On Periodic Solutions of Delay Differential Equations with Impulses

The purpose of this paper is to study the nonlinear distributed delay differential equations with impulses effects in the vectorial regulated Banach spaces R([−r, 0],Rn). The existence of the periodic solution of impulsive delay differential equations is obtained by using the Schäffer fixed point theorem in regulated spaceR([−r, 0],Rn).


Introduction
In this paper, we will investigate the existence of periodic solutions for vectorial distributed delay differential equations with impulses in regulated Banach spaces.More precisely, the prototype of this delay differential equations with impulses, is of the form dx(t) dt = −λx(t) + f (t, x t ), a.e.t ∈ [0, ω + τ], λ > 0, ω > 0, x(t j ) = x(t − j ), and x(t + j ) − x(t j ) = h j (x(t j )), ∀j = 1, . . ., l, x 0 (θ) = ϕ(θ), θ ∈ [−τ, 0], with x t (θ) = x(t + θ), θ ∈ [−τ, 0], τ > 0 and where x and ϕ are R n -valued functions on [−τ, ω], and [−τ, 0], respectively.The Equation ( 1) is a nonlinear delay differential equation.More details about this type of equations can be found in [1].Moreover, we assume that (i) h j ∈ C(R n , R n ), j = 1, . . ., l, (ii) {t 1 , t 2 , • • • , t l } is an increasing family of strictly positive real numbers, (iii) there exist δ > 0 and T < ∞, such that for any j = 1, . . ., l − 1, we have We call (2) the impulses equation where, x(t − j ) (resp.x(t + j )) denotes the limit from the left (resp.from the right) of x(t), as t tends to t j .This type of differential equations without delay was initiated in 1960's by Milman and Myshkis [2,3].This problem started to be popular mostly in Eastern Europe in the years 1960-1970, with special attention during the seventies of the last century.Later on, several investigations and important monographs appeared with more details, which show the importance of studying such systems, see for example [4][5][6][7][8][9][10][11].In recent years, many investigations have arisen with applications to life sciences, such that the periodic treatment of some biomedical applications, where the impulses correspond to administration of a drug treatment at certain given times [12][13][14][15].However, comparatively speaking, not much has been done in the study of impulsive functional differential equations in regulated vectorial space, taking into account the general theory of functional analysis and having an acceptable hypothesis that can be used in real life applications, see [12] for more details.
Let us first introduce for each τ > 0, the regulated Banach space R = R([−τ, 0], R n ), given by: R = ϕ : [−τ, 0] → R n : ϕ has left and right limits at every points of [−τ, 0] , endowed with the following norm We will make the following assumptions We set the initial value problem as follows Problem 1.Let ϕ be an element of R. We want to find a function x defined on [−τ, ω + τ] such that x satisfies (1)-(3).
We consider the nonlinear impulsive delay differential equation in R as x(t j ) = x(t − j ), and x(t + j ) − x(t j ) = h j (x(t j )), ∀j = 1, . . ., l, x 0 (θ) = ϕ(θ), θ ∈ [−τ, 0] and x(0 The aim of this paper is to extend the main results related to the existence of the ω-periodic solutions for ordinary differential equations with impulses presented by Li et al. [16] and Nieto [17].These papers contain references which provide additional reading on this topic, i.e., differential equations with impulses by using the fixed point theory.

Existence and Uniqueness of Solution
Let us start first by introducing some related definitions and lemmas.
has a unique solution.
Proof.We set S = {y ∈ C([0, t 1 ], R n ), y(0) = x(0 + ) = ξ}.Let us define the operator T by For each y ∈ S, we consider the Nemytski operator F, defined by where Then, we get Define, the norm of any function y in S by where ρ is a fixed positive constant greater than K + λ.We have for each y 1 (t) and y 2 (t) in S, and hence Since K+λ ρ < 1, then, T is a contraction on S, and the result follows immediately.Proof.The proof follows by using the last lemma.
By passing to the limit as t goes to t − j , and by solving the delay differential Equation ( 14) on the interval [t j−1 , t j ), we have Then, by taking into account the impulses condition (15), we have f (s, e −λ(s+θ) z s )e λs ds + e λt j h j (e −λt j z(t j )), Consequently, we can rewrite the last equations in more general form for all t > 0 where z(0 + ) = x(0 + ) = ξ, and Now, we will try to involve the u k s.To this end, we will take the limit from the left of the Formula (17) as t tends to t k > 0, we obtain Substituting the last expression into (18), we have In particular, we have {j : 0 ≤ t j < t 1 } = ∅, and therefore By using, the Equation ( 16), we can rewrite the Equation ( 17) as z t (θ) = ξ + t+θ 0 f (s, e −λ(s+θ) z s )e λs ds + ∑ 0≤t j <t+θ e λt j u j , t + θ / ∈ {t k } k≥0 , and t + θ ≥ 0, (19) and by using x(t) = e −λt z(t), we have for t + θ / ∈ {t k } k≥1 , and t + θ ≥ 0 e −λ(t+θ−t j ) u j .
Using ( 12) and ( 13), we get where starting from Remark 1. Taking into account the conditions (II)-(III), we have u t ∈ R, ∀t ∈ [0, ω + τ], and t → x t is a regulated function, because the functions t → ϕ 0 t , and t → H 0 t are regulated.
In the next section, we will investigate the existence of the periodic solution(s) for the delay differential equation with impulses (1)-(3) using Schäffer's fixed point theorem [19].

Existence of Periodic Solutions
Let us consider the Poincaré operator, given by: where x ω (ϕ) is the solution of the delay differential equation with impulses (1)- (3).It is clear that if the Poincaré operator J admit a fixed point, then (1)-( 3) has a ω-periodic solution.The following lemma is useful to prove the main theorem.
(2) If 1 − (c + 1)e −λτ = 0, and then, we have the existence of infinitely many τ-periodic solutions.Now, we can consider for each t ≥ −τ and ω ≥ τ, the Poincaré operator J : R → R defined by where and, starting from It is clear, that, the ω-periodic solutions in R of (1)-( 3) are exactly the fixed points of the Poincaré operator J, i.e., J ϕ = ϕ.
The following theorem, is known as the Schäffer's fixed point theorem [19], which can be found for example in Deimling's book [20].
Theorem 1. [19][20][21][22] Let X be a normed space, F a continuous mapping of X into X, such that the closure of F (B) is compact for any bounded subset B of X.Then either: (i) the equation x = λF x has a solution for λ = 1, or (ii) the set of all such solutions x, for 0 < λ < 1, is unbounded.
Before, we state the main theorem of our work, we will need the following lemma.Lemma 6.Let f : [0, ω + τ] × R → R n be a map satisfying (I) and (II), where ω ≥ τ, and h j , j = 1, . . ., l are bounded and satisfy the condition (III).Then, the Poincaré operator J : R → R is completely continuous.
Proof.Let B ⊂ R be a bounded set and ϕ ∈ B. Then by using the condition (I), we have Therefore, there exist two constants M and M such that where and starting from and, we have which imply that J(B) is uniformly bounded.For each t ≥ 0, there exists n ∈ N * such that t ∈ [t n , t n+1 ), and for any θ, θ ∈ [−r, 0], one can obtain for any ϕ ∈ Therefore, for each t ∈ [t n , t n+1 ), we will have as | θ − θ | goes to 0, J ϕ(θ) − J ϕ( θ) goes to 0, which imply that the Poincaré operator J(B) is equicontinuous.Using Arzelà-Ascoli's theorem, we conclude that the Poincaré operator J is completely continuous.Now, we are ready to state the main result of our work, related to the existence of ω-periodic solution(s) of ( 1)-(3).Theorem 2. Let f : [0, ω + τ] × R → R n be a map satisfying (I) and (II), where ω ≥ τ, and h j , j = 1, . . ., l are bounded and satisfy the condition (III).Then, the nonlinear impulsive problem (1)-( 3), has at least one ω-periodic solution in R.
If, there exists a constant C < 1, such that This contradiction implies, the uniqueness of the ω-periodic solution of (1)-(3).

Conclusions
The method described in this work presents new challenges for more investigation on more realistic models; such as the extension of the ascorbic acid model [12] and H IV model [13,14].Taking into account the delay effect on respective compartments [23][24][25].This kind of work, will need more investigation on modeling validation effort, keeping a close eye on the real life data in order to have a more realistic model.The explicit solutions presented in the technical Lemma 4 and methods of proving the existence of periodic solutions are very useful for further future investigations.