Communications on Applied Nonlinear Analysis 

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Formulation of Impulsive Differential Equations with Time-Dependent 

Continuous Delay 
 

Venkatachalapathi Uday1, Dr. Gautam Kumar Rajput2 
1Research Scholar, Department of Mathematics, Sunrise university, Alwar, Rajasthan 

2Associate professor, Department of Mathematics, Sunrise University, Alwar, Rajasthan 

 

Article History: 

Received: 06-08-2024 

Revised: 25-09-2024 

Accepted: 08-10-2024 

Abstract:  

Research in impulsive delay differential equations has been undergoing some 

exciting growth in recent times. This to a large extent can be attributed to the 

quest by mathematicians in particular and the science community as a whole to 

unveil nature the way it truly is. The realization that differential equations, in 

general, and indeed impulsive delay differential equations are very important 

models for describing the true state of several real-life processes/phenomena may 

have been the tunic. One can attest that in most human processes or natural 

phenomena, the present state is most often affected significantly by their past 

state and those that were thought of as continuous may indeed undergo abrupt 

change at several points or even be stochastic. In this study, a special strictly 

ascending continuous delay is constructed for a class of system of impulsive 

differential equations. It is demonstrated that even though the dynamics of the 

system and the delay have ideal continuity properties, the right side may not 

even have limits at some points due to the impact of past impulses in the present. 

The integral equivalence of the formulated system of equations is also obtained 

via a scheme similar to that of Perron by making use of certain assumptions. 

Keywords: Impulsive, Differential Equation, Continuous Delay, Integral 

Equivalence 

 

1. Introduction and Statement of Problem 

The theory of impulsive delay differential equations (IDE) is based on the behaviour of processes or 

phenomena which undergo abrupt changes in their state and past events affect the current behaviour 

(delay). These kinds of processes are best described by coupled systems of differential equations, 

either starting with delay differential equations and adding impulses or starting with impulsive 

differential equations and adding delay arguments. Whichever is the case, a coupled problem is 

obtained and the structure is radically changed. Several of the properties of solutions in ordinary, 

delay, or impulsive differential equations are no longer sustained. 

The theory of impulsive delay differential equations has been relatively less developed because of 

significant technical and theoretical difficulties, and as such only a few pieces of literature are 

available. However, interest is on the increase largely due to the fact that a lot of everyday 

phenomena in Sciences, Economics, Engineering, Space sciences, and control systems are modeled 

by impulsive delay differential equations. In particular, Ballinger's Ph.D thesis and his subsequent 

work provide a good working tool for further research work in this area, especially, as it relates to 

existence, uniqueness, boundedness, continuation, and stability of solutions of Impulsive Delay 

Differential Equations (IDDE). This happens to be a fusion of two areas - Delay Differential 

Equations, and Impulsive Differential Equations. Several evolution processes in Sciences, 



Communications on Applied Nonlinear Analysis 

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Engineering, Technology, Economics, etc., are modeled by impulsive differential equations with 

delays. 

In ordinary or delay differential equations, the solutions are continuously differentiable, at least once 

or more, whereas impulsive differential equations possess non-continuous (piecewise continuous) 

solutions. Intuitively, it is expected that the coupled problems of impulsive delay differential 

equations should possess piecewise continuously differentiable solutions. To some extent, our 

intuition may be true if the delays are discrete. When continuous delays interplay with impulses, the 

story may become significantly different. This surge in the number of discontinuous points creates 

several problems as it pertains to existence, stability/instability of solutions, just to mention a few. 

Since the continuity properties of the solutions play fundamental roles in the analysis of their 

behaviors, the techniques used to handle the solutions of impulsive delay differential equations are 

basically different from those of Ordinary Differential Equations, Impulsive Differential Equations 

and Delay Differential Equations. However, some basic concepts are still valid. 

In this study, a system of first-order impulsive differential equations with continuous time-dependent 

delays and fixed moments of impulse is formulated. Some recent results in impulsive delay 

differential equations with constant impulsive jumps. 

Preliminaries 

The theory of impulsive systems was developed not long ago as an independent area of mathematical 

analysis. The development arose out of curiosity to develop a mathematical framework that truly 

describes physical and biological processes as they occur in nature. Prior to this noble development, 

scientists had often made an underlying assumption that the behaviors of physical and biological 

systems described by ordinary differential equations is continuous and integrable in some sense. It 

was observed that the state of a system is susceptible to changes, and in some processes, these 

changes are often characterized by short time perturbations (impulses) whose durations are negligible 

when compared with the total duration of their entry time evolution. 

IDEs are adequate mathematical models for the description of evolution processes characterized by 

the combination of continuous and jump changes of their state. For the continuous change of such 

processes, ordinary differential equations are used, while the moments and the magnitude of the 

jumps are given by the jump conditions. Impulsive systems are systems whose states are 

characterized by small perturbations (impulses) in the form of jumps. 

IDEs are usually defined by a pair of equations - an ordinary differential equation to be satisfied 

during the continuous portion of the evolution, and difference equations defining the change of state 

at the discrete impulsive points. This is the main formulation of early scholars such as Bainov, 

Simeonov, Lakshminkatham, Gopalsamy, Zhang, among others. Solutions are usually considered to 

be piecewise continuously differentiable functions with discontinuities occurring at the impulsive 

times. 

 Impulsive differential equations with fixed moments of impulsive effects have the form: 

{
𝑥 ′(𝑡)= 𝑓(𝑡, 𝑥(𝑡)), ⬚∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑥(𝑡𝑘)= 𝑓(𝑡𝑘, 𝑥(𝑡𝑘)), ⬚∀𝑡𝑘 ∈ 𝑆
 

where 𝑇 ⊂ 𝑅, (𝑡, 𝑥) ∈ Ω ⊂ 𝑅 × 𝑅𝑛 and the real numerical sequence 𝑆 = {𝑡𝑘}𝑘=1
∞  increases and has 

no finite accumulation point. In the case of unfixed moments of impulsive effects, the impulse points 

may be time and state dependent. That is, 𝑡𝑘: = 𝑡𝑘(𝑡, 𝑥(𝑡)). When the function 𝑡𝑘 depends on the 

state of the given system, it is said to have impulses at variable times. This is reflected in the fact that 

different solutions will tend to undergo impulses at different times. However, if the functions 𝑡𝑘 are 



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all constants, the system is said to have impulses at fixed times which implies that all solutions 

undergo impulse actions at the same time. 

It is observed that the question of the existence of solutions of the system is non-trivial when 

impulses occur at variable times. The precise notion of what a solution is must be carefully stated. It 

is fairly clear that solutions should be piecewise continuous and in fact piecewise continuously 

differentiable (or piecewise absolutely differentiable when considering generalized types of 

solutions). A solution will undergo simple jump discontinuity when it intersects impulse hyper-

surfaces. Even after focusing on a particular class of relations 𝑡(𝑠, 𝑥(𝑠)) = 0 given by impulse hyper-

surfaces, impulsive differential equations still exhibit some unusual behaviour [3]. In this study, 

focus will be placed only on those equations with fixed moments of impulse effects. 

 Be as it may, to obtain or discuss the solution of an impulsive differential equation, certain 

peculiarities of the model must be taken into cognizance. It assume that for 𝑡 ∈ 𝑇 ∖ 𝑆, the solution 

𝑥(𝑡) of the earlier stated equation is determined by the ordinary differential equation 𝑥 ′(𝑡) =
𝑓(𝑡, 𝑥(𝑡)). For 𝑡 ∈ 𝑆, a change by jump of the solution 𝑥(𝑡) occurs so that 𝑥(𝑡𝑘

−) = 𝑥(𝑡𝑘) and 

𝑥(𝑡𝑘
+) = 𝑥(𝑡𝑘) + Δ𝑥(𝑡𝑘) = 𝑥(𝑡𝑘) + 𝑓(𝑡𝑘, 𝑥(𝑡𝑘)). After the jump, at the moment 𝑡 = 𝑡𝑘, the solution 

𝑥(𝑡) of the system coincides with the solution 𝑦(𝑡) of the initial value problem [15]: 

{
𝑦 ′(𝑡) = 𝑓(𝑡, 𝑦(𝑡)),𝑡𝑘 < 𝑡 ≤𝑘+1

Δ𝑦(𝑡𝑘) = 𝑥(𝑡𝑘
+),𝑡 = 𝑡𝑘 ∈ 𝑆

 

 This simply means that after the jump at 𝑡 = 𝑡𝑘, a new function 𝑦(𝑡) takes over control from 

𝑥(𝑡). 

 Let 𝑇 ⊂ 𝑅 be a set of time points and let our processes take place in 𝑅𝑛. Also, let these 

processes be described by 𝑥: 𝑇 → 𝑅𝑛 state functions, assuming that they may be influenced by past 

events defined by delay functions ℎ𝑗: 𝑇 → 𝑅+. The properties of these functions will be specified 

later as progress is made. 

 Now, let 𝑔‾: = (𝑔1, 𝑔2, … , 𝑔𝑚), �̂�: = (𝑔, 𝑔, … , 𝑔) ∈ 𝑅𝑚 and 

𝑓 ∘ 𝑥‾: = (𝑓(𝑥1), 𝑓(𝑥2), … , 𝑓(𝑥𝑚)) it then follows that: 

𝑓(𝑡, 𝑥(𝑡), 𝑥(𝑡 − ℎ𝑙(𝑡)), 𝑥(𝑡 − ℎ2(𝑡)), … … , 𝑥(𝑡 − ℎ𝑚(𝑡))) = 𝑓 (𝑡, 𝑥(𝑡), �̂� ∘ (�̂� − ℎ‾ ∘ �̂�))  (1)  

 In the course of this work, it shall be assumed that (𝑎, 𝑏): = 𝑇 ⊂ 𝑅 is a non-empty open 

subset of 𝑅. For asymptotic investigation, at least 𝑏 = ∞ is assumed. Let 𝑆: = {𝑡𝑘}𝑘=1
∞  or 𝑆: =

{𝑡𝑘}𝑘=−∞
∞  be an increasing sequence of numbers (to be referred to as impulse times or points) with at 

most two condensation points. Let 𝐷∗: = 𝑆 × 𝑅𝑛 and 𝐷: = 𝑇 × 𝑅(𝑚+𝑙)×𝑛. Also, let 𝑓: 𝐷 → 𝑅𝑛 and 

𝑓∗: 𝐷∗ → 𝑅𝑛 be continuous functions fulfilling Lipschitz condition in 𝑥 ∈ 𝐷 and 𝑥 ∈ 𝐷∗, 

respectively, at each fixed 𝑡 ∈ 𝑇. Let ℎ𝑖: 𝑇 → 𝑅+ be continuous ascending delay function such that 

ℎ𝑖(𝑡) ≤ 𝑡 − 𝑎, ∀𝑡 ∈ 𝑇, ⬚∀𝑙 ≤ 𝑖 ≤ 𝑚. Then from the notation in equation (1), a system of first-order 

impulsive differential equations with continuous delay is of the form: 

{
𝑥 ′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥(𝑡 − ℎ𝑙(𝑡)), … , 𝑥(𝑡 − ℎ𝑚(𝑡))), ∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑥(𝑡𝑘) = 𝑥(𝑡𝑘 + 0) − 𝑥(𝑡𝑘 − 0) = 𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘)), ∀𝑡𝑘 ∈ 𝑆
(2) 

 From the notation in (1), it can be written in a more compact form as: 

{
𝑥 ′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥 ∘ (�̂� − ℎ‾ ∘ �̂�)), ⬚∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑥(𝑡𝑘) = 𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘)), ⬚∀𝑡𝑘 ∈ 𝑆
(3) 



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 Let 𝑓:Ω → 𝑅𝑛,Ω ⊂ 𝑇 × 𝑅𝑛 be continuous and 𝑥 at least once differentiable. Let ℎ𝑗 ∈ 𝐶(𝑡) be 

continuous delay functions, 𝑙 ≤ 𝑗 ≤ 𝑚 and ℎ‾ ∘ �̂� = (ℎ𝑙(𝑡), ℎ2(𝑡), … , ℎ𝑚(𝑡)) be a continuous delay 

vector, then 𝑟 = sup
𝑡∈(𝑎,𝑏)

 max1≤𝑗≤𝑛  {ℎ𝑗(𝑡)} is called the delay constant (where the delays are 

discrete, that is, = max1≤𝑗≤𝑛  {ℎ𝑗} ). 

 Let equation (2) or equation (3) be given subject to the initial or history function 

𝑥(𝑡) = 𝜙(𝑡), 𝑡0 − 𝑟 ≤ 𝑡 ≤ 𝑡0 (4) 

 where 𝑥(𝑡𝑘) = 𝑥(𝑡𝑘 − 0) and 𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘)) prescribes the jump at each impulse point 𝑡𝑘 ∈ 𝑆, 

then equation (2) or (3) is called a first-order impulsive delayed differential equation with continuous 

delays. Subject to equation (4) it is called an initial value or function problem. 

 Let 𝐴 and 𝐵 be real 𝑛 by 𝑛 matrix functions with components in 𝐶(𝑎, 𝑏); let �⃗� be a vector 

with 𝑛 components in 𝐶(𝑎, 𝑏) and �̃�𝑘 be an 𝑛 by 𝑛 matrix function on 𝑆, then a system of linear 

impulsive differential equation with continuous delays is defined as: 

{
𝑥 ′(𝑡) = 𝐴(𝑡) + 𝐵(𝑡)�̂� ∘ (�̂� − ℎ‾�̂�) + �⃗�, ∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑥(𝑡𝑘) = �̃�𝑘𝑥(𝑡𝑘), ∀𝑡𝑘 ∈ 𝑆
(5) 

 If �⃗� is identically zero, equation (5) is called a homogeneous equation and is given by: 

{
𝑥 ′(𝑡) = 𝐴(𝑡)𝑥(𝑡) + 𝐵(𝑡)�̂� ∘ (�̂� − ℎ‾�̂�), ∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑥(𝑡𝑘) = �̃�𝑘𝑥(𝑡𝑘), ∀𝑡𝑘 ∈ 𝑆
(6) 

Main Results 

Consider the system of equations given by 

{
𝑥 ′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥(𝑡 − ℎ(𝑡))), ∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑥(𝑡𝑘) = 𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘)), ∀𝑡𝑘 ∈ 𝑆
(7) 

 which is similar to the system (6). This system assumes that for 𝑡 ∈ 𝑇 ∖ 𝑆, the solution 𝑥(𝑡) is 

determined by the delay differential equation 𝑥 ′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑥(�̂� − ℎ‾ ∘ (�̂�))) and for 𝑡 ∈ 𝑆, a 

change by jump of the solution 𝑥(𝑡) occurs so that 𝑥(𝑡𝑘
−) = 𝑥(𝑡𝑘) and𝑥(𝑡𝑘

+) = 𝑥(𝑡𝑘) +

𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘))[1 − 5]. 

 Here, a special strictly ascending continuous delay for the system of equations in (6) is 

constructed. 

 The construction will take several steps as shown below: 

Step 0: Let 𝜌 ∈ (𝑡𝑘, 𝑡𝑘 + 1) for a fixed 𝑘 ∈ 𝑍. It is assumed here that (𝑡𝑘, 𝑡𝑘+1) ⊂ (𝜌 − 𝜋, 𝜌 + 𝜋), 

otherwise, (𝑡𝑘, 𝑡𝑘+1) will be replaced by (𝛼𝑘, 𝛼𝑘+1) = (𝑡𝑘, 𝑡𝑘+1) ∩ (𝜌 − 𝜋, 𝜌 + 𝜋). Moreover, let 

𝑡𝑠 ∈ 𝑆, 𝑡𝑠 < 𝑡𝑘 be an impulse point in the past. 

Step 1: Let 𝜙0: (𝑡𝑘, 𝑡𝑘+1) → 𝑅+be defined by 

∅0(𝑡): = {
𝜌𝜖(𝑡𝑘, 𝑡𝑘+1), ∅0(𝜌𝜌) ≔ 𝑠𝑖𝑛

1

|𝑡−𝜌|
  ∀𝑡 ≠ 𝜌

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 

Remark 3.1. 𝜙0 has roots at 𝑡 = 𝜌 ±
1

𝑘𝜋
,  𝑘 ∈ 𝑁. 

Step 2: Let 𝜙𝑙(𝑡): =∣ 𝑡 − 𝜌3𝜙0(𝑡),  ∀𝑡, 𝜌 ∈ (𝑡𝑘, 𝑡𝑘+1). Then its derivative is: 



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𝑑𝜙𝑙(𝑡)

𝑑𝑡
= {

 3|𝑡 − 𝜌|2𝜙0(𝑡) − |𝑡 − 𝜌|cos 
1

|𝑡 − 𝜌|
,  𝑡 ≠ 𝜌

0  otherwise 

 

∀𝑡, 𝜌 ∈ (𝑡𝑘, 𝑡𝑘+1), which is bounded by 3(𝑡𝑘+1 − 𝑡𝑘)2. 

Remark 3.2 In addition to Remark 3.1,  1 is continuously differentiable in (𝑡𝑘, 𝑡𝑘+1). 

Step 3: Let us consider another function: 

𝝓2(𝑡) = (𝑡 − 𝑡𝑘)2(𝑡 − 𝑡𝑘+1)2𝝓𝑙(𝑡),  ∀𝑡 ∈ (𝑡𝑘, 𝑡𝑘+1) 

 This is continuously differentiable in (𝑡𝑘, 𝑡𝑘+1), its derivative is 0 at 𝑡𝑘 and 𝑡𝑘+𝑙, and hence, it 

has a continuously differentiable extension to 𝑇. 

 Remark 3.3 In addition to Remark 3.2, 𝜙2(𝑡) fulfils 𝜙2(𝑡𝑘) = 𝜙
2′

(𝑡𝑘) = 𝜙2(𝑡𝑘+1) =

𝜙
2′

(𝑡𝑘+1) = 0. Hence 

𝜙(𝑡) = {
𝜙2(𝑡), 𝑡 ∈ (𝑡𝑘, 𝑡𝑘+1)

0, otherwise

ℎ𝜀(𝑡) = 1 − 𝜙𝜀(𝑡) = 𝑡 − (𝑡𝑠 +
𝜀

𝑀 + 1
(𝑡 − 𝑡𝑘)2(𝑡 − 𝑡𝑘+1)2|𝑡 − 𝜌|3𝜙0(𝑡)) , ∀𝑡 ∈ (𝑡𝑘, 𝑡𝑘+1)

 

and 𝑡𝑠 ∈ 𝑆; 

then the following can be asserted: 

 Remark 3.5 Based on Remark 3.4, ℎ𝜀(𝑡) is continuously differentiable in [𝑡𝑘, 𝑡𝑘+1] and its 

derivative is greater than 1 − 𝜀 > 0, hence it is strictly ascending. 

Theorem 3.1 The delay ℎ𝜀(𝑡) ≤ 𝑡 is a continuous strictly ascending function such that the composite 

function 𝑓(𝑡, 𝑥(𝑡), 𝑥(𝑡 − ℎ𝜀(𝑡))) has no limit at 𝜌 for 𝑡𝑠 < 𝑡𝑘 < 𝑡 < 𝑡𝑘+1. 

Proof: Note that 𝑡 − ℎ𝜀(𝑡) = 𝜙𝜀(𝑡) and 𝜙𝜀(𝑡) passes 𝑡𝑠 at 𝑡 = 𝜌 ±
1

𝑘𝜋
, 0 < 𝑘 < ∞. Hence 𝑥(𝑡 −

ℎ𝜀(𝑡)) has a jump of the size of 𝑓∗(𝑡𝑠, 𝑥(𝑡𝑠)) at each 𝑡 = 𝜌 ±
1

𝑘𝜋
, 0 < 𝑘 < ∞. Thus 𝑓(𝑡, 𝑥(𝑡), 𝑥(𝑡 −

ℎ𝜀(𝑡))) has no limit at 𝜌 ∈ (𝑡𝑘, 𝑡𝑘+1). 

 Discussion 3.1 By using continuous delays, the continuity of 𝑓(𝑡, 𝑥(𝑡)𝑥(𝑡 − ℎ(𝑡))) can no 

longer be relied on as it is currently assumed in the literature since the delay maps the discontinuity 

of x at 𝑡𝑠 into the interval (𝑡𝑘, 𝑡𝑘+1) as an impact of the event at 𝑡𝑠 on the dynamics of 𝑓(𝑡, 𝑥(𝑡)𝑥(𝑡 −
ℎ(𝑡))) at present. In other words, positive increasing continuous functions as delays can be found 

such that 𝑓(𝑡, 𝑥(𝑡)𝑥(𝑡 − ℎ(𝑡))) is discontinuous/has no limit at some point(s) in (𝑡𝑘, 𝑡𝑘+1). 

 Furthermore, in this study, the integral equivalence of the formulated system of impulsive 

delay differential equations is to be obtained for the purpose of analysis of its qualitative properties. 

In order to achieve this, the following underlying assumptions are employed: 

Assumtion 3.1 

• When 𝑡 ∉ 𝑆, equation (2) reduces to a delay differential equation and solution is obtained 

from 𝑥 ′(𝑡) = 𝑓(𝑡, 𝑥(𝑡), �̂� ∘ (𝑡 − ℎ‾ ∘ �̂�)), 𝑡 ∈ 𝑇 ∖ 𝑆; 

• 𝑓 is continuous in Ω - an open subset of 𝑇 × 𝑅(𝑚+𝑙)×𝑛; 

• For each 𝑡𝑘 ∈ 𝑆, 𝑥 is left continuous at 𝑡𝑘, i.e. 𝑥(𝑡𝑘
−) = 𝑥(𝑡𝑘), and there is a jump change at 

each of these impulse points given by is a continuously differentiable extension of 𝜙2(𝑡) from 
(𝑡𝑘, 𝑡𝑘+1) to 𝑅. 



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Step 4: Let us define yet another function by 𝜙𝜀(𝑡) = 𝑡𝑠 +
𝜀

𝑀+1
𝜙2(𝑡)∀𝑡 ∈ (𝑡𝑘, 𝑡𝑘+1)⬚ and ⬚𝑡𝑠 ∈ 𝑆, 

where 𝑀 = max𝑡∈𝑡𝑘,𝑡𝑘+1𝐽  max{|𝜙2(𝑡)|, |𝜙2 ′(𝑡)|}. 

Remark 3.4 In addition to Remark 3.3, 𝝓𝜀 fulfils thecondition that 

 max
𝑡∈[𝑡𝑘,𝑡𝑘+1]

 max {𝜙𝜀(𝑡)|, |𝜙𝜀⬚′(𝑡) ∣} < 𝜀 

Step 5: Let  0.5>ε>0. 

𝑥(𝑡𝑘
+) = 𝑥(𝑡𝑘) + Δ𝑥(𝑡𝑘) = 𝑥(𝑡𝑘) + 𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘)) (8)

iv) After the jump at the moments 𝑡 = 𝑡𝑘, 𝑘 = 0,1,2, …, the solution 𝑥(𝑡) of equation (2) coincides 

with the solution 𝑦(𝑡) of the initial function problem 

𝑦 ′(𝑡) = 𝑓(𝑡, 𝑦(𝑡), �̂� ∘ (𝑡 − ℎ‾ ∘ �̂�)), 𝑡𝑘 < 𝑡 ≤ 𝑡𝑘+1

𝑦(𝑡) = 𝑥(𝑡), 𝑡 < 𝑡𝑘 and 𝑦(𝑡𝑘) = 𝑥(𝑡𝑘) + 𝑓∗(𝑡𝑘, 𝑥(𝑡𝑘))
(9) 

v) The function 𝑓(𝑡, 𝑥(𝑡)𝑥(𝑡 − ℎ(𝑡))) is continuous in 𝑥 for each fixed 𝑡 in (𝑡𝑘, 𝑡𝑘+1) and 

measurable in 𝑡 for each fixed 𝑥. 

 To enable us follow the content of this work smoothly, it is necessary to define some basic 

terms, concepts, notations and lemmas that may be used in the sequel. 

Definition 3.1 Let 𝐴 ⊂ 𝐵 be non-empty and 𝑓: 𝐵 → 𝑅, then 𝑓|𝐴(𝑥) = 𝑓(𝑥), ∀𝑥 ∈ 𝐴 is called the 

restriction of 𝑓 to 𝐴. 

Definition 3.2 PC [T, Rn] = 𝑓 | 𝑓 ∶ 𝑇 → Rn, 𝑓 | [𝑡𝑗,𝑡𝑗+1]
⊂𝐶[(𝑡𝑗 , 𝑡𝑗+1], 𝑅𝑛] ⇒ ∃𝑓(𝑡𝑗 + 0) ∈ 𝑅∀ − ∞ <

𝑗 < ∞. That is, 𝑓 restricted to (𝑡𝑗 , 𝑡𝑗+1) is continuous. 

Definition 3.3𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛] = 𝑓⌊⬚𝑎,𝑏]: 𝑓 ∈ 𝑃𝐶[𝑇, 𝑅𝑛], where [𝑎, 𝑏] ⊂ 𝑇. 

Notation 3.1 Let 

i) [𝑎, 𝑏] = [𝑎, 𝑡𝑗𝑎
] ∪ [𝑡𝑗𝑎

, 𝑡𝑗] ∪ [𝑡𝑗 , 𝑡𝑗+1] ∪ [𝑡𝑗+1, 𝑡𝑗+2] ⋯ … ∪ [𝑡𝑗𝑏−1, 𝑡𝑗𝑏
] ∪ [𝑡𝑗𝑏

, 𝑏]; 

ii) 𝐴𝑗𝑎−1 = [𝑎, 𝑡𝑗𝑎
], 𝐴𝑗 = [𝑡𝑗, 𝑡𝑗+1], ∀𝑗𝑎 ≤ 𝑗 < 𝑗𝑏⬚ and 𝐴𝑗𝑏

= [𝑡𝑗𝑏
, 𝑏]. 

iii) 𝐶[𝐴𝑗 , 𝑅𝑛] denote the set of continuous 𝑅𝑛-valued functions on the closed and bounded 

interval 𝐴𝑗, where 𝑗𝑎 − 𝑙 ≤ 𝑗 ≤ 𝑗𝑏. 

Lemma 3.1 If 𝑓 ∈ 𝑃𝐶[𝑇, 𝑅𝑛] and [𝑎, 𝑏] ⊂ 𝑇, then‖𝑓‖[𝑎,𝑏] = sup
𝑥∈[𝑎,𝑏]

 ‖𝑓(𝑥)‖ is a norm and the 

ordered pair, {𝑃𝐶([𝑎, 𝑏], 𝑅𝑛), ‖ ∗ ‖[𝑎,𝑏]} is a Banach space. 

Proof: 

i) Since [𝑎, 𝑏] ⊂ 𝑇 is a closed and bounded interval such that 𝑠0 < 𝑎 < 𝑏 < 𝑠1, 𝑆[𝑎,𝑏] = [𝑎, 𝑏] ∩ 𝑆 is a 

finite set since S has no condensation point in [a, b]. Let 𝑆[𝑎,𝑏] = [𝑎, 𝑏] ∩ 𝑆 = {𝑡𝑗}
𝑗=𝑗𝑎−1

𝑗𝑏
⊂ [𝑎, 𝑏] ⊂

𝑇⬚, then 

 

ii) If 𝑎, 𝑏 ∈ 𝑆, then 𝑎 = 𝑡𝑗𝑎
 and 𝑏 = 𝑡𝑗𝑏

. Hence, [𝑎, 𝑡𝑗𝑎
] = {𝑡𝑗𝑎

} ⊂ [𝑡𝑗𝑎
, 𝑡𝑗𝑎+1]⬚ and ⬚[𝑡𝑗𝑏

, 𝑏] =

{𝑡𝑗𝑏
} ⊂ [𝑡𝑗𝑏−1, 𝑡𝑗𝑏

]. Otherwise, if 𝑎 ∉ 𝑆, then [𝑎, 𝑡𝑗𝑎
] is a closed bounded interval and 

‖𝑓().‖isacontinuousfunctiononit. 



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If 𝑏 ∉ 𝑆, then [𝑡𝑗𝑏
, 𝑏] is a closed and bounded interval and 

‖𝑓().‖hasacontinuousextensiononitbytheexistingright limit 𝑓(𝑡𝑗𝑏
+ 0). Finally, 

‖𝑓().‖hasacontinuousextension from (𝑡𝑗 , 𝑡𝑗+1] to [𝑡𝑗, 𝑡𝑗+1] by the existence of the right limit 𝑓(𝑡𝑗 +

0), ∀𝑗𝑎 ≤ 𝑗 < 𝑗𝑏. Thus 

 iii) ‖𝑓‖𝑗𝑎−1 = sup
𝑥∈[𝑎,𝑡𝑗𝑎]

 ‖𝑓(𝑥)‖
(10) 

and 

‖𝑓‖𝑗𝑏
= sup

𝑥∈[𝑡𝑗𝑏
,𝑏]

 ‖𝑓(𝑥)‖
(11) 

are finite by condition (ii). Again, 

‖𝑓‖[𝑎,𝑏] = sup
𝑥∈[𝑎,𝑏]

 ‖𝑓(𝑥)‖ = max
𝑗𝑎−<𝑗<𝑗𝑏

 ‖𝑓‖𝑗 (12) 

iv) Since 𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛] ⊂ 𝐿∞[[𝑎, 𝑏], 𝑅𝑛]⬚ and 𝐿∞[[𝑎, 𝑏], 𝑅𝑛] is a Banach space with essential sup 

norm, It follows that ‖ ⋅ ‖[𝑎,𝑏] is a norm, where 𝐿∞[[𝑎, 𝑏], 𝑅𝑛] is called a Banach space. 𝑓 ∈

𝐿∞[[𝑎, 𝑏], 𝑅𝑛] with ‖𝑓‖∞ = 0 ⇒ 𝑓 = 0 almost everywhere and not everywhere. Hence the real 

Banach space is the factor space 

𝐿∞[[𝑎, 𝑏], 𝑅𝑛] with {𝑓 ∣ 𝑓 ∈ 𝐿∞[[𝑎, 𝑏], 𝑅𝑛], ‖𝑓‖∞ = 0} (13) 

However, 

𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛] ∩ {𝑓 ∣ 𝑓 ∈ 𝐿∞[[𝑎, 𝑏], 𝑅𝑛], ‖𝑓‖∞ = 0} = {0}hence, the supremum so defined gives a 

norm on 𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛] which is a closed linear subspace of 𝐿∞[[𝑎, 𝑏], 𝑅𝑛]. 

 Now, let the sequence {𝑓𝑗}
𝑗=1

∞
⊂ 𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛] ⊂ 𝐿∞[[𝑎, 𝑏], 𝑅𝑛] be convergent to 𝑓 ∈

𝐿∞[[𝑎, 𝑏], 𝑅𝑛]. 

Then ⬚ from ⬚‖ℎ‖[𝑎,𝑏] = sup
𝑥∈𝑎,𝑏]

 ‖ℎ(𝑥)‖ = max𝑗𝑎≤𝑗≤𝑗𝑏
 ‖ℎ‖𝑗 ≥ ‖ℎ‖𝑘, ∀𝑗𝑎 − 𝑙 ≤ 𝑘 ≤ 𝑗𝑏⬚, ⬚ it 

follows that 𝑓𝑗 → 𝑓 ∈ 𝐿∞[[𝑎, 𝑏], 𝑅𝑛] for 𝑗 → ∞ in ‖ ⋅ ‖[𝑎,𝑏] ⇒ 𝑓𝑗 → 𝑓 ∈ 𝐶𝑘[[𝑎, 𝑏], 𝑅𝑛] in ‖ ⋅ ‖𝑘∀𝑗𝑎 −

𝑙 ≤ 𝑘 ≤ 𝑗𝑏 

 Hence the sequence of functions in 𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛] is continuous and convergent to f 

uniformly in each interval 𝐴𝑘, and this proves that f is continuous on these intervals ∀𝑗𝑎 − 𝑙 ≤ 𝑘 ≤
𝑗𝑏. Thus 𝑓 ∈ 𝑃𝐶[[𝑎, 𝑏], 𝑅𝑛]. 

Lemma 3.2 Let f ∈ 𝑃𝐶𝑙[𝑇, 𝑅𝑛]. Then 

𝑓|[𝑡𝑗,𝑡𝑗+1] ⊂ 𝐶𝑙 ((𝑡𝑗 , 𝑡𝑗+1], 𝑅𝑛)  and 𝑓|[𝑡𝑗,𝑡𝑗+1](𝑡) = 𝑓(𝑡𝑗 + 0) + ∫  
𝑡

𝑡𝑗
 𝑓 ′(𝑠)𝑑𝑠∀𝑡 ∈ (𝑡𝑗 , 𝑡𝑗+1], −∞ < 𝑗 <

∞(14)Proof: This follows from properties of indefinite integrals. Now, to establish the qualitative 

properties of equations (2) and (4) such as the existence of solution of the initial function problem, 

the continuous dependence of the solutions on the initial function (4) and stability of the solutions, 

the integral equivalence of the problem is adopted. 

Using Lemma 3.2 and notation (1.1) equation (2) is re-written as: 

𝑥‾ ∘ �̂� = 𝑥(𝑡𝑘) + ∫  
𝑡

𝑡𝑘

 𝑓(𝑠, 𝑥(𝑠), �̂� ∘ (�̂� − ℎ‾ ∘ �̂�))𝑑𝑠 + ∑  

𝑡𝑚≤𝑡𝑘<𝑡

 𝑓∗(𝑡𝑚, 𝑥(𝑡𝑚)) (15) 



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where 𝑓∗(𝑡𝑚, 𝑥(𝑡𝑚)) = 𝑥(𝑡𝑚
+ ) − 𝑥(𝑡𝑚) is the prescribed jump at the impulse point 𝑡𝑚 ∈ 𝑆, ∀𝑘 =

1,2,3, … and 𝑡 > 𝑡𝑘. With the formulation (15) above, some conditions are imposed on 𝑓 and 𝑥 to 

ensure the existence and uniqueness of solutions and the continuous dependence of solutions on the 

initial function. To obtain the stability of solutions, the problem is re-formulated slightly by 

separating the linear components of 𝑓 from the non-linear component as follows: 

   (16) 

where 𝑓 and 𝑓∗ are respectively the linear and non-linear components of 𝑓. This formulation enables 

us to develop a scheme for stability of solutions similar to that of Perron. In his work [27], Perron 

discussed the stability and asymptotic stability of solutions of 

𝑥 ′(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) + 𝑓∗(𝑡, 𝑥(𝑡)) (17) 

with respect to the stability and asymptotic stability of the solution of the homogeneous linear system 

𝑦 ′(𝑡) = 𝑓(𝑡, 𝑦(𝑡)) (18) 

In line with Perron's formulation, in place of relation (16), the system 

{
𝑦 ′(𝑡) = 𝑓(𝑡, 𝑦(𝑡), �̂� ∘ (�̂� − ℎ‾ ∘ �̂�))∀𝑡 ∈ 𝑇 ∖ 𝑆

Δ𝑦(𝑡𝑘) = 𝑓∗(𝑡𝑘, 𝑦(𝑡𝑘))∀𝑡 ∈ 𝑆
(19) 

will be examined subject to the initial condition in equation (4), where 𝑓∗(𝑡𝑘, 𝑦(𝑡𝑘)) prescribes the 

jump at 𝑡𝑘. Therefore, the integral equivalence of this is: 

𝑦‾ ∘ �̂� = 𝑦(𝑡𝑘) + ∫  
𝑡

𝑡𝑘

 𝑓(𝑠, 𝑦(𝑠), �̂� ∘ (�̂� − ℎ‾ ∘ �̂�))𝑑𝑠 + ∑  

𝑡𝑚⊴𝑘<

 𝑓∗(𝑡𝑚, 𝑦(𝑡𝑚)) (20) 

Conclusion 

The solutions of impulsive differential equations with the general concept of delays are, however, 

fundamentally different from those of ordinary differential equations. It is observed that the 

derivative of the solutions become discontinuous even at non-impulse points and so ceases to be 

differentiable. Also, the forcing function has no limit even when the delays are strictly ascending and 

continuously differentiable functions. Here, it has been shown that it is possible to define a 

continuous ascending delay function whose derivative exists everywhere on an interval 𝐼𝑘 extendible 

to 𝑇, whereas the right-hand side of equation (2) does not have limits at certain other points which 

are not even impulse points. The integral equivalence of the formulated system of impulsive delay 

differential equations has also been obtained. 

References 

[1] Abasiekwere, U. A., Esuabana, I. M., Isaac, I. O., Lipcsey, Z, Classification of Non-Oscillatory Solutions of 

Nonlinear Neutral Delay Impulsive Differential Equations, Global Journal of Science Frontier Research: 

Mathematics and Decision Sciences (USA), Volume 18, Issue 1, 49-63, 2018, doi:10.17406/GJSFR. 

[2] Abasiekwere, U. A., Esuabana, I. M., Asymptotic behaviour of nonoscillating solutions of neutral delay differential 

equations of the second order with impulses, Journal of Mathematical and Computational Science, Vol 8, No 5 

(2018), 620-629, doi: 10.28919/jmcs/3765. 

[3] Arino, O. and Pituk, M. (2001), More on linear differential systems with small delays, Journal of Differential 

Equations, Vol 17, pp. 381-407. 



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ISSN: 1074-133X 

Vol 32 No. 1s (2025) 

 

53 
 

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[4] Bastinec, J., Diblik, J. and Smarda, Z. (2000), Convergence tests for one scalar differential equation with vanishing 

delay, Arch. Math. (Brno), 36, pp. 405-414. CDDE Issue. 

[5] Bainov, D. D. and Simeonov, P. S. (1995), Impulsive Differential Equations - Asymptotic Properties of the 

Solutions, World Scientific Pub. Coy. Pte. Ltd, Singapore. 

[6] Bereketoglu, H. and Karakoc, F. (2008). Asymptotic, constancy for impulsive delay differential equations. Dynamic 

Systems and Applications, 17, pp. 71-84. 

[7] Bainov, D. D. and Simeonov, P. S. (1989), Systems with Impulse Effects, Stability, Theory and Applications, Ellis 

Horwood Series in Mathematics and its Applications. 

[8] Coddington, A. E. and Levinson, N. (1955), Theory of Ordinary Differential Equations, McGraw-Hill Book 

Company, New York. 

[9] Dubeau, F. and Karrakchou, J. (2002). State-dependent impulsive delay differential equations, Applied Mathematics 

Letters, 15, pp. 333-338. 

[10] Esuabana, I. M. and Abasiekwere, U. A., On stability of first order linear impulsive differential equations, 

International Journal of Statistics and Applied, Mathematics, Volume 3, Issue 3C, 231-236, 2018. 

[11] Gopalsamy, K. (1992), Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer 

Academics, Dordrecht. 

[12] Liu, X. and Ballinger, G. (2002). Existence and continuability of solutions for differential equations with delays and 

statedependent impulses, Nonlinear Analysis TMA, 51, pp. 633-647. 

[13] Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S. (1989), Theory of Impulsive Differential Equations, 

World Scientific Publishing Company Limited, Singapore. 

[14] Oyelami, B. O. (1999), On military model for impulsive reinforcement functions using exclusion and 

marginalization techniques, Nonlinear Analysis, Vol 35, pp. 947-958. 

[15] Oyelami, B., Ale, S. O., Ogidi, J. A., and Onumanyi, P. (2003), Impulsive hiv - 1 model in the presence of 

antiretroval drugs using b transform method, Proceedings of African Mathematical Union, 1, pp. 62-76. 

[16] Simeonov, P. S. and Bainov, D. D. (1988). Differentiability of solutions of systems with impulse effects with 

respect to initial data and parameters, Proceedings of Edinburgh Maths Society, 31, pp. 353-368. 

[17] Samaoilenko, A. M. and Perestyuk, J. J. J. (1995), Impulsive Differential Equations, World Scientific, New York. 

[18] UgbohJ. A., EsuabanaI. M., Existence and Uniqueness Result for a Class of Impulsive Delay Differential Equations, 

International Journal of Chemistry, Mathematics and Physics, Vol. 2 (4), 27-32, 2018, doi:10.22161/ijcmp.2.4.1. 

[19] Yan, J. (2004), Oscillation properties of a second order impulsive delay differential equation, Comp. and Math. with 

Applications, 47, pp. 253-258.