jltinfin Hence f

with delayed arguments is measurable and notcontinuous function of t in [a b]

(3) Continuous descending delay leads to bijectivemapping of the impulse points hence in this casethere are no accumulation points of the images ofimpact points but the statement about measurableright side remains valid

Conclusion 1 Examples 1 and 3 can be handled with thehelp of the existence theorems such as CauchyndashPeanorsquos orPiccardndashLindelofrsquos [9] since the discontinuity points have noaccumulation points +e second example however re-quires limit theorems and additional reasoning If Example 2

is combined with the construction of Cantorrsquos triadic set[39] then we get a set of discontinuity points of continuumcardinality +is means that alternative approach may benecessary to handle such initial value problems

2 Extended Impulsive Differential Equationsand Existence of Their Solution

+e extended impulsive differential equations meanchanging some basic assumptions used in Bainovrsquos model asdescribed in equation (1) or in rewritten form in equations(3) and (10) Major changes include the time control of theimpulses may have infinite discontinuities but has to be ofbounded variation on every closed bounded interval and thesystem dynamics is measurable as a function of the time andnot necessarily continuous

21 e Extended Impulsive Differential Equations Let theprocess evolve in an open time interval T sub R and in an openset T timesΩ0≕Ω sub T times Rn Let f g Ω⟶ Rn be measurablefunctions in the time variable t for each fixed spatial valuex isin Ω0 and continuousLipschitz-continuous functions inthe spatial variable x for each fixed t isin T


Let τ T⟶ R+ be a singular ascending function of thetime parameter t as the singular ldquoimpulse timerrdquo It is im-portant to see that τ may have a countably infinite set ofjump points where the total lengths of these jumps must bebounded on any closed bounded interval Using equation(8) we can rewrite the RadonndashNikodym derivatives in termsof characteristic functions of the sets Nλ

λ andNτλ as follows

d λλ + τλ1113872 1113873

d]λdλλ

d]λ+dτλ

d]λ 1rArr

dλλ

d]λ χNλ

λand

dτλ

d]λ χNτ

λ

(17)

Putting these into equation (10) and changing 1113957g with gwe get the extended impulsive differential equation in t-scaleas

x(t) x0 + 1113938t

t0f(s x(s))χNλ

λ+ g(s x(s))χNλ

τ1113874 1113875d]λ t isin T


+e integral transformations discussed in Section 235will give a similar result in both ]-scale and τ-scale We willhandle the ]-scale representation first Let f] ≔((f(s x(s))χNλ

λ) middot 1113954μ]λ N]

λ⟶ Rn and g] ≔ ((g(s x(s))

χNλτ) middot 1113954μ]λ N]

τ⟶ Rn +en let

h] ≔ f

]χN]λ

+ g]χN]

τ a b]1113858 1113859 timesΩ0⟶ R

n(19)

be the measurable right side of the extended impulsivedifferential equation in ]-scale

We will use the notations [a bλ] sub T for t-scale [a b]]

for the generated ]-scale and [a bτ] for τ-scale to get theadvantages of compact sets

In Section 22 we will discuss the main results of thispaper which is formulation of the extension of Car-atheodoryrsquos existence theorem for the extended impulsivedifferential equations with measurable right side +e basisof our discussion is the approach presented in pg 43 in [9]

22 Caratheodoryrsquos eorem We present Caratheodoryrsquosexistence theorem in Rn as it is presented in the cited pages42-43 for one dimension

We are considering a process on an open setS sub Ω sub R times Rn Let f S⟶ Rn be a function not neces-sarily continuous

Problem (E) find an interval I sub [a b] and an absolutecontinuous function φ I sub (a b)⟶ Rn such that

(t φ(t)) isin S

φprime(t) f(tφ(t)) almost all t isin I(20)

+en the function φ I⟶ Rn is a solution of equation(20) in the extended sense

Caratheodoryrsquos existence theorem [9] targets findinga solutions to problem (E) with an initial value(t0 ξ) isin Ωφ(t0) ξ where the right side is a measurablefunction of t for each fixed x isin Ω0 on T timesΩ0 whereemptyne (α β) T sub R andΩ0 sub Rn are open sets Car-atheodoryrsquos condition for the existence of the solution is theexistence of a local positive integrable dominant m (t0 minus

c t0 + c) sub T⟶ R+ 0 and εgt 0 such that f(t x)le

m(t) forall(t x) isin (t0 minus c t0 + c) times Bε(ξ) +is conditionguarantees that for any measurable curve φ (t0 minus c

t0 + c)⟶ Bε(ξ) the measurable function f(tφ(t))

t isin (t0 minus c t0 + c) is integrable in the intervals (t0 minus c

t0) and (t0 t0 + c) by f(tφ(t)) lem(t) forallt isin (t0 minus c

t0 + c)

Definition 1 Let a point (t0 ξ) isin Ω be selected and letRδε(t0 ξ) ≔ (t0 minus δ  t0 + δ) times Bε(ξ) sub Ω 0lt δ ε be a cyl-inder Let f Ω⟶ Rn be a measurable function +en wewill call f locally t-integrable at a point (t0 ξ) isin Ω if thereexists a cylinderRδε(t0 ξ) sub Ω 0lt δ  ε and an dominatingintegrable function (D I F) m (t0 minus δ t0 + δ)⟶ R+ 0

to f on the cylinder Rδε(t0 ξ) such that f(t x)lem(t)forall(t x) isinRδε(t0 ξ)

Theorem 1 (Caratheodory) Let f S⟶ Rn be measur-able in t for each fixed x and let it be continuous in x for eachfixed t forall(t x) isin S Let (t0 ξ) isin S be a fixed point and leta cylinder Rδε(t0 ξ) sub S exist with a dominating integrablefunction (DIF) m (t0 minus δ  t0 + δ)⟶ R+ 0 to f on thecylinderRδε(t0 ξ) en there exists a solution φ of problem(E) in an extended sense in an interval (t0 minus β  t0 + β)

0lt βle δ such that (t  φ(t)) isinRδε(t0 ξ)forallt isin (t0 minus β  t0 +

β) and φ(t0) ξCaratheodory actually proved the existence on an in-

terval [t0 t0 + β) interval and used this result to prove theexistence on (t0 minus β t0] by using suitable transformations ofsymmetry

Using Caratheodoryrsquos theorem we can prove the existenceof solution of the extended impulsive differential equationequation on ]-scale with right side (19) precisely

Corollary 2 Let f] N]λ timesΩ0⟶ Rn and g] N]

τtimes

Ω0⟶ Rn hence let h] ≔ f]χN]λ

+ g]χN]τbe measurable in σ

for each fixed x and let it be continuous in x for each fixed σforall(σ x) isin Ω Let (σ0 ξ0) isin Ω be a fixed point and let a cyl-inder Rδε(σ0 ξ0) sub Ω exist with a DIFm [σ0  σ0 + δ)⟶ R+ 0 onRδε(σ0 ξ0) to h]en thereexists an interval [σ0  σ0 + β) 0lt βle δ for the equation


φ(σ) ξ0 + 1113946σ

σ0f](v φ(v))χN]

λ+ g

](v φ(v))χN]

τ1113874 1113875d] ξ0 + 1113946

σ

σ0h](vφ(v))d] (21)

such that equation (21) has a solution φ in that interval suchthat (σ  φ(σ)) isinRδε(σ0 ξ)forallσ isin [σ0  σ0 + β) and φ(σ0)

ξ0

Corollary 3 Let h] defined by equation (15) beB]

λ[ab]]c-measurable as a function of σ for each fixed x

forall(σ x) isin [a b]] timesΩ0 in addition to the conditions of Cor-ollary 2 en the solution of initial value problem φ(σ0)

ξ0 (σ0 ξ0) isin N]λ timesΩ0 for equation (17) exists on an interval

[σ0 σ0 + β) for a suitable βgt 0 and xλ ≔ φ(μλ+) is a solutionof the initial value problem xλ(t0) ξ0 with t0 1113954μλ(σ0)imposed on the equation

φ μλ+(t)1113872 1113873 ξ0 + 1113946μλ+(t)

σ0h](vφ(v))d]]λ ξ0 + 1113946

t

t0

h] μλ+(s)φ μλ+(s)1113872 11138731113872 1113873d]λrArr

xλ(t) ξ0 + 1113946t

t0

hλ

1113954μλ μλ+(s) xλ(s)1113872 1113873d]λ ξ0 + 1113946t

t0

hλ

s xλ(s)( 1113857d]λ1113888

(22)

is identity follows from equation (13) Details will bediscussed later

Remark 1 Note that the condition t0 notin Dλ used in Corollary3 is in the Bainovian initial value problem (4) +erefore theBainovian case with measurable right side is covered by thissimple example

Note also that h] can beB([a b]] ])-measurable whichis not covered in this corollary

Remark 2 With Corollary 2 the discussion about generalexistence theorem has been finished +e rest of this paperwill target to include the solutions of equations withB([a b]] ])-measurable right sides and the formulation ofthe conditions for the existence of solutions in terms of themeasurable functions fλ and gτ Some issues will have to beclarified about the initial value problems

As shown in the example solutions of the impulsivedifferential equations are obtained from the absolute con-tinuous solutions on the ]-scale with the help of suitabletransformations We will develop some extensions of themappings μλλminus μλλ+ and 1113954μλ

Uniqueness the solution of an initial value problem isunique if the right side of the differential equation fulfills localLipschitz condition [9] Although there are other conditionsfor uniqueness we will demonstrate our presentation on thisconditionWe concluded in Section 14 about continuation ofsolutions that for impulsive differential equations the so-lution of an initial value problem (t0φ(t0)) (t0 ξ0) isin Ωexists on an interval [t0 t0 + δ) sub T if the conditions of one ofthe existence theorems hold forall(t0 ξ0) isin Ω

Let us consider the impact of this condition on an ex-ample (originating from [9]) Let the differential equation beas follows

yprime

0 minus infinlt tlt minus 1

2y

t minus 1le tlt 0 y isin R

0 0le tltinfin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

+e differential equation fulfills local Lipschitz conditionforall(t y) isin R2 on an interval [t t + δt) δt gt 0

Let I ≔ (minus infin minus 1)cup[0infin) then for any (t0 y0) isin I times

R L 1 is a suitable choice as Lipschitz constant in[t0 t0 + δt0

) times Bδt0(y0) sub [t0 t0 + δt0

] times Bδt0(y0) sub I times R

with a suitable δt0gt 0

If (t0 y0) isin [minus 1 0) times R and [t0 t0 + δt0) sub [t0 t0+

δt0] sub [minus 1 0) then L max 2|t| | (t y) isin [t0 t0 + δt0

]times1113966

Bδt0(y0) will serve as the Lipschitz constant +e right side

fulfills a local Lipschitz condition in a suitable neighbour-hood [t0 t0 + δt0

) times Bδt0(y0) at any (t0 y0) isin R2 Hence no

solution trajectory split into two or more trajectories at anypoint tgt t0

However the right side does not fulfill Lipschitz con-dition in any interval (a 0] alt 0 +e solutions of all initialvalue problems y(minus 1) y1 isin R will pass throughφ(0 minus 1 y1) 0 by the formula φ(t minus 1 y1) (y1(minus 1)2)t2+erefore merging of solution trajectories can occur whilesplitting of trajectories is excluded by having local Lipschitzcondition at each point (t y) isin R2 in an interval [t t+

δt) times Bδt(y) However note that if φ1(t1) y1 andφ2(t2)

y2 isin R (t1 y1)ne (t2 y2) then the two global solutions aredifferent by the initial value problems even if φ1(s) φ2(s)

holds at an sgtmax t1 t21113864 1113865 isin RrArrφ1(t) φ2(t) foralltge s Hencethe global solutions of two different initial value problemsare two different trajectories (not necessarily disjointtrajectories)


23 Timescales and eir Density Functions +is sectionsummarises the concepts which will serve as the basis ofmost of our coming discussion and were developed in [10]and partly in [40] We use the notations and conceptsformulated in Section 11 and our starting point will be theintegral equation (4)

We showed that given [a bλ] Nλλ Nλ

τ λλ and τλ we

obtain [a b]] N]λ N]

τ and ]]λ We now show the reverseorder

231 e t-Scale from ]-Scale We show now that[a b]] N]

λ N]τ and ] determines [a bλ] Nλ

λ Nλτ λ

λ and τλSince N]

λ andN]τ are measurable sets and [a b]] is bounded

the characteristic functions of N]λ andN]

τ are ]-integrableLet the time scale interval be [a bλ] with bλ ≔ a + 1113938

b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]

λd] isin a bλ1113858 1113859 foralls isin a b]1113858 1113859

μλλminus (s) ≔ inf 1113954μminus 1λ ( s ) isin a b]1113858 1113859 foralls isin a bλ1113858 1113859

μλλ(s) ≔ μλλ+(s) ≔ sup 1113954μminus 1λ ( s ) isin a b]1113858 1113859 foralls isin a bλ1113858 1113859


]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]

τ1113874 1113875d]] a + λλ([a s)) + τλ([a s)) foralls isin a bλ1113858 1113859

(24)

where ]λ is right continuous Since ]λ is strictly ascendingNλ

λ (]λ)minus 1(N]λ) andNλ

τ (]λ)minus 1(N]τ) and τλ is singular

with respect to λλ and λλ is singular with respect to τλ by

λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)

SH ≔ Dλ ≔ t | μλminus (t)ne μλ+(t) t isin [a bλ]1113966 1113967 is thecountable set of discontinuity points in t-scale From nowon we will use Dλ in the place of SH

Hence based on equation (22) withh] xλ ≔ φ(μλ+) and t0 ≔ 1113954μλ(σ0) and by equation (17)χNλ

λd]λ dλλ and χNλ

τd]λ dτλ gives us a t-scale-based

impulsive differential equation

xλ(t) ξ0 + 1113946μλ+(t)

σ0h] μλminus (v) xλ(v)1113872 1113873d]]λ

ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ

1113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981absolute continuous

+ g] μλ+(s) xλ(s)1113872 1113873dτλ

1113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981singular

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)

It is important to note that the ]-scale concepts definea second impulsive system with τ

232 e τ-Scale from ]-Scale We show now that[a b]] N]

λ N]τ and ] determine [a bτ] Nτ

λ Nττ λτ and ττ

As stated in Section 231 the characteristic functions ofN]

λ andN]τ are ]-integrable Let the time scale interval be

[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]

τd] isin a bτ1113858 1113859 foralls isin a b]1113858 1113859

μτminus (s) ≔ inf 1113954μminus 1τ ( s ) isin a b]1113858 1113859 foralls isin a bτ1113858 1113859

μτ(s) ≔ μτ+(s) ≔ sup 1113954μminus 1τ ( s ) isin a b]1113858 1113859 foralls isin a bτ1113858 1113859

1113954μτ middot μττminus (t) 1113954μτ middot μττminus (t) id abτ[ ]

]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]

τ1113874 1113875d]] a + λτ([a s)) + ττ([a s)) foralls isin a bτ1113858 1113859

(27)


where ]τ is right continuousSince ]τ is strictly ascending hence bijective Nτ

λ

(]τ)minus 1(N]λ) Nτ

τ (]τ)minus 1(N]τ) and

λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)

which means ττ is singular with respect to λτ and λτ issingular with respect to ττ

Dτ ≔ t | μτminus (t)ne μτ+(t) t isin [a bτ]1113966 1113967 is the countable setof discontinuity points in τ-scale Hence based on equation(22) with h] xτ ≔ φ(μτ+) and ϑ0 ≔ 1113954μτ(σ0) and based onequations (17) and (27) χNτ

λd]τ dλτ and χNτ

τd]τ dττ

gives us a τ-scale-based impulsive differential equation

xτ(ϑ) ξ0 + 1113946μτ+(ϑ)

σ0h] μτminus (v) xτ(v)1113872 1113873d]]τ

ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ

1113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981absolut continuous

⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)

233 Interpretation of the Two Representations We need aninterpretation of the two representations (26) and (29) of theBainovian impulsive systems +e impulsive process de-scribed by equations (1) and (4) is composed from a processwith f-dynamics and a process with g-dynamics In t-scaleequation (26) the fully described process with f-dynamics(absolute continuous component) is exposed to impulsesgenerated by the (singular) impulse generator with g-dy-namics +e second τ-scale representation equation (29)gives a full description of the (absolute continuous) impulsegenerator with g-dynamics while being exposed to the(singular) counter impacts caused by the process with f-dynamics +us these systems operate in action processreaction-counter action processes which is a deep principlein interactions in sciences

+e ]-scale representation presents both processes in fulldetails in a time-sharing system +e ]-scale process ispresented in equation (30) +e ]-scale process is absolutecontinuous and makes it possible to prove existence theo-rems using Caratheodoryrsquos techniques

In our presentation we use the model in equations (1)(4) and (10) +e f]-dynamics operates in intervals[μλminus (ti) μλ+(ti+1)) sub N]

λforallti ti+1 isin Dλ while the g]-dy-namics operates in intervals [μλminus (ti) μλ+(ti)) sub N]

τ

forallti isin Dλ Hence the ]-scale consists of connected intervalsalternating between f]-dynamics and g]-dynamics +eactions of thef]-dynamics and g]-dynamics are as indicatedby the column headings in equation (30)+e impulse pointsselected are ti ti+1 ti+2 and ti+3 isin Dλ ti lt ti+1 lt ti+2 lt ti+3

f] χN]τ

0 τ const g] χN]λ

0 t const

middot middot middot μλλ+ ti( 1113857 μλλminus ti+1( 11138571113960 1113873 ⟶ μλλminus ti+1( 1113857 μλλ+ ti+1( 11138571113960 1113873 ⟶

μλλ+ ti+1( 1113857 μλλminus ti+2( 11138571113960 1113873 ⟶ μλλminus ti+1( 1113857 μλλ+ ti+2( 11138571113960 1113873 ⟶

μλλ+ ti+2( 1113857 μλλminus ti+3( 11138571113960 1113873 ⟶ μλλminus ti+2( 1113857 μλλ+ ti+3( 11138571113960 1113873 middot middot middot

ti ti+1 ti+2 ti+3 isin Dλ

(30)

+is scheme of operation follows the rules of timedchess game +e players are f]-dynamics and g]-dynamics+eir clocks are t and τ respectively If a solution ofequations (1) and (4) is φ [ti ti + δ)⟶Ω with a suitableδ gt ti+3 minus ti then f] plays on the interval [μλλ+(ti) μλλminus (ti+1))

for a period of ti+1 minus ti t-time while g] waiting with stoppedτ-time till f] produces φ(ti+1 minus 0) Actions of f] are un-known to g] At ti+1 the game switches to g] the t-clockstops τ-clock operates and g] performs its job for a periodof μλλ+(ti+1) minus μλλminus (ti+1) of τ-time +e actions of g] are notknown by f] +is continues in this order until the solutionexits Note that the intervals are left-closed right-open by

the fact that the new player starts to play at the leftmostpoint of its domain

+erefore in the t-scale process all g-actions are hiddenand each g-interval appears in the form of jump In the caseof τ-scale representation all f-intervals appear as jumpsHence the two equations are completely symmetric andform a pair of impulsive differential equations

234 Initial Value Problem for a Pair of Impulsive Differ-ential Equations From the interpretation of a pair of im-pulsive differential equation follows that the concept of


initial value problem as presented in the Bainovian model(1) (4) and (10) requires some clarifications +e Bainovianmodel discussed in Section 233 has form (4) or more likely(10) and the initial time t0 isin [a bλ]Dλ cannot be a dis-continuity point From the analysis of equation (30) followsthat any change between f]⟶ g] or g]⟶ f] takes placestarting from the leftmost point of the domaininterval of thenew dynamics Hence [μλλ+(ti) μλλminus (ti+1)) sub N]

λ is in thedomain of f] and [μλλminus (ti+1) μλλ+(ti+1)) sub N]

τ is in the do-main of g] in the [ti ti+1] interval forallti ti+1 isin Dλ +ereforeforallσ0 isin [a b]] either σ0 isin N]

λrArr1113954μλ(σ0) t0 isin Nλλ sub [a bλ] or

σ0 isin N]τrArr1113954μτ(σ0) ϑ0 isin Nτ

τ sub [a bτ]Using the left closed right open intervals in equation (30)

we obtain a pair of impulsive differential equations such thatany initial value problem (σ0 ξ) isin [a b]] timesΩ0φ(σ0) ξhas a solution on an interval [σ0 σ0 + δ0) sub N]

λ if σ0 isin N]λ or

has a solution on an interval [σ0 σ0 + δ0) sub N]τ if σ0 isin N]

τ Finally an initial value problem will give initial valueproblems on the t-scale and on the τ-scale as follows Since

the solution is xλ φ middot μλ+ and xτ φ middot μτ+ we simply candefine the initial value problems as follows

σ0 isin N]λ rArrt0 ≔ 1113954μλ σ0( 1113857 and σ0 μλ+ t0( 1113857 ξ0λ ≔ φ μλ+ t0( 11138571113872 1113873 ξ0

ϑ0 ≔ 1113954μτ σ0( 1113857 ξ0τ ≔ φ μτ+ ϑ0( 11138571113872 1113873 hence let σ0τ ≔ μτ+ ϑ0( 1113857

(31)

Similarly for the case σ0 isin N]τ

σ0 isin N]τ ϑ0 ≔ 1113954μτ σ0( 1113857 and σ0 μτ+ ϑ0( 1113857rArrξ0τ ≔ φ μτ+ ϑ0( 11138571113872 1113873 ξ0

t0 ≔ 1113954μλ σ0( 1113857 ξ0λ ≔ φ μλ+ t0( 11138571113872 1113873 hence let σ0λ ≔ μλ+ t0( 1113857(32)

235 Integral Transformations among t- τ- and ]-Scales+e details of the assertions in this section come from paper[10]

We summarize the mappings between t-scale to ]-scalelisted in Section 231 (Table 1)

+e mappings τλ and ]λ are not continuous ascendingfunctions with a common set of discontinuity points Dλ+erefore the measures τλ and ]λ are defined on the sem-iring P][abλ]c [s t) | s t isin [a bλ]Dλ sle t1113864 1113865 and themeasures τλ([s t)) ≔ τλ(t) minus τλ (s) and ]λ[(s t)) ≔ ]λ(t) minus ]λ(s)forall [s t) isin P] [a bλ] c can be extended toB][abλ]c σ(P][abλ]c)

+e mappings μλλminus μλλ+ [a bλ]⟶ [a b]] map the setof discontinuity points Dλ into the set of left-closed right-open intervals

D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ


Moreover the mappings μλλminus μλλ+ are bijective on the setof continuity points [a b]]D]

λ and μλminus (t) μλ+(t)

forallt isin [a bλ]Dλ+e mappings μλλminus μλλ+ [a bλ]⟶ [a b]] transform



by their continuity points withP]

][ab]]c≔ [s t) | s t isin [a b]]D]

λ1113864 1113865+e measure ]]λ on P]

][ab]]cis defined by

]]λ([u v)) ≔ v minus u μλminus (1113954μλλ(v)) minus μλminus (1113954μλλ(u)) ]λ(1113954μλλ([u

v)))forall[u v) isin P]][ab]]c

Also if [s t) isin P][abλ]c then

[μλminus (s) μλminus (t)) isin P]][ab]]c

and ]]λ([μλminus (s) μλminus (t)))

μλminus (t) minus μλminus (s) ]λ([s t)) Let the smallest σ-algebracontaining the semiring P]

λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c

) with the extended measure ]]λ on itFrom this follows that if h [a b]]⟶ R is

B]λ[ab]]c

-measurable and ]]λ-integrable then

1113946μλλ+

(t)

ahd]]λ 1113946

t

ah middot μλλminus d]λ forallt isin a bλ1113858 1113859 (35)

Conversely if h [a bλ]⟶ R is Bλ[ab]]c-measurable

and ]λ-integrable then

1113946σ

ah middot 1113954μλd]

]λ 1113946

1113954μλ(σ)

ahd]λ forallσ isin a b]1113858 1113859D

]λ

(36)

+ese are some of the main conclusions from paper [10]presented here in a condensed form

We summarize the mappings between τ-scale and]-scale listed in Section 232 (Table 2)

+e mappings λτ and ]τ are not continuous ascendingfunctions with a common set of discontinuity points Dτ +erefore the measures λτ and ]τ are defined on the sem-iring P][abτ]c [s t) | s t isin [a bτ]Dτ sle t1113864 1113865 and themeasures λτ([s t)) ≔ λτ(t) minus λτ(s) and ]τ([s t)) ≔ ]τ(t) minus

]τ(s) forall[s t) isin P][abτ]c can be extended to B][abτ]c

σ(P][abτ]c)+emappings μττminus μττ+ [a bτ]⟶ [a b]]map the set of

discontinuity points Dτ into the set of left-closed right-openintervals

D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)


Table 1+emappings in the first column are strictly ascending leftand right continuous versions of ]λ while the mapping in thesecond column is absolute continuous and ascending

[a bλ]⟶ [a b]][a bλ]⟷ [a b]]

[a b]]⟶ [a bλ]Relation

μλλminus

μλλ ≔ μλλ+

1113954μλ1113954μλ middot μλλminus id[abλ]

1113954μλ middot μλλ+ id[abλ]


D]τ ≔ cup

tisinDτ

μτminus (t) μτ+(t)1113960 1113873 sub a b]1113858 1113859 (38)

Moreover the mappings μττminus and μττ+ are bijective on the

set of continuity points [a b]]D]τ and μτminus (t) μτ+(t)

forallt isin [a bτ]Dτ +e mappings μττminus μττ+ [a bτ]⟶ [a b]] transform

[s t) isin P][abτ]c⟶ []τ(s) ]τ(t)) [μττminus (s) μττminus (t))

[μττ+(s) μττ+(t)) isin P]][ab]]c



λ1113864 1113865+e measure ]]τ on P]

][ab]]cis defined by ]]τ([u v)) ≔

v minus u μτminus (1113954μττ(v)) minus μτminus (1113954μττ(u)) ]τ(1113954μττ([u v)))forall[u v)

isin P]][ab]]c

Also if [s t) isin P][abτ]c then [μτminus (s) μτminus (t))

isin P]][ab]]c

and ]]τ([μτminus (s) μτminus (t))) μτminus (t) minus μτminus (s)

]τ([s t)) Let the smallest σ-algebra containing the semiringP]

τ[ab]]cbe B]

τ[ab]]c≔ σ(P]

τ[ab]]c) with the extended

measure ]]τ on itFrom this follows that if h [a b]]⟶ R is

B]τ[ab]]c

-measurable and ]]τ-integrable then

1113946μττ+(t)

ahd]]τ 1113946

t

ah middot μττminus d]τ forallt isin a bτ1113858 1113859 (39)

Conversely if h [a bτ]⟶ R is Bτ[ab]]c-measurable

and ]τ-integrable then

1113946σ0

ah middot 1113954μτd]

]τ 1113946

1113954μτ σ0( )

ahd]τ forallσ isin a b]1113858 1113859D

]τ

(40)

+ese are some of the main conclusions of paper [10]presented here in a condensed form

+e details about measures and RadonndashNikodym de-rivatives summarised below are in paper [10]

Let B([a b]] ]) denote the Borel sets on [a b]] and let] be the Lebesgue measure on the σ-algebra B([a b]] ])

+enB]

λ[ab]]csubB([a b]] ]) andB]

τ[ab]]csubB([a b]] ])

Let f [a b]]⟶ Rn be a B([a b]] ])-measurable]-integrable function Let ]fλ(A) ≔ 1113938

Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A

fd]forallA isinB]τ[ab]]c

be signed measuresabsolute continuous with respect to the measures ]]λ and ]

]τ

respectively +en their RadonndashNikodym derivatives withrespect to ]]λ and ]

]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A

1113954f]λd]]λ forallA isinB

]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A

1113954f]τd]]τ forallA isinB

]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)

Combining equations (35) and (41) gives

1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a

1113954f]λ middot μλλminus d]λ forallt isin a bλ1113858 1113859

(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a

1113954f]τ middot μττminus d]τ forallϑ isin a bτ1113858 1113859

(44)

It was proved in paper [40] that the RadonndashNikodymderivatives 1113954f]λ and 1113954f]τ fulfill the relations

f(s) 1113954f]λ(s) aes isin a b]1113858 1113859D]λ

f(s) 1113954f]τ(s) aes isin a b]1113858 1113859D]τ

(45)

which implies the assertions below as follow ups of theindicated equation (45)

1113946μλλ+

(t)

af times χ ab][ ]D]

λd] 1113946

μλλ+(t)

a

1113954f]λ times χ ab][ ]D]λd]]λ

(43)1113946μλλ+

(t)


λd]]λ

1113946t

af times χ abλ[ ]Dλ

1113874 1113875 middot μλλminus d]λ forallt isin a bλ1113858 1113859

(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a

1113954f]τ times χ ab][ ]D]τd]]τ

(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ

af times χ abτ[ ]Dτ

1113874 1113875 middot μττminus d]τ forallϑ isin a bτ1113858 1113859

(47)

Table 2+emappings in the first column are strictly ascending leftand right continuous versions of ]τ while the mapping in thesecond column is absolute continuous and ascending

[a bτ]⟷[a b]][a bτ]⟷ [a b]]

[a b]]⟶ [a bτ]Relation

μττminus

μττ ≔ μττ+

1113954μτ1113954μτ middot μττminus id[abτ]

1113954μτ middot μττ+ id[abτ]


+is relation enables us to transform the solution of a ]-scale differential equation into solutions of a t-scaleτ-scaleimpulsive differential equations

+e function f can be written as f f times χ[ab]]D]λ

+ f times

χD]λ

by [a b]] ([a b]]D]λ)cupD

]λ andempty ([a b]]D

]λ)cap

D]λ

Similarly f f times χ[ab]]D]τ

+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD

]τ andempty ([a b]]D

]τ)capD

]τ

+erefore equations (46) and (47) can be rewritten as

1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa

f times χ abτ[ ]Dτ+

1113956f times χD]

τ1113872 1113873]τ1113874 1113875 forallϑ isin a bτ1113858 1113859

(48)

+e sets in D]λ and in D]

τ are atoms as described inLemma 29 in [10] in detail Since an atom inB]

λ[ab]]cdoes

not have any proper measurable subset in B]λ[ab]]c

but thesame set is a nonatomic measurable set inB([a b]] ]) f canbe integrated on it by ] Similarly an atom inB]

τ[ab]]cdoes

not have any proper measurable subset in B]τ[ab]]c

but thesame set is a nonatomicmeasurable set inB([a b]] ]) and fcan be integrated on it by ] Hence foralltj isin Dλ the set[μλminus (tj) μλ+(tj)) isin D]

λ is an atom Similarly forallτj isin Dτ theset [μτminus (τj) μτ+(τj)) isin D]

τ is an atom +erefore theRadonndashNikodym derivatives in equation (48) can be writtenas follows

1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857

μλminus tj( 1113857fd]

μλ+ tj1113872 1113873 minus μλminus tj1113872 1113873

1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857

μτminus τj( 1113857fd]

μτ+ τj1113872 1113873 minus μτminus τj1113872 1113873

(49)

24 Existence of the Solutions on t-scale andon τ-Scale In thissection we want to formulate the existence theorem for thepair of impulsive differential equations on the t-scale andτ-scale

First we assume that Corollary 2 is true +e initial valueproblem in equation (21) has a solution φ [σ0 σ0+β)⟶Rδε(σ0 ξ)φ(σ0) ξ

Applying the statements in equations (43) and (44) toequation (21) we get that the solution in t-scale can bexλ(t) φ middot μλλ+(t) t isin [t0 1113954μλ(σ0 + β)) t0 1113954μλ(σ0) and inτ-scale it can be xτ(ϑ) φ middot μττ+(ϑ) ϑ isin [ϑ0 1113954μτ(σ0 + β))

ϑ0 1113954μτ(σ0) which transformed ]-scale solutions We haveto prove that they fulfill the initial value problems withrespect to the t-scale and τ-scale versions and that also theyfulfill the respective impulsive differential equations

+e initial value problem φ(σ0) ξ on the ]-scaleproblem fulfills either σ0 isin N]

λ which is detailed out inequation (31) or it fulfills σ0 isin N]

τ which is detailed out in(32) Hence the two cases of initial value problems fulfilledby the pair of impulsive differential equations are as follows

When σ0 isin N]λ t0 ≔ 1113954μλ(σ0) and ξ0λ ≔ φ(μλ+(t0)) ξ

and ϑ0 ≔ 1113954μτ(σ0) and ξ0τ ≔ φ(μτ+(ϑ0))When σ0 isin N]

τ ϑ0 ≔ 1113954μτ(σ0) and ξ0τ ≔ φ(μτ+(ϑ0)) ξwhile t0 ≔ 1113954μλ(σ0) and ξ0λ ≔ φ(μλ+(t0))

Note that the following equations (31) and (32) for theinitial value problems are given in t-scale uniformly (t0 ξ0λ)

and similarly for τ-scale uniformly (ϑ0 ξ0τ) but the valuesare obtained differently according to equations (31) and (32)(the difference is whether ξ0λ ξ or ξ0τ ξ)

In these equations we used that μλ+ Nλλ⟶ N]

λ andμτ+ Nτ

τ⟶ N]τ which are bijective strictly ascending

mappings Hence the initial conditions are fulfilledSolution 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 h]

φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0

1113954h]]λ μλλminus1113872 1113873d]λ forallt isin a bλ1113858 1113859

(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0

1113954h]]τ μττminus1113872 1113873d]τ forallϑ isin a bτ1113858 1113859

(51)


Let us split h] on ]-scale with D]λ into a component on

continuity points h]λc ≔ h] times χ[ab]]D

]λand a component on

discontinuity points h]λd ≔ h] times χD]

λ We can do this with

discontinuity points D]τ on ]-scale also A component on

continuity points is h]τc ≔ h] times χ[ab]]D

]τ and a component

on discontinuity points is h]τd ≔ h] times χD]

τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)

Let us put the definition h] ≔ f]χN]λ

+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)

Considering N]λ sub [a b]]D

]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]

λ1113980radicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradic1113981h]λc

+ g]

times χD]λ1113980radicradicradic11139791113978radicradicradic1113981

h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]

τ1113980radicradicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradicradic1113981h]τc

+ f]

times χD]τ1113980radicradicradic11139791113978radicradicradic1113981

h]τd

(54)

Let us apply the relations in equations (50) and (51) toh]λd and h]

τd as expressed in equation (54) (cases ofdiscontinuity)

1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]

times χD]λ(v φ(v))d]

1113946μλλ+

(t)

σ0

1113956g] times χD]

λ1113874 1113875

]λ(v) times χ ab][ ]D]

λd]]λ 1113946

t

t0


λ1113874 1113875

]λmiddot μλλminus d]λ forallt isin a bλ1113858 1113859

(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]

times χD]τ(v φ(v))d] 1113946

μττ+(ϑ)

σ0

1113956f] times χD]

τ1113872 1113873]τ

(v) times χ ab][ ]D]τd]]τ

1113946ϑ

ϑ0


τ1113872 1113873]τ

middot μττminus d]]τ forallϑ isin a bτ1113858 1113859

(56)

Let us apply equations (46) and (47) to the componentshλc and hτc using equation (54)

1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)

σ0h]λc(v φ(v))d]]λ

1113946t

t0

f]χN]

λmiddot μλλminus + g

]χN]τ

times χ ab][ ]D]λ∘μλλminus1113874 1113875d]λ forallt isin a bλ1113858 1113859

(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0

1113955h]τc]τ(v) times χ ab][ ]D]

τd]]τ (43) 1113946

μττ+(ϑ)

σ0h]τc(vφ(v))d]]τ

1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]

τ∘μττminus + g

]χN]τ∘μττminus1113874 1113875d]τ forallϑ isin a bτ1113858 1113859

(58)

where equation (45) is obtained from Corollary 2 and+eorem 1 in [40]

+e RadonndashNikodym derivatives of h]λd and h]

λd re-mains to be determined +e domains of theRadonndashNikodym derivatives are countable unions of pair-wise disjoint atoms of the σ-algebrasB]

λ[ab]]candB]

τ[ab]]c

respectively as defined in equations D]λ in (11) D]

τ in (37)+e lists of these atoms are defined D]

λ in (12) and D]τ in

(38) +is means that the RadonndashNikodym derivatives are

fully determined on D]λ if they are determined on the in-

tervals in D]λ and they are fully determined onD]

τ if they aredetermined on each interval in D]

τ Let impulse time points tj isin Dλ and τj isin Dτ be selected

and let φ [t0 t0 + β)⟶ Bε0(x0) be the solution of equation(21)

Based on equation (54) h]λd and h]

τd is replaced byg] times χD]

λandf] times χD]

τ respectively in equation (49) +is

leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857

μλminus tj( 1113857g]d]

μλ+ tj1113872 1113873 minus μλminus tj1113872 1113873φ μλλ+ tj1113872 11138731113872 1113873 minus φ μλλminus tj1113872 11138731113872 1113873

μλλ+ tj1113872 1113873 minus μλλminus tj1113872 1113873≕ 1113954g0

]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857

μτminus τj( 1113857f]d]

μτ+ τj1113872 1113873 minus μτminus τj1113872 1113873φ μττ+ tj1113872 11138731113872 1113873 minus φ μττminus tj1113872 11138731113872 1113873

μτ+ τj1113872 1113873 minus μτminus τj1113872 1113873≕ 1113954f0

]τ τjφ1113872 1113873

forallτj isin Dτ

(60)

Let T([a b]]) denote all the solution trajectories withdomains as subsets of [a b]]

With equations (59) and (60) 1113954g0]λ is defined on the

interval [μλλminus (tj)) [μλλ+(tj)) foralltj isin Dλ and 1113954f0]λ is defined on

the interval [μττminus (τj)) [μττ+(τj)) foralltj isin Dτ hence 1113954g0]λ is

defined on D]λ and 1113954f0

]τ is defined on D]

τ +en we define1113954g]λ amp1113954f

]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ

1113896 forall(σφ) isin a b]1113858 1113859 times T a b]1113858 1113859( 1113857

1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨

⎩ forall(σφ) isin a b]1113858 1113859 times T a b]1113858 1113859( 1113857

(61)

+ese jumps in this case are dependent on the solutionand the time parameter is based on [a b]] If however theright sides fulfill Lipschitz condition then the solution isdetermined by tjφ(tj)) hence 1113954gλ(tjφ) and 1113954fτ(τjφ) arefully determined by the initial value problems φ(tj)

y (tj y) isin tj1113966 1113967 timesΩ0 and φ(τj) y (τj y) isin τj1113966 1113967 timesΩ0

hence the jumps can be written 1113954gλ(tjφ)⟶ 1113954gλ(tjφ(tj))

and 1113954fτ(τjφ(τj))⟶ 1113954fτ(τjφ(τj)) which is Bainovrsquos for-mulation Combining equations (55) and (57)equation and(56) about h]

λc and h]τc with h]

λd and h]τd and equations (59)

and (60) on the RadonndashNikodym derivatives results inequations

φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0

f] μλλminus φ μλλminus1113872 11138731113872 1113873χNλ

λ+ g

] μλλminus φ μλλminus1113872 11138731113872 1113873χNλτ

times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0

1113954g]λ μλλminus φ1113872 1113873d]λ forallt isin a bλ1113858 1113859

(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ

ϑ0f] μττminus φ μττminus1113872 11138731113872 1113873χNτ

λtimes χ abτ[ ]Dτ

+ g] μττminus φ μττminus1113872 1113873χNτ

τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0

1113954f]τ μλτminus φ1113872 1113873d]τ forallϑ isin a bτ1113858 1113859

(63)


Let us put into equations (62) and (63) the expressionsxλ ≔ φ(μλλminus ) andxτ ≔ φ(μττminus ) and from equation (17)χNλ

λd]λ dλ and χNτ

τd]τ dτ +is leads to

xλ(t) ξ0λ + 1113946t

t0

f] μλλminus xλ1113872 1113873dλλ

1113980radicradicradicradicradicradic11139791113978radicradicradicradicradicradic1113981absolute continuous

+ 1113946t

t0

g] μλλminus xλ1113872 1113873 times χ abλ[ ]Dλ1113980radicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradic1113981

continuous singular

+ 1113954gλ μλλminus φ1113872 11138731113980radicradicradicradic11139791113978radicradicradicradic1113981

pure jumping singular

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠dτλ forallt isin a bλ1113858 1113859

(64)

xτ(ϑ) ξ0τ + 1113946ϑ

ϑ0g] μττminus xτ1113872 1113873dττ


+ 1113946ϑ

ϑ0f] μττminus xτ1113872 1113873 times χ abτ[ ]Dτ1113980radicradicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradicradic1113981

continuous singular

+ 1113954fτ μττminus φ1113872 11138731113980radicradicradicradic11139791113978radicradicradicradic1113981


⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠dλτ forallϑ isin a bτ1113858 1113859

(65)

Conclusion 2 We proved that from the statement ofCorollary 2 follows the existence of the solution of the pairof impulsive differential equations Important to note thatthe function f is fully known in t-scale and in ]-scalewhereas the function g is fully known in τ-scale and ]-scale+erefore we will prove that if fλ is locally t-integrableforall(t x) isin [a bλ] timesΩ0 and gτ is locally τ-integrable forall(ϑ x) isin[a bτ] timesΩ0 then the corresponding h] [a b]]⟶ Rn islocally ]-integrable forall(σ x) isin [a b]] timesΩ0 (see the definitionof the concept of local t-integrability in Definition 1)

Lemma 1 Let [a b] sub R be a closed bounded interval and letψ [a b] timesΩ0⟶ Rn be measurable in its variable t isin [a b]

for any fixed x isin Ω0 and let it be continuous in its variablex isin Ω0 for each fixed t isin [a b] If ψ is locally integrable ateach (t0 x0) isin [a b] timesΩ0 then there exists an 0lt ε isin R andan integrable dominator mx0

[a b]⟶ R+ 0 such thatψ(t x)lemx0

(t) forall(t x) isin [a b] times Bε(x0) sub [a b] timesΩ0forall(t0 x0) isin [a b] timesΩ0

Proof Let ψ be extended to 0 outside [a b] sub R Let(t0 x0) isin [a b] timesΩ0 By the formulation of the lemmaexistRδt0 x0 εt0 x0

(t0 x0) andmt0 x0 (t0 minus δt0 x0

t0 + δt0 x0)⟶ R+

0 such that ψ(t x)lemt0 x0(t) forall(t x) isin (t0 minus δt0 x0

t0+

δt0 x0) times Bεt0 x0

(x0) +en [a b] sub cupsisin[ab](s minus δsx0 s + δsx0

)Since [a b] is compact a finite subsystem (tj x0) | 1le1113966

jleN can be selected such that [a b] sub cupNj1(tj minus δtjx0 tj +

δtjx0) covers the interval [a b] Extending mtjx0

by zerooutside (tj minus δtjx0

tj + δtjx0) forall1le jleN we can form

a positive integrable dominator mx0(t) ≔ max mtjx0

(t) | 1le1113882

jleN forallt isin [a b] and can get a positive εx0≔ min εtjx0

| 11113882

le jleN such that the positive dominator

mx0 [a b]⟶ R+ 0 fulfills ψ(t x)lemx0

(t) forall(t x)

isin [a b] times Bεx0(x0) sub [a b] timesΩ0

+eorem for the existence of solutions of a pair ofimpulsive differential equations let us denote the functions fand g on t-scale by fλ Nλ

λ⟶ Rnandgλ Nλτ⟶ Rn on

τ-scale by fτ Nτλ⟶ Rnandgτ Nτ

τ⟶ Rn and on ]-scaleby f] N]

λ⟶ Rnandg] N]τ⟶ Rn Let h] ≔ (fλ middot 1113954μλ)

χN]λ

+ (gτ middot 1113954μτ)χN]τ [a b]]⟶ Rn

Lemma 2 e following statements are equivalent

Statement 1 the mapping fλ is locally t-integrableforall(t x) isin [a bλ] timesΩ0 and the mapping gτ is locallyτ-integrable forall(ϑ x) isin [a bτ] timesΩ0Statement 2 the mapping h] is locally ]-integrableforall(σ x) isin [a b]] timesΩ0

Proof Statement 1 rArr Statement 2

(1) fλ is l Statement 2 the mapping h] is locally]-integrable forall(σ x) isin [a b]] timesΩ0 and locally t-in-tegrable forall(t0 x0) isin [a bλ] timesΩ0rArrexist0lt ελx0

isin R

andmλx0 [a bλ]⟶ R+ 0 such that fλ(t

x)lemλx0(t)forall(t x) isin [a bλ] times Bελx0

(x0) sub [a bλ]

timesΩ0forallx0 isin Ω0 by Lemma 1(2) gτ is locally τ-integrable forall(ϑ0 x0) isin [a bτ]times

Ω0rArrexist 0lt ετx0isin R andmτx0

[a bτ]⟶ R+ 0

such that gτ(ϑ x)lemτx0(ϑ)forall(ϑ x) isin [a bτ]times

Bετx0(x0) sub [a bτ] timesΩ0forallx0 isin Ω0 by Lemma 1

(3) By point 1 in [a bλ] let x0 isin Ω0 then exist0lt ελx0isin R

such that fλ(t x)lemλx0(t) forall(t x) isin [a bλ]times

Bελx0(x0) sub [a bλ] timesΩ0rArrfλ (1113954μλ(σ) x)lemλx0

(1113954μλ


(σ)) forall(σ x) isin [a b]]timesBελx0(x0) sub [a b]] timesΩ0

Hence with f](σ x) ≔ fλ(1113954μλ(σ) x) timesχN]λ(σ)

f](σ x)lemλx0(1113954μλ (σ)) times χN]

λ(σ)forall(σ x) isin N]

λtimes

Bελx0(x0) sub [a b]] timesΩ0

(4) By point 1 in [a bτ] let x0 isin Ω0 then exist0lt ετx0isin R


Bετx0(x0) sub [a bτ] timesΩ0rArrgτ(1113954μτ(σ) x)lemτ

x0(1113954μτ(σ)) forall(σ x)isin [a b]] times Bετx0(x0) sub [a b]]times

Ω0 Hence with g](σ x) ≔ gτ(1113954μτ(σ) x) times χN]τ(σ)

g](σ x)lemτx0(1113954μτ(σ)) times χN]

τ(σ) forall(σ x) isin N]

λtimes

Bετx0(x0) sub [a b]] timesΩ0

(5) By point 3 f](σ x)lemλx0(1113954μλ(σ)) times χN]

λ(σ) in

N]λ times Bελx0

(x0) and by point 4 g](σ x)lemτx0

(1113954μτ(σ)) times χN]τ(σ) hold in N]

λ times Bετx0(x0) +en

with m]x0(σ) ≔ max mλx0

(1113954μλ(σ)) times χN]λ(σ)1113882

mτx0(1113954μτ(σ)) times χN]

τ(σ)forallσ isin [a b]] and with

ε]x0≔ min ελx0

ετx01113966 1113967 we obtain that h](σ x)

f](σ x) + g](σ x)lem]x0(σ) forall(σ x) isin [a b]]times

Bε]x0(x0)

Statement 2rArr Statement 1 assume thatexist0lt ε]x0

isin R andm]x0 [a b]]⟶ R+ 0 forallx0 isin Ω0 such

that h](σ x)lem]x0(σ)forall(σ x) isin [a b]] times Bε]x0

(x0) whereh] ≔ (f]χN]

λ+ g]χN]

τ) [a b]] timesΩ0⟶ Rn

(1) h](σ x)lem]x0(σ)rArrh](μλminus (t) x)lem]x0

(μλminus

(t)) holds forall(t x) isin [a bλ] times Bε]x0(x0) Multiplying

both sides by the characteristic function of Nλλ we

obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g

] μλminus (t) x1113872 1113873χNλτ(t)1113874 1113875χNλ

λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)

lem]x0μλminus (t)1113872 1113873forall(t x) isin a bλ1113858 1113859 times Bε]x0

x0( 1113857

(66)

which proves the case for the t-scale +e case forτ-scale is word for word the same as the case of t-scaleand is left to the reader

Conclusion 3 With this we proved that the initial valueproblem prescribed for a pair of impulsive differentialequations has a solution if Caratheodoryrsquos condition holdsfor fλ Nλ

λ⟶ Rn and for gτ Nττ⟶ Rn +is condition is

equivalent to the condition of Corollary 2 We state a finalversion of the existence theorem

Condition 1

C1 let fλ (t x) isin Nλλ timesΩ0⟶ fλ (t x) isin Rn and

gτ (ϑ x) isin Nττ timesΩ0⟶ gτ(ϑ x) isin Rn be measurable

functions of t and ϑ for each fixed x respectively andlet they be continuous in x for each fixed t and ϑrespectively forall(t x) isin Nλ

λ timesΩ0 andforall(ϑ x) isin Nττ timesΩ0

C2 let f](σ x) ≔ fλ(1113954μλ(σ) x) andg](σ x) ≔gτ(1113954μτ(σ) x)forall(σ x) isin [a b]] timesΩ0 +en by conditionC1 h](σ x) ≔ f](σ x)χN]

λ(σ) + g](σ x)χN]

τ(σ) is

measurable in σ for each fixed x and it is continuous inx for each fixed σ forall(σ x) isin [a b]] timesΩ0C3 let fλ and gτ be locally t- and τ-integrable on theirrespective domains or equivalently let h] be locally]-integrable on [a b]] timesΩ0

Theorem 2 Let the mappings fλ gτ and h] fulfill Condition1 C1 C2 and C3 Let (σ0 ξ0) isin [a b]] timesΩ0 be a fixed pointand let a cylinder Rδε(σ0 ξ0) sub [a b]] timesΩ0 exist witha DIF m [σ0  σ0 + δ)⟶ R+ 0 on Rδε(σ0 ξ0) to h]en there exists an interval [σ0  σ0 + β) 0lt βle δ such thatequation (21) has a solution φ in that interval such that(σ  φ(σ)) isinRδε(σ0 ξ)forallσ isin [σ0  σ0 + β) and φ(σ0) ξ0

Moreover let t0 1113954μλ(σ0) isin [a bλ] and ϑ0

1113954μτ(σ0) isin [a bλ] Let xλ ≔ φ middot μλλ+ [t0 1113954μλ(σ0 + β))⟶Bε(ξ0) and let xτ ≔ φ middot μττ+ [ϑ0 1113954μτ(σ0 + β))⟶ Bε(ξ0) Letxλ(t0) ≔ φ(μλλ+(t0))≕ξ0λ and let xτ(ϑ0) ≔ φ(μττ+

(ϑ0))≕ ξ0τ +en the following equations hold

xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ

1113954μτ μλλminus (s)1113872 1113873 xλ(s)1113872 1113873 times χ abλ[ ]Dλ+ 1113954g

] μλλminus (s)1113872 11138731113874 1113875dτλ forallt isin t0 1113954μλ σ0 + β( 11138571113858 1113857

xτ(ϑ) ξ0τ + 1113946ϑ

ϑ0gτ η xτ(η)( 1113857dττ

+ 1113946ϑ

ϑ0fλ

1113954μλ μττminus (η)1113872 1113873 xτ(η)1113872 1113873 times χ a bτDτ[ ] + 1113954f]μττminus (η)1113872 11138731113874 1113875dλτ forallϑ isin ϑ0 1113954μλ σ0 + β( 11138571113858 1113857

(67)


Proof In the theorem f](σ x) fλ(1113954μλ(σ) x) andg](σ x) gτ(1113954μτ(σ) x) by 21 C2 +en f](μλλminus (t) x)

fλ(1113954μλ(μλλminus (t)) x) fλ(id[abλ](t) x) fλ(t x) and g](μττminus

(ϑ) x) gτ(1113954μτ(μττminus (ϑ)) x) gτ(id[abτ](t) x) gτ (ϑ x)where we applied the identities listed in equations (24)and (27)

Conclusion 4 +e Bainovian impulsive processes describethe movements of a process (f-dynamics) under the impulsesof another process (g-dynamics) In this paper we de-veloped a technique to split such a system into two impulsiveprocesses One is the Bainovian process moving with f-dynamics under the impulses of the process with g-dy-namics while the other is the process with g-dynamicsunder the impacts of f-impulses on it +e relationship is anactionreaction interaction between the two processes +isgives a pair of impulsive differential equation with the re-lationship between them analysed We established an ex-istence theorem for impulsive differential equations withright side being a measurable function of time which fa-cilitates the analysis of delayed impulsive differentialequations On the contrary trajectories may be connectedtogether by impulse effects +is will make the flow of so-lutions to be a tree structure instead of a connected flow as inordinary differential equations +e leaves of the tree consistof trajectories coming from discontinuity points withouthistory+is tree is directed with orientation from the leavesto the root which is called in-tree or antiarborescence [41]+is gives wide range of modelling facilities by enabling oneto model and study mixing new generations in addition tostudying flows of solutions

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] D D Bainov and P S Simeonov Impulsive DifferentialEquationsndashAsymptotic Properties of the Solutions WorldScientific Pub Coy Pte Ltd Singapore 1995

[2] H G Ballinger Qualitative theory of impulsive delay differ-ential equations PhD thesis University of Waterloo Wa-terloo Canada 1999

[3] V Lakshmikantham D D Bainov and P S Simeonoveory of Impulsive Differential Equations World ScientificPublishing Company Limited Singapore 1989

[4] B O Oyelami ldquoOn military model for impulsive re-inforcement functions using exclusion and marginalizationtechniquesrdquo Nonlinear Analysis eory Methods and Ap-plications vol 35 no 8 pp 947ndash958 1999

[5] B O Oyelami and S O Ale ldquoSolutions of impulsive diffusionand Von-Foerster-Makendrick models using the B-trans-formrdquo Applied Mathematics vol 4 no 12 pp 1637ndash16462013

[6] B O Oyelami ldquoOn singular impulsive systems using iso-morphism decomposition methodrdquo Asian Journal of Math-ematics and Computer Research vol 11 pp 78ndash96 2016

[7] A M Samoilenko and N A Perestyuk Impulsive DifferentialEquations World Scientific Publishing Company Ltd Sin-gapore 1995

[8] I M Esuabana and J A Ugboh ldquoSurvey of impulsive dif-ferential equations with continuous delayrdquo InternationalJournal of Mathematics Trends and Technology vol 60 no 1pp 22ndash28 2018

[9] A E Coddington and N Levinson eory of OrdinaryDifferential Equations McGrawndashHill Book Company NewYork Ny USA 1955

[10] Z Lipcsey I M Esuabana J A Ugboh and I O IsaacldquoIntegral representation of functions of bounded variationrdquoHindawi Journal of Mathematics vol 2019 Article ID1065946 11 pages 2019

[11] R F Bass Real Analysis for Graduate Students Measure andIntegration eory Createspace Ind Pub Scotts Valley CAUSA 2011

[12] U A Abasiekwere I M Esuabana I O Isaac and Z LipcseyldquoExistence theorem for linear neutral impulsive differentialequations of the second orderrdquo Communications in AppliedAnalysis vol 22 no 2 2018

[13] A S Abdel-Rady A M A El-Sayed S Z Rida and I AmeenldquoOn some impulsive differential equationsrdquo MathematicalSciences Letters vol 1 no 2 pp 105ndash111 2012

[14] D D Bainov and I M Stamova ldquoExistence uniqueness andcontinuability of solutions of impulsive differential-differenceequationsrdquo Journal of Applied Mathematics and StochasticAnalysis vol 12 no 3 pp 293ndash300 1999

[15] I M Esuabana U A Abasiekwere J A Ugboh andZ Lipcsey ldquoEquivalent construction of ordinary differentialequations from impulsive systemsrdquo Academic Journal ofMathematical Sciences vol 4 no 8 pp 77ndash89 2018

[16] B Li ldquoExistence of solutions for impulsive fractional evolu-tion equations with periodic boundary conditionrdquo Advancesin Difference Equations vol 2017 no 236 Article ID 2362017

[17] M J Mardanov N I Mahmudov and Y A Sharifov ldquoEx-istence and uniqueness theorems for impulsive fractionaldifferential equations with the two-point and integralboundary conditionsrdquoe Scientific World Journal vol 2014Article ID 918730 8 pages 2014

[18] M J Mardonov Y A Sharifov and K E Ismayliov ldquoExis-tence and uniqueness of solutions for non-linear impulsivedifferential equations with threendashpoint boundary conditionsrdquoE-Journal of Analysis and Mathematics vol 1 pp 21ndash28 2018

[19] D Zhang and B Dai ldquoExistence of solutions for nonlinearimpulsive differential equations with dirichlet boundaryconditionsrdquo Mathematical and Computer Modelling vol 53no 5-6 pp 1154ndash1161 2011

[20] A Anokhin L Berezansky and E Braverman ldquoStability oflinear delay impulsive differential equationsrdquo DynamicalSystems and Applications vol 4 pp 173ndash187 1995

[21] A Anokhin L Berezansky and E Braverman ldquoExponentialstability of linear delay impulsive differential equationsrdquoJournal of Mathematical Analysis and Applications vol 193no 3 pp 923ndash941 1995

[22] C T H Baker C A H Paul and D R Wille ldquoIssues in thenumerical solution of evolutionary delay differential equa-tionsrdquo Advances in Computational Mathematics vol 3 no 3pp 171ndash196 1995


[23] HWille and F Karakoc ldquoAsymptotic constancy for impulsivedelay differential equationsrdquo Dynamic Systems and Applica-tions vol 17 pp 71ndash84 2008

[24] L Berezansky and E Braverman ldquoImpulsive stabilization oflinear delay differential equationsrdquo Dynamic Systems Appli-cations vol 5 pp 263ndash276 1996

[25] L Berezansky and E Braverman ldquoExponential boundednessof solutions for impulsive delay differential equationsrdquo Ap-plied Mathematics Letters vol 9 no 6 pp 91ndash95 1996

[26] B Du and X Zhang Delay Dependent Stability Analysis andSynthesis for Uncertain Impulsive Switched System with MixedDelays Hindawi Publishing CorporationndashDiscrete Dynamicin Nature and Society London UK 2011

[27] F Dubeau and J Karrakchou ldquoState-dependent impulsivedelay-differential equationsrdquo Applied Mathematics Lettersvol 15 no 3 pp 333ndash338 2002

[28] J R Graef M K Grammatikopoulos and P W SpikesldquoAsymptotic properties of solutions of nonlinear neutral delaydifferential equations of the second orderrdquo Radovi Mate-maticki vol 4 pp 133ndash149 1988

[29] M K Grammatikopoulos G Ladas and A MeimaridouldquoOscillations of second order neutral delay differentialequationsrdquo Radovi Matematicki vol 1 pp 267ndash274 1985

[30] I O Isaac and Z Lipcsey ldquoLinearized oscillations in nonlinearneutral delay impulsive differential equationsrdquo Journal ofModern Mathematics and StatisticsndashMedwell Journal-sndashPakistan vol 3 no 1 pp 1ndash7 2009

[31] I O Isaac and Z Lipcsey ldquoOscillations in linear neutral delayimpulsive differential equations with constant coefficientsrdquoCommunications in Applied Analysis vol 14 no 2 pp 123ndash136 2010

[32] I O Isaac and Z Lipcsey ldquo+e existence of positive solutionsto neutral delay impulsive differential equationsrdquo Commu-nication in Applied Analysis vol 16 no 1 pp 23ndash46 2012

[33] I O Isaac Z Lipcsey and U Ibok ldquoLinearized oscillations inautonomous delay impulsive differential equationsrdquo BritishJournal of Mathematics amp Computer Science vol 4 no 21pp 3068ndash3076 2014

[34] Q Wang and X Liu ldquoImpulsive stabilization of delay dif-ferential systems via the Lyapunov-Razumikhin methodrdquoApplied Mathematics Letters vol 20 no 8 pp 839ndash845 2007

[35] A Weng and J Sun ldquoImpulsive stabilization of second-orderdelay differential equationsrdquo Nonlinear Analysis Real WorldApplications vol 8 no 5 pp 1410ndash1420 2007

[36] J Yan ldquoOscillation properties of a second-order impulsivedelay differential equationrdquo Computers amp Mathematics withApplications vol 47 no 2-3 pp 253ndash258 2004

[37] A Zhao and J Yan ldquoAsymptotic behavior of solutions ofimpulsive delay differential equationsrdquo Journal of Mathe-matical Analysis and Applications vol 201 no 3 pp 943ndash9541996

[38] G Ballinger and X Liu ldquoExistence and uniqueness results forimpulsive delay differential equationsrdquo DCDIS vol 5pp 579ndash591 1999

[39] B S Nagy Introduction to Real Functions and OrthogonalExpansions Oxford University Press Oxford UK 1965

[40] Z Lipcsey I M Esuabana J A Ugboh and I O IsaacldquoAbsolute continuous representation of functions of boundedvariationrdquo In press 2019

[41] J C Fournier Graphs eory and Applications WileyndashISTEHoboken NJ USA 2013


are handled in impulsive differential equations with con-tinuous right side

+e analysis of problems arising from delayed equationsand their handling leads us to the necessity of the formu-lation of the extended concept of impulsive differentialequations and analysis of the difference between the Bai-novian and extended models

11 Systems Described by Impulsive Differential Equations+e Bainovian model for the simplest case is as follows Letthe process evolve in a period of time T (T (α β) sub R is anopen interval) Let Ω0 sub Rn be an open set and Ω ≔ T timesΩ0Let f Ω⟶ Rn be an at least continuous mapping whichin additionmay fulfill local Lipschitz condition in its variablex isin Rn for each fixed t forall(t x) isin Ω Let H sub Z be an infinitesubset ofZ (H N orH Zwill be used)+en let the timesequence SH tk1113864 1113865kisinH sub T be increasing without accumu-lation points in T and tk⟶ α k⟶ minus infinand tk⟶ β k⟶infin (equivalently forallm M isin T mltM [m M]capSH sub T is a finite set) Let g SH times Rn⟶ Rn becontinuous and may fulfill Lipschitz condition in its variablex forall(tk x) isin Ω +en the controlling impulsive differentialequation is given by

xprime(t) f(t x(t)) forallt isin TSH

Δx tk( 1113857 g tk x tk( 1113857( 1113857 foralltk isin SH

⎧⎨

⎩ (1)

where (t x(t)) isin Ω+e impulsive differential equation (1) can be re-

written as an integral equation We define an ascendingstep function τ R⟶ Z with unit jumps at the impulsepoints

τ(t) ≔ k minus 1 if tkminus 1 le tlt tkforalltk isin SZ (2)

If H N then τ(t) 0foralltlt t1

Corollary 1 e function τ R⟶ R is a singular ascendingfunction in t which means as an ascending function it isdifferentiable almost everywhere and

dτ(t)

dt 0 almost everywhere (3)

The singular ascending function τ defines a singularmeasure τ on the Borel sets of R +e domain of the functiong is extended to the 1113957g Ω⟶ Rn from the set SH timesΩ0 sub Ω

With measures τ and 1113957g equation (1) can be rewritten inan integral form

x(t) x0 + 1113946t

t0

(f(s x(s))ds + 1113957g(s x(s))dτ) t0 lt t isin T


+e technical details of these facts are discussed in [10]

Equation (4) has two measures therefore it does notlook like an integral equation of an (impulsive) differentialequation We will change the parametrization of thisequation

Let ]λ(t) ≔ t + τλ(t) id[abλ](t) + τλ(t) (τλ represents τas defined in Corollary 1) which is a strictly ascendingfunction ]λ [a bλ]⟶ [a b]] with b] ≔ bλ + τλ(bλ)minus

τλ(a) As an ascending function ]λ has a left- and a right-continuous version

μλλminus (t) ≔ ]λ(t minus 0) forallt isin a bλ1113858 1113859

μλλ+(t) ≔ ]λ(t + 0) forallt isin a bλ1113858 1113859(5)

+e mappings defined in (5) give us an increasingfunction 1113954μλ [a b]]⟶ [a bλ] defined as follows

1113954μλ(t) ≔ s forallt isin a b]1113858 1113859 exists isin a bλ1113858 1113859st t isin μλλminus (s) μλλ+(s)1113960 1113961


(6)

Note that 1113954μλ is one-to-one on the set of continuity points([μλλminus (s) μλλ+(s)] is singleton if μλλminus (s) μλλ+(s) ) +e as-cending function id[abλ](t) is continuous in [a bλ] while τand ] may have a countable set of discontinuity points

Dλ ≔ t | τλ(t minus 0)ne τλ(t + 0)1113966 1113967 t | ]λ(t minus 0)ne ]λ(t + 0)1113966 1113967

(7)

Hence τλ and ]λ are continuous in [a bλ]Dλ

12 Absolute Continuity and Singularity Let ]λ denote themeasure defined by the strictly ascending function ]λ +eintegral in equation (4) is the sum of integrals with twomeasures λλ and τλ Both measures λλ and τλ are absolutecontinuous with respect to ]λ therefore both can be writtenas an integral of the RadonndashNikodym derivatives [11] Withnotations ρλ dλλd]λ and ρτ dτλd]λ the following im-portant properties are formulated

Let Nλτ ≔ (dτλd]λ)gt 01113864 1113865 andNλ

λ ≔ (Nλτ)prime +en

λλ(Nλτ) 0 τλ(Nλ

λ) and

ρλ + ρτ d λλ + τλ1113872 1113873d]λ dτλd]λ + dλλd]λ 1 (8)

Moreover let

N]λ ≔ 1113954μminus 1

λ Nλλ1113872 1113873 sub a b]1113858 1113859

N]τ ≔ 1113954μminus 1

λ Nλτ1113872 1113873 sub a b]1113858 1113859

N]λcupN

]τ a b]1113858 1113859

N]λcapN

]τ empty

(9)

+ese RadonndashNikodym derivatives enable us to rewriteequation (4) using one measure ]λ as follows


x(t) x0 + 1113938t

t0f(s x(s))

dλλ

d]λ+ 1113957g(s x(s))

dτλ

d]λ1113888 1113889d]λ t isin T


(10)






D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (11)


D]λ ≔ cup

tisinDλ








[a b]]D]λ









1113938μλ+(t)

af middot 1113954μλd]]λ 1113938

t


1113938σa

fd]]λ 11139381113954μλ(σ)



]λ ab][ ]c

1113874 1113875

(13)
















(14)


x(t) x0 + 1113938t


x t0( 1113857 x0 t0 isin (a c)SH(15)








u(t) ≔c +

(minus 1)j


2tj+1 minus t1113872 1113873


0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(16)














d λλ + τλ1113872 1113873

d]λdλλ

d]λ+dτλ

d]λ 1rArr

dλλ

d]λ χNλ

λand

dτλ

d]λ χNτ

λ

(17)


x(t) x0 + 1113938t

t0f(s x(s))χNλ

λ+ g(s x(s))χNλ

τ1113874 1113875d]λ t isin T






τ⟶ Rn +en let

h] ≔ f

]χN]λ

+ g]χN]

τ a b]1113858 1113859 timesΩ0⟶ R

n(19)








(t φ(t)) isin S









t0 + c)









τtimes





φ(σ) ξ0 + 1113946σ

σ0f](v φ(v))χN]

λ+ g

](v φ(v))χN]

τ1113874 1113875d] ξ0 + 1113946

σ

σ0h](vφ(v))d] (21)


ξ0






φ μλ+(t)1113872 1113873 ξ0 + 1113946μλ+(t)

σ0h](vφ(v))d]]λ ξ0 + 1113946

t

t0

h] μλ+(s)φ μλ+(s)1113872 11138731113872 1113873d]λrArr

xλ(t) ξ0 + 1113946t

t0

hλ

1113954μλ μλ+(s) xλ(s)1113872 1113873d]λ ξ0 + 1113946t

t0

hλ

s xλ(s)( 1113857d]λ1113888

(22)








yprime


2y


0 0le tltinfin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)









]times1113966












τ λλ and τλ we





λ Nλτ λ

λ and τλSince N]




b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]





]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]


(24)





λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)






xλ(t) ξ0 + 1113946μλ+(t)


ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ


+ g] μλ+(s) xλ(s)1113872 1113873dτλ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)







[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]





]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]


(27)



λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References












































x(t) x0 + 1113938t

t0f(s x(s))

dλλ

d]λ+ 1113957g(s x(s))

dτλ

d]λ1113888 1113889d]λ t isin T


(10)






D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (11)


D]λ ≔ cup

tisinDλ








[a b]]D]λ









1113938μλ+(t)

af middot 1113954μλd]]λ 1113938

t


1113938σa

fd]]λ 11139381113954μλ(σ)



]λ ab][ ]c

1113874 1113875

(13)
















(14)


x(t) x0 + 1113938t


x t0( 1113857 x0 t0 isin (a c)SH(15)








u(t) ≔c +

(minus 1)j


2tj+1 minus t1113872 1113873


0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(16)














d λλ + τλ1113872 1113873

d]λdλλ

d]λ+dτλ

d]λ 1rArr

dλλ

d]λ χNλ

λand

dτλ

d]λ χNτ

λ

(17)


x(t) x0 + 1113938t

t0f(s x(s))χNλ

λ+ g(s x(s))χNλ

τ1113874 1113875d]λ t isin T






τ⟶ Rn +en let

h] ≔ f

]χN]λ

+ g]χN]

τ a b]1113858 1113859 timesΩ0⟶ R

n(19)








(t φ(t)) isin S









t0 + c)









τtimes





φ(σ) ξ0 + 1113946σ

σ0f](v φ(v))χN]

λ+ g

](v φ(v))χN]

τ1113874 1113875d] ξ0 + 1113946

σ

σ0h](vφ(v))d] (21)


ξ0






φ μλ+(t)1113872 1113873 ξ0 + 1113946μλ+(t)

σ0h](vφ(v))d]]λ ξ0 + 1113946

t

t0

h] μλ+(s)φ μλ+(s)1113872 11138731113872 1113873d]λrArr

xλ(t) ξ0 + 1113946t

t0

hλ

1113954μλ μλ+(s) xλ(s)1113872 1113873d]λ ξ0 + 1113946t

t0

hλ

s xλ(s)( 1113857d]λ1113888

(22)








yprime


2y


0 0le tltinfin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)









]times1113966












τ λλ and τλ we





λ Nλτ λ

λ and τλSince N]




b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]





]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]


(24)





λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)






xλ(t) ξ0 + 1113946μλ+(t)


ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ


+ g] μλ+(s) xλ(s)1113872 1113873dτλ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)







[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]





]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]


(27)



λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References


















































(14)


x(t) x0 + 1113938t


x t0( 1113857 x0 t0 isin (a c)SH(15)








u(t) ≔c +

(minus 1)j


2tj+1 minus t1113872 1113873


0 otherwise

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(16)














d λλ + τλ1113872 1113873

d]λdλλ

d]λ+dτλ

d]λ 1rArr

dλλ

d]λ χNλ

λand

dτλ

d]λ χNτ

λ

(17)


x(t) x0 + 1113938t

t0f(s x(s))χNλ

λ+ g(s x(s))χNλ

τ1113874 1113875d]λ t isin T






τ⟶ Rn +en let

h] ≔ f

]χN]λ

+ g]χN]

τ a b]1113858 1113859 timesΩ0⟶ R

n(19)








(t φ(t)) isin S









t0 + c)









τtimes





φ(σ) ξ0 + 1113946σ

σ0f](v φ(v))χN]

λ+ g

](v φ(v))χN]

τ1113874 1113875d] ξ0 + 1113946

σ

σ0h](vφ(v))d] (21)


ξ0






φ μλ+(t)1113872 1113873 ξ0 + 1113946μλ+(t)

σ0h](vφ(v))d]]λ ξ0 + 1113946

t

t0

h] μλ+(s)φ μλ+(s)1113872 11138731113872 1113873d]λrArr

xλ(t) ξ0 + 1113946t

t0

hλ

1113954μλ μλ+(s) xλ(s)1113872 1113873d]λ ξ0 + 1113946t

t0

hλ

s xλ(s)( 1113857d]λ1113888

(22)








yprime


2y


0 0le tltinfin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)









]times1113966












τ λλ and τλ we





λ Nλτ λ

λ and τλSince N]




b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]





]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]


(24)





λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)






xλ(t) ξ0 + 1113946μλ+(t)


ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ


+ g] μλ+(s) xλ(s)1113872 1113873dτλ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)







[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]





]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]


(27)



λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References














































d λλ + τλ1113872 1113873

d]λdλλ

d]λ+dτλ

d]λ 1rArr

dλλ

d]λ χNλ

λand

dτλ

d]λ χNτ

λ

(17)


x(t) x0 + 1113938t

t0f(s x(s))χNλ

λ+ g(s x(s))χNλ

τ1113874 1113875d]λ t isin T






τ⟶ Rn +en let

h] ≔ f

]χN]λ

+ g]χN]

τ a b]1113858 1113859 timesΩ0⟶ R

n(19)








(t φ(t)) isin S









t0 + c)









τtimes





φ(σ) ξ0 + 1113946σ

σ0f](v φ(v))χN]

λ+ g

](v φ(v))χN]

τ1113874 1113875d] ξ0 + 1113946

σ

σ0h](vφ(v))d] (21)


ξ0






φ μλ+(t)1113872 1113873 ξ0 + 1113946μλ+(t)

σ0h](vφ(v))d]]λ ξ0 + 1113946

t

t0

h] μλ+(s)φ μλ+(s)1113872 11138731113872 1113873d]λrArr

xλ(t) ξ0 + 1113946t

t0

hλ

1113954μλ μλ+(s) xλ(s)1113872 1113873d]λ ξ0 + 1113946t

t0

hλ

s xλ(s)( 1113857d]λ1113888

(22)








yprime


2y


0 0le tltinfin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)









]times1113966












τ λλ and τλ we





λ Nλτ λ

λ and τλSince N]




b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]





]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]


(24)





λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)






xλ(t) ξ0 + 1113946μλ+(t)


ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ


+ g] μλ+(s) xλ(s)1113872 1113873dτλ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)







[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]





]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]


(27)



λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References












































φ(σ) ξ0 + 1113946σ

σ0f](v φ(v))χN]

λ+ g

](v φ(v))χN]

τ1113874 1113875d] ξ0 + 1113946

σ

σ0h](vφ(v))d] (21)


ξ0






φ μλ+(t)1113872 1113873 ξ0 + 1113946μλ+(t)

σ0h](vφ(v))d]]λ ξ0 + 1113946

t

t0

h] μλ+(s)φ μλ+(s)1113872 11138731113872 1113873d]λrArr

xλ(t) ξ0 + 1113946t

t0

hλ

1113954μλ μλ+(s) xλ(s)1113872 1113873d]λ ξ0 + 1113946t

t0

hλ

s xλ(s)( 1113857d]λ1113888

(22)








yprime


2y


0 0le tltinfin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)









]times1113966












τ λλ and τλ we





λ Nλτ λ

λ and τλSince N]




b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]





]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]


(24)





λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)






xλ(t) ξ0 + 1113946μλ+(t)


ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ


+ g] μλ+(s) xλ(s)1113872 1113873dτλ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)







[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]





]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]


(27)



λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References














































τ λλ and τλ we





λ Nλτ λ

λ and τλSince N]




b]

aχN]

λd]

1113954μλ(s) ≔ a + 1113938s

aχN]





]λ(s) a + 1113938μλλ+

(s)

a1d]] a + 1113938

μλλ+(s)

aχN]

λ+ χN]


(24)





λλ Nλτ1113872 1113873 0 τλ N

λλ1113872 1113873 (25)






xλ(t) ξ0 + 1113946μλ+(t)


ξ0 + 1113946t

t0

h] μλ+(s) xλ(s)1113872 1113873d]λ

ξ0 + 1113946t

t0

f] μλ+(s) xλ(s)1113872 1113873dλλ


+ g] μλ+(s) xλ(s)1113872 1113873dτλ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallt isin a bλ1113858 1113859

(26)







[a bτ] with bτ ≔ a + 1113938b]

aχN]

τd]

1113954μτ(s) ≔ a + 1113938s

aχN]





]τ(s) a + 1113938μττ+(s)

a1d]] a + 1113938

μττ+(s)

aχN]

λ+ χN]


(27)



λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References













































λ



λτ Nττ( 1113857 0 ττ N

τλ( 1113857 (28)




τd]τ dττ


xτ(ϑ) ξ0 + 1113946μτ+(ϑ)


ξ0 + 1113946ϑ

ϑ0h] μτ+(s) xτ(s)1113872 1113873d]τ

ξ0 + 1113946ϑ

ϑ0f] μτ+(s) xτ(s)1113872 1113873dλτ


+ g] μτ+(s) xτ(s)1113872 1113873dττ


⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠ forallϑ isin a bτ1113858 1113859

(29)





τ


f] χN]τ


0 t const





(30)




















(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References

























































(31)








D]λ ≔ μλminus (t) μλ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dλ1113882 1113883 (33)


D]λ ≔ cup

tisinDλ









λ1113864 1113865+e measure ]]λ on P]








λ[ab]]cbe B]

λ[ab]]c≔

σ(P]λ[ab]]c


B]λ[ab]]c


1113946μλλ+

(t)

ahd]]λ 1113946

t




1113946σ


]λ 1113946

1113954μλ(σ)


]λ

(36)







D]τ ≔ μτminus (t) μτ+(t)1113960 1113873

11138681113868111386811138681113868 t isin Dτ1113882 1113883 (37)



[a bλ]⟶ [a b]][a bλ]⟷ [a b]]


μλλminus

μλλ ≔ μλλ+




D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References












































D]τ ≔ cup

tisinDτ









λ1113864 1113865+e measure ]]τ on P]



isin P]][ab]]c


isin P]][ab]]c



τ[ab]]cbe B]

τ[ab]]c≔ σ(P]



B]τ[ab]]c


1113946μττ+(t)

ahd]]τ 1113946

t




1113946σ0


]τ 1113946

1113954μτ σ0( )


]τ

(40)




+enB]




Afd]forallA isinB]

λ[ab]]c

and ]fτ(A) ≔ 1113938A



]τ


]τ give

1113954f]λ ≔d]fλ

d]]λ⟺1113946

Afd]

1113946A


]λ ab][ ]c

1113954f]λ isin L1 ]λB]λ ab][ ]c1113874 1113875

(41)

1113954f]τ ≔d]fτ

d]]τ⟺1113946

Afd]

1113946A


]τ ab][ ]c

1113954f]τ isin L1 ]]τ B]τ ab][ ]c1113874 1113875

(42)


1113946μλλ+

(t)

afd] 1113946

μλλ+(t)

a

1113954f]λd]]λ 1113946

t

a


(43)


1113946μττ+(ϑ)

afd] 1113946

μττ+(ϑ)

a

1113954f]τd]]τ 1113946

ϑ

a


(44)




(45)


1113946μλλ+

(t)


λd] 1113946

μλλ+(t)

a


(43)1113946μλλ+

(t)


λd]]λ

1113946t



(46)

1113946μττ+(ϑ)


τd] 1113946

μττ+(ϑ)

a


(43)1113946μττ+(ϑ)


τd]]τ

1113946ϑ



(47)


[a bτ]⟷[a b]][a bτ]⟷ [a b]]


μττminus

μττ ≔ μττ+






+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References














































+ f times

χD]λ



]λ)cap

D]λ


+ f times χD]τ

by [a b]]

([a b]]D]τ)cupD


]τ)capD

]τ


1113938μλλ+

(t)

afd] 1113938

μλλ+(t)


λ+ f times χD]

λ1113874 1113875d] 1113938

t


+1113956

f times χD]λ

1113874 1113875]λ


1113938μλλ+

(t)

afd] 1113938

μττ+(ϑ)


τ+ f times χD]

τ1113874 1113875d] 1113938

ϑa


1113956f times χD]


(48)



λ[ab]]cdoes



τ[ab]]cdoes





1113956f times χD]

λ1113874 1113875

]λtj1113872 1113873 ≔

1113938μλ+ tj( 1113857



1113956f times χD]

τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857



(49)














λ andμτ+ Nτ




φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h](vφ(v))d] ξ0 + 1113946

μλλ+(t)

σ0

1113954h]]λ(v)d]]λ

ξ0λ + 1113946t

t0


(50)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h](v φ(v))d] ξ0 + 1113946

μττ+(ϑ)

σ0

1113954h]]τ(v)d]]τ

ξ0τ + 1113946ϑ

ϑ0


(51)









]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References



















































]τ and a component


τ

h]

h]λc + h

]λd

h]

h]τc + h

]τd

(52)


+ g]χN]τ

intoequation (52)

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

λ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

λ

h] ≔ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χ ab][ ]D]

τ+ f

]χN]λ

+ g]χN]

τ1113874 1113875 times χD]

τ

(53)


]λ andN]

τ sub [a b]]D]τ

N]λcapN

]τ empty and D]

λ sub N]τ andD

]τ sub N]

λ leads to

h] ≔ f

]χN]λ

+ g]χN]

τtimes χ ab][ ]D]


+ g]


h]λd

h] ≔ f

]χN]λ

times χ ab][ ]D]τ

+ g]χN]


+ f]


h]τd

(54)



1113946μλλ+

(t)

σ0hλdd] 1113946

μλλ+(t)

σ0g]


1113946μλλ+

(t)

σ0


λ1113874 1113875


λd]]λ 1113946

t

t0


λ1113874 1113875


(55)

1113946μλλ+

(t)

σ0hτdd] 1113946

μττ+(ϑ)

σ0f]


μττ+(ϑ)

σ0


τ1113872 1113873]τ


1113946ϑ

ϑ0


τ1113872 1113873]τ


(56)


1113946μλλ+

(t)

σ0h]λc(vφ(v))d] 1113946

μλλ+(t)

σ0

1113954h]]λ(v) times χ ab][ ]D]

λd]]λ

(43)1113946μλλ+

(t)


1113946t

t0

f]χN]


]χN]τ


(57)

1113946μττ+(ϑ)

σ0h]τc(v φ(v))d] 1113946

μττ+(ϑ)

σ0


τd]]τ (43) 1113946

μττ+(ϑ)


1113946ϑ

ϑ0f]χN]

λtimes χ ab][ ]D]



(58)




λ[ab]]candB]

τ[ab]]c












λandf] times χD]


leads to



λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References













































λ1113874 1113875

]λtjφ1113872 1113873 ≔

1113938μλ+ tj( 1113857




]λ tjφ1113872 1113873

foralltj isin Dλ

(59)


τ1113872 1113873]τ

τj1113872 1113873 ≔1113938μτ+ τj( 1113857




]τ τjφ1113872 1113873

forallτj isin Dτ

(60)








]τ as follows

1113954g]λ(σφ) ≔

1113954g0]λ(σφ) 1113954μλ(σ) isin Dλ

0 σ notin D]λ


1113954f]τ(σφ) ≔

1113954f0]τ(σφ) 1113954μτ(σ) isin Dτ

0 σ notin D]τ

⎧⎨


(61)








φ μλλ+(t)1113872 1113873 ξ0 + 1113946μλλ+

(t)

σ0h]λc + h

]λd1113872 1113873(vφ(v))d]

ξ0λ + 1113946t

t0


λ+ g


times χ abλ[ ]Dλ1113874 1113875d]λ

+ 1113946t

t0


(62)

φ μττ+(ϑ)1113872 1113873 ξ0 + 1113946μττ+(ϑ)

σ0h]τc + h

]τd1113872 1113873(v φ(v))d]

ξ0τ + 1113946ϑ




τ1113872 11138731113874 1113875d]τ

+ 1113946ϑ

ϑ0


(63)





xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References















































xλ(t) ξ0λ + 1113946t

t0



+ 1113946t

t0


continuous singular




(64)

xτ(ϑ) ξ0τ + 1113946ϑ



+ 1113946ϑ


continuous singular




(65)








t0 + δt0 x0)⟶ R+


t0+









(t) | 1le1113882


| 11113882



(t) forall(t x)







χN]λ






isin R



(x0) sub [a bλ]



[a bτ]⟶ R+ 0






(1113954μλ






λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References
















































λtimes









λtimes



λ(σ) in

N]λ times Bελx0





(1113954μλ(σ)) times χN]λ(σ)1113882



ε]x0≔ min ελx0



Bε]x0(x0)





λ+ g]χN]



(μλminus



obtain that

h] μλminus (t) x1113872 1113873χNλ

λ

f]χN]

λ+ g

]χN]τ

1113874 1113875 μλminus (t) x1113872 1113873χNλλ

f] μλminus (t) x1113872 1113873χNλ

λ(t) + g


λ(t)

f] μλminus (t) x1113872 1113873χNλ

λ(t)


x0( 1113857

(66)





Condition 1







τ(σ) is






xλ(t) ξ0λ + 1113946t

t0

fλ

s xλ(s)( 1113857dλλ

+ 1113946t

t0

gτ



xτ(ϑ) ξ0τ + 1113946ϑ


+ 1113946ϑ

ϑ0fλ


(67)






Data Availability




References
















































Data Availability




References












































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