A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type

In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated arguments investigated before as particular cases. We also give some examples showing the applicability of our results.


Introduction †
Occasionally, natural phenomena must be modeled using differential equations that may have discontinuous solutions, such as a piecewise constant, or the impulsive effect must be present.Some examples of such modeling can be found in the works of S. Busenberg and K. Cooke [7] (where the authors modeled vertical transmission diseases) and L. Dai and M.C.Singh [12] (oscillatory motion of springmass systems subject to piecewise constant forces such Ax([t]) or A cos([t])).The last work studied the motion of mechanisms modeled by where [•] is the greatest integer function.(See [11]).
In the 70's, A. Myshkis [15] studied differential equations with deviating arguments (h(t) ≤ t, such as h(t) = [t] or h(t) = [t − 1]).The Ukrainian mathematician M. Akhmet generalized those systems, introducing differential equations of the form y ′ (t) = f (t, y(t), y(γ(t))), ( . This class of differential equations is known as Differential Equations with Piecewise Constant Argument of Generalized Type (DEPCAG).They have continuous solutions, even though γ is discontinuous.If we assume continuity of the solutions of (1.1), integrating from t n to t n+1 , we define a finite-difference equation, so we are in the presence of a hybrid dynamic (see [3,17]).For example, taking γ(t) = t+l h h with 0 ≤ l < h, we have t + l h h = nh, when t ∈ I n = [nh − l, (n + 1) h − l).
Then, we see that γ(t) − t ≥ 0 ⇔ t ≤ nh and γ(t) − t ≤ 0 ⇔ t ≥ nh.Hence, we have . Now, if an impulsive condition is defined at {t n } n∈Z , we are in the presence of the Impulsive differential equations with piecewise constant argument of generalized type (IDEPCAG) (see [2]), where x(t − n ) = lim t→t − n x(t), and J n is the impulsive operator (see [18]).
When the piecewise constant argument used in a differential equation is explicit, it will be called DEPCA (IDEPCA if it has impulses).

An elementary and illustrative example of IDEPCA.
Consider the scalar IDEPCA 3) can be written as In the following, we will assume t 0 = 0. Now, integrating on [n, n + 1) from n to t we see that (1.5) Next, assuming continuity at t = n + 1, we have Applying the impulsive condition to the last expression, we get the following finitedifference equation x((n + 1)) = (αβ)x(n).
The discrete and the continuous parts of the system are dependent.For example, A stable continuous part (associated with the coefficient α) can be unstabilized by the discrete part (associated with the parameter β).See [18].
In the next table, we describe some of the behavior of the solutions of (1.7): Behavior of solutions Condition Table 1.Behavior of solutions of (1.7)  1.1.Why study IDEPCAG?: impulses in action.Example 1.Let the following scalar linear DEPCA and the scalar linear IDEPCA where a(t) is a continuous locally integrable function and The solution of (1.8) is x(t) = x 0 , ∀t ≥ τ.I.e., all the solutions are constant (see [17]).On the other hand, as we will see, the solution of (1.9) is where k(t) = k is the only integer such that t ∈ [k, k + 1].Hence, all the solutions are nonconstant if c j ̸ = 1 and c j ̸ = 0, for all j ≥ k(τ ).This example shows the differences between DEPCA and IDEPCA systems.The discrete part of the system can greatly impact the whole dynamic, determining the qualitative properties of the solutions.In [9], K.L. Cooke and J. Wiener were the first to obtain a fundamental matrix for a scalar DEPCA's using the delayed piecewise constant arguments γ(t) . Also, they considered the very interesting scalar DEPCA Also, in [19], K.L. Cooke and S.M. Shah studied the DEPCA Then, in [8], K.L. Cooke and J. Wiener studied the mixed-type piecewise constant argument γ(t) = 2 t+1 2 and considered the DEPCA Additionally, in [22], K.L. Cooke and A.R. Aftabizadeh considered the mixed-type piecewise constant argument γ(t) = m t+k m where 0 < k < m, k, m, n ∈ Z + , and they studied the DEPCA 1.2.2.Variation of parameters formula for a DEPCA.
In [13] (1991), N. Jayasree and S.G.Deo were the first to consider the advanced and delayed parts of the solutions studying the equation obtaining a variation of parameters formula for this DEPCA, in terms of the homogeneous linear DEPCA associated: where ϕ and Ψ are the fundamental solutions of x ′ (t) = ax(t) and y ′ (t) = ay(t) + by(2[(t + 1) /2]) respectively.
In [14] (2001), Q. Meng and J. Yan obtained a variation of parameters formula for the differential equation where a(t), b(t) and f (t) are locally integrable functions on [0, ∞), g(t) is a piecewise constant function defined by g(t) = np for t ∈ [np − l, (n + 1)p − l) with n ∈ N and p, l positive constants such that p > l.The authors studied the oscillation and asymptotic stability properties of the solutions.
In [1] (2008), M. Akhmet considered the DEPCAG for systems and W is the fundamental matrix of the homogeneous linear DEPCAG Later, in [17] (2011), M. Pinto gave a new DEPCAG variation of parameters formula.This time, the author considered the delayed and advanced intervals defined by the general piecewise constant argument ζ k X(t k+1 , s)g(s, z(s), z(γ(s)))ds In the DEPCAG theory, decomposing the interval I n into the advanced and delayed subintervals is critical.As we will see, it is necessary for the forward or backward continuation of solutions.
For the IDEPCA case, In [16] (2012), G. Oztepe and H. Bereketoglu studied the scalar IDEPCA They proved the convergence of the solutions to a real constant when t → ∞, and they showed the limit value in terms of x 0 , using a suitable integral equation.They concluded the following expression for the solutions of (1.12) For the IDEPCA case, in [6] (2023), K-S.Chiu and I. Berna considered the following impulsive differential equation with a piecewise constant argument and where a(t) ̸ = 0, b(t) and f (t) are real-valued continuous functions, p < l and d k ∈ R − {1}.The authors obtained criteria for the existence and uniqueness, a variation of parameters formula, a Gronwall-Bellman inequality, stability and oscillation criteria for solutions for (1.13) and (1.14).
To our knowledge, there is no variation formula for impulsive differential equations with a generalized constant argument.As we have shown, some authors have studied just some particular cases before.

Aim of the work
We will get a variation of parameters formula associated with IDEPCAG system extending the particular case treated in [6] and the general results of the DE-PCAG case studied in [17] to the IDEPCAG context.

Preliminaires
Let PC(X, Y ) be the set of all functions r : X → Y which are continuous for t ̸ = t k and continuous from the left with discontinuities of the first kind at t = t k .Similarly, let PC 1 (X, Y ) the set of functions s : X → Y such that s ′ ∈ PC(X, Y ).

Definition 1 (DEPCAG solution
, where y(t − k ) denotes the left-hand limit of the function y at t k .Let the IDEPCAG system: Let the following hypothesis hold: (H1) Let η 1 , η 2 : R → [0, ∞) locally integrable functions and where ∥ • ∥ is some matricial norm.(H2) In the following, we mention some useful results: an integral equation associated with (2.1) and two Gronwall-Bellman type inequalities necessary to prove the uniqueness and stability of solutions.

Variation of parameters formula for IDEPCAG
In this section, we will construct a variation of parameters formula for the IDE-PCAG system where y ∈ R n×1 , t ∈ R, F (t) ∈ R n×1 is a real valued continuous matrix, A(t), B(t) ∈ R n×n are real valued continuous locally integrable matrices, C k , D k ∈ R n×n , (I + C k ) invertible ∀k ∈ Z, where I n×n = I is the identity matrix and γ(t) is a generalized piecewise constant argument.This time, we will consider the advanced and the delayed intervals in our approach.
First, we will find the fundamental matrix for the linear IDEPCAG Then, we will give the variation of parameters formula for (4.1).
Let Φ(t, s), t, s ∈ R, with Φ(t, t) = I the transition (Cauchy) matrix of the ordinary system x 3) We will assume the following hypothesis: (H3) Let Consider the following matrices where Remark 2. It is important to notice the following facts: a) As a consequence of (H3), J(t k , ζ k ) and J(t k+1 , ζ k ) are invertible ∀k ∈ Z, and Additionally, setting t 0 := τ we will assume that J −1 (τ, γ(τ )) exists.b) Also, due to (H3) and the Gronwall inequality, we have
We adopt the following convention: Also, we will assume γ(τ ).We will adopt the following notation: We will construct the fundamental matrix for system (4.6).Let t, τ ∈ I k = [t k , t k+1 ) for some k ∈ Z.In this interval, we are in the presence of the ordinary system w ′ (t) = A(t)w(t) + B(t)w(ζ k ).
So, the unique solution can be written as Keeping in mind Hence, we get (4.9)Then, by the definition of E(t, τ ) = Φ(t, τ )J(t, τ ), we have (4.10) Now, from (4.7) working on (4.12) Then, setting This expression corresponds to a finite-difference equation.Then, by solving it, we get Finally, by (4.14) and the impulsive condition, we have Hence, considering τ = t k in (4.14) and applying (4.15) we get The last equation is the solution of (4.6) on [τ, t).
We call to the expression (4.17) the fundamental matrix for (4.6) for t ∈ I k(t) and τ ∈ I k(τ ) .
Remark 3. We use the decomposition of In fact, we can rewrite (4.17) in terms of the advanced and delayed parts using (4.13): for t ∈ I k(t) and τ ∈ I k(τ ) .
Remark 4. a) Considering B(t) = 0, we recover the classical fundamental matrix of the impulsive linear differential equation (see [18]).b) If C k = 0, ∀k ∈ Z, we recover the DEPCAG case studied by M. Pinto in [17].
c) If we consider γ(t) = p t + l p , with p < l, we recover the IDEPCA case studied by K-S.Chiu in [6].

The variation of parameter formula for IDEPCAG.
Let the IDEPCAG y ), then the unique solution of (4.18) is Now, if we consider t = τ in (4.19) we have and, by (H3), we get the following estimation for y(ζ k ) , (4.21) Next, taking the left-side limit to the last expression, we have (4.22) Applying the impulsive condition, we get Therefore, considering τ = t k in the last expression we have , which corresponds to a non-homogeneous linear difference equation, where Recalling that we get the discrete solution of (4.18): or, written in terms of (4.17), Now, considering τ = t k in (4.21) we have Finally, replacing y(t k(t) ) by (4.23) and rewriting in terms of (4.17), we get the variation of parameters formula for IDEPCAG (4.18): where W is the fundamental matrix of (4.6).
(2) Let the IDEPCAG We see that Φ(t, s) = I, E(t, s) = J(t, s) and J(t, s) = I + t s B(u)du, where I is the identity matrix.Hence the fundamental matrix for (4.31) is given by for t ∈ I k(t) and τ ∈ I k(τ ) .This case is very important because it is used for the approximation of solutions of differential equations considering γ(t) = t h h, with h > 0 fixed.
Also, if is the fundamental matrix for (4.32) with t ∈ I k(t) and τ ∈ I k(τ ) .
The solution for (4.33) is given by e A(t−s) f (s)ds.
(1) We recover the variation of parameters concluded in [17] when D r = C r = 0.
(2) Also, our result implies the variation of constant formulas given in section 1.2

Some Examples of Linear IDEPCAG systems
In [16], H. Bereketoglu and G. Oztepe studied the following linear IDEPCAG where γ(t) is some piecewise constant argument of generalized type, A(t) is a continuous locally integrable matrix, D : N → R is such that D k ̸ = 0, ∀k ∈ N. The authors originally considered the cases γ 1 (t) = [t + 1], and γ 2 (t) = [t − 1].Hence, t k = k, ζ 1,k k = k + 1 and ζ 2,k = k − 1, respectively.Let Φ(t) be the fundamental matrix of the ordinary differential system x ′ (t) = A(t)x(t). (5.2) It is well known that Φ −1 (t) is the fundamental matrix of the adjoint system associated with (5.2).So, it satisfies   Remark 6.This is the IDEPCAG case for the well-known differential equation studied by K.L Cooke and J.A. Yorke in [10].The authors investigated the following delay differential equation (DDE): x ′ (t) = g(x(t)) − g(x(t − L)), where x(t) denotes the number of individuals in a population, the number of births is g(x(t)), and L is the constant life span of the individuals in the population.Then, the number of deaths g(x(t − L)).Since the difference g(x(t)) − g(x(t − L)] means the change of the population.Therefore x ′ (t) corresponds to the growth of the population at instant t.
Hence, the solution of (5.The piecewise constant used in this example was introduced in [21] to study the approximation of solutions of differential equations (under some stability assumptions and taking h → 0.)

Conclusions
In this work, we gave a variation of parameters formula for impulsive differential equations with piecewise constant arguments.We analyzed the constant coefficients case and gave several examples of formulas applied to some concrete piecewise constant arguments.We extended some cases treated before and showed the effect of the impulses in the dynamic.
1.1)where γ(t) is a piecewise constant argument of generalized type.In order to define such γ, let (t n ) n∈Z and (ζ n ) n∈Z such that t n < t n+1 , ∀n ∈ Z with lim n→∞ t n = ∞, lim n→−∞ t n = −∞ and ζ n ∈ [t n , t n+1 ].Then, γ(t) = ζ n , if t ∈ I n = [t n , t n+1) .I.e., γ(t) is a step function.An elementary example of such functions is γ(t) = [t] which is constant in every interval [n, n + 1[ with n ∈ Z (see (1.3)).If a piecewise constant argument is used, the interval I n is decomposed into an advanced and delayed subintervals I n = I + n I − n , where I + n = [t n , ζ n ] and (i) x ′ (t) exists at each point t ∈ R with the possible exception at the times t k , k ∈ Z, where the one side derivative exists.(ii)x(t)satisfies (1.1) on the intervals of the form (t k , t k+1 ), and it holds for the right derivative of x(t) at t k . is continuous onI k = [t k , tk+1 ) with first kind discontinuities at t k , k ∈ Z, where y ′ (t) exists at each t ∈ R with the possible exception at the times t k , where lateral derivatives exist (i.e.y(t) ∈ PC 1 ).A continuous function x(t) is a solution of (1.1) if: Let γ − (t) = t k and γ + (t) = t k+1 , for all t ∈ I k = [t k , t k+1 ).I.e., we are considering the completely delayed and advanced general piecewise constant arguments.Then, taking in account Remark 3, the solution of (4.18) for both cases y − (t) and y + (t) respectively are: + (t, s) + W − (t, s).