Formulation of Impulsive Differential Equations with Time-Dependent Continuous Delay

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.


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 have been relatively less developed because of significant technical and theoretical difficulties, and as such only a few pieces of literature are available [1][2]. 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 [1,[3][4][5]. 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) [3]. This happens to be a fusion of two areas -Delay Differential Equations, and Impulsive Differential Equations. Several evolution processes in Sciences, Engineering, Technology, Economics, etc., are modeled by impulsive differential equations with delays [1,[3][4][5][6].
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 Time-Dependent Continuous Delay 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 behaviour, 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 can be seen in [7][8][9][10][11][12][13][14].

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 behaviour 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 shorttime perturbations (impulses) whose durations are negligible when compared with the total duration of their entry time evolution [15,3,[16][17][18].
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 [15,[19][20][21]. Impulsive systems are systems whose states are characterized by small perturbations (impulses) in the form of jumps [22][23][24].
These equations are classified into two categories: those with fixed moments of impulsive effects (moments of jumps are previously fixed), and those with unfixed moments of impulsive effects (moments of jump occur when certain space-time relations are satisfied) [15,18,20,25].
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 [3,26].
Impulsive differential equations with fixed moments of impulsive effects have the form: S ={ t } ∞ 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, k k t :=t ( t ,x( t )) . When the function k t 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 k t are 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 ′ For t S , ∈ a change by jump of the solution . + After the jump, at the moment k t=t , the solution x( t ) of the system coincides with the solution ( ) y t of the initial value problem [15]: This simply means that after the jump at k t=t , 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 n R . Also, let these processes be described by n x :T R → state functions, assuming that they may be influenced by past events defined by delay functions j h :T R .
The properties of these functions will be specified later as progress is made. Now, let .. , f ( x )) it then follows that: .
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  (1), a system of first-order impulsive differential equations with continuous delay is of the form: From the notation in (1), it can be written in a more compact form as: Let ).

Let equation (2) or equation (3) be given subject to the initial or history function
x(t )) prescribes the jump at each impulse point k t 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 g be a vector with n components in C(a, b) and k A ɶ be an n by n matrix function on S, then a system of linear impulsive differential equation with continuous delays is defined as:

t )= A( t ) B( t )x ( t ht ) g , t T \ S x( t )= A x( t ), t S.
∆ ɶ If g is identically zero, equation (5) is called a homogeneous equation and is given by:

Main Results
Consider the system of equations given by ( ) 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( t h ( t ))
′ − and for t S , ∈ a change by jump of the solution x( t ) occurs so that k k 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 π ρ π + ⊂ − + , otherwise, k k 1 ( t ,t ) + will be replaced by Step 2: Let . Then its derivative is: Step 3: Let us consider another function: t (t ,t ).

t,x( t )x( t h( t ))
− is discontinuous/has no limit at some point(s) in k k 1 ( t ,t ) + . 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 i) When t S , ∉ equation (2) reduces to a delay differential equation and solution is obtained from ii) f is continuous in Ω -an open subset of ( m 1) n k k , and there is a jump change at each of these impulse points given by x(t )).
iv) After the jump at the moments k k 0,1,2 . t=t , ,.. = , the solution x(t) of equation (2) coincides with the solution y(t) of the initial function problem
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.   (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: