A NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE OF PERIODIC SOLUTIONS OF LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS WITH DISTRIBUTED DELAY

A necessary and sufficient condition is established for the existence of periodic solutions of linear impulsive differential equations with distributed delay.

1. Introduction and preliminaries.Impulsive delay differential equations can suitably model various evolutionary processes that exhibit both delay and impulse characteristics.In particular, they provide a natural description of the motion of several real world processes which, on one hand, depends on the processes history that often turns out to be the cause of phenomena substantially affecting the motion and, on other hand, is subject to short time perturbations whose duration is almost negligible.Such processes are often investigated in various fields of science and technology, such as physics, population dynamics, ecology, biological systems, optimal control, etc.For more details see [5,15,18,19,20,23,24,25,26,28] and reference quoted therein.
It is well known that (see eg., [17]) the nonhomogeneous linear equation has periodic solutions if and only if ω 0 y T (t)f (t) dt = 0 (1) for all periodic solutions y(t) of period ω of the adjoint equation where A ∈ C(R, R n×n ) and f ∈ C(R, R n ) are periodic functions of period ω.
In his remarkable monograph [10], Halanay extended the above result to linear delay differential equations of the form where A, B ∈ C(R, R n×n ) and f ∈ C(R, R n ) are periodic functions of period ω and τ > 0 is a fixed real number.It was shown that the required condition involves the same integral (1).Indeed,  ( Our purpose in this paper is to carry out the above result to a type of linear impulsive delay differential equations.Let η : R + × R → R n×n be a kernel function (cf.[11]) satisfying the following conditions: (a) η(t, s) is normalized so that η(t, s ) There exists a function V (t) > 0 bounded for t ≥ 0 such that the total variation of η(t, s) in s on [−τ, 0] for t ≥ 0 is not larger than V (t).We shall consider impulsive differential equation with distributed delay of the form where the following conditions are satisfied: (c) η(t, s) is continuous in t ∈ [0, ∞) uniformly for s ∈ [−τ, 0] and f (t) is continuous on (−∞, ∞); (d) η(t, s) and f are ω periodic functions in t, τ < ω; (e) {A i0 }, {A ik }, {f i } are p periodic sequences in i, j < p is a fixed positive integer; (f) {θ i } is an increasing real sequence satisfying lim i→∞ θ i = ∞ and θ i+p = θ i + ω.Conditions (c)-(f) guarantee that (A) is an ω periodic equation and in particular if x(t) is a solution then so is x(t + ω).By a solution of (A) on an interval J, we mean a function x which is defined on J such that x is continuous on J except possibly at θ i ∈ J for i ∈ Z, where x(θ ) and that x satisfies (A) on J.Under the above conditions (even less) one can easily show that for given σ ≥ 0 and φ ∈ P LC[σ − τ, σ] there is a unique solution x(t) of (A) which satisfies where P LC[σ − τ, σ] denotes the set of piecewise left continuous functions defined on [σ − τ, σ] having a finite number of discontinuity points of the first kind at θ i , i ∈ Z.
Existence of periodic solutions of delay differential equations (without impulses) and impulsive differential equations (without delays) has been extensively investigated in the literature, see [7,9,10,11,16,22,27] and [3,4,8,12,18,21] respectively.However, very rare investigations have been achieved in the direction of the corresponding theory of impulsive delay differential equations, see for instance the papers [6,13,14,19,20,29].It is to be noted that our equation (A) differs from those considered in the literature not only it is more general but also it allows delay terms in the impulse conditions.Such impulse conditions are more natural for delay differential equations.
2. Some auxiliary assertions.The discussion in this section stems from [1].We shall construct a function analogous to (3) with respect to the equations considered in this paper.It turns out that function (3) should be modified as follows where Function (5) will enable us to construct the adjoint equation for (A) and derive their solution representations.These results have been provided in [1].Thus, we shall state them here without proofs.We should note that no periodicity assumption is required in this section.Consider the equations is a solution of (B) and y(t) is a solution of where < x(t), y(t) > is defined by (5).
[1] Let X(t, α) be a fundamental matrix of (B) and σ ≥ 0 a real number.If x(t) is a solution of (A), then Definition 2. A matrix solution Y (t, α) of (C) satisfying Y (α, α) = I and Y (t, α) = 0 for t > α is said to be a fundamental matrix of equations (C).
[1] Let Y (t, α) be a fundamental matrix of (C) and σ ≥ 0 a real number.If y(t) is a solution of (C), then Corollary 1.
[1] Let X(t, α) be a fundamental matrix of (B) and Y (t, α) be a fundamental matrix of (C).Then 3. The main result.Let x(t) = x(t; ϕ) be the solution of equation (A) defined for t ≥ 0 such that x(t) coincides with ϕ on [−τ, 0].The periodicity of the equation implies that x(t + ω; ϕ) is likewise a solution of the equation defined for t + ω ≥ τ .If this solution coincides with ϕ in [−τ, 0], then on the basis of the uniqueness theorem it follows that x(t + ω; ϕ) = x(t; ϕ) for all t ≥ −τ and the solution is periodic.Thus the periodicity condition of the solution is written as is periodic if and only if W ϕ = ϕ, i.e., ϕ is a fixed point of W .Let z(t) = z(t; ϕ) be the solution of the homogeneous equation (B) defined for t ≥ 0 such that z(t) = ϕ(t) on [−τ, 0].Then by the representation of solutions (7), the periodicity condition reads as Let y(t) = y(t; ψ) be the solution of (C) defined for t ≤ ω + τ such that y(t) = ψ(t) on [ω, ω + τ ].Similarly, we conclude that if y(t − ω; ψ) coincides with ψ in [ω, ω + τ ] then y(t−ω; ψ) = y(t; ψ) and hence the solution is periodic.From the representation of solutions (8) and identity (9), we obtain For the sake of convenience, we also use the notation for matrix functions Ψ and Φ defined on [−τ, 0] provided that multiplication is possible.Note that < Ψ(s), Φ(s) > could be a number, or a vector or a matrix depending on the sizes of Ψ and Φ.
Lemma 1.For any matrix functions N, M, L ∈ R n×n , we have The proof of the above lemma is straightforward and can be achieved similarly as in [10] by following the same arguments used in [2, Lemma 2].
With this notation, the operator U can be written as If we define Ũ φ =< φ(η), X(ω + η, s + τ ) > T , then in view of Lemma 1 we obtain If ρ is an eigenvalue of Ṽ , then there exists a nonzero solution of the equation where φ(s) = ψ(s + ω + τ ), s ∈ [−τ, 0].The right side of the above equation is nothing but Ũ φ.Thus the eigenvalues of the operators Ũ and Ṽ coincide and in addition, if ψ is an eigenfunction for Ṽ , then φ = ψ(s + ω + τ ) is an eigenfunction for Ũ .
The following lemma is needed to prove the main result.
Lemma 2. Equations (B) and (C) have the same number of linearly independent periodic solutions of period ω > τ .
Proof.Using the notation introduced above and proceeding as in [10, p.414] one can conclude that ρϕ(s) − U ϕ(s) = 0 and ρ φ(s) − Ũ φ(s) = 0 have the same number of linearly independent solutions, which implies in particular the fact that U and Ũ have the same eigenvalues, hence if ρ is a multiplier of the equation, then 1/ρ is a multiplier of the adjoint equation.
We are now in a position to state and prove the main result of this paper.
Theorem 4. A necessary and sufficient condition for the existence of periodic solutions of period ω of equation (A) is that for all periodic solutions y(t) of period ω of the adjoint equation (C).
In view of Lemma 2, it is clear that has solutions if and only if < φ(α), F (α) >= 0 (16) for all solutions φ of ρ φ(s)− Ũ φ(s) = 0. Therefore it suffices to show that (16) holds under condition (14).To see this we first observe from (10) that which is clearly zero by our assumption (14).The proof is complete.