Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients

We study the asymptotic behaviour of the solutions of a class of 
 linear neutral delay differential equations with discrete delay 
 where the coefficients of the non neutral part are periodic 
 functions which are rational multiples of all time delays. We show 
 that this technique is applicable to a broader class where the 
 coefficients of the neutral part are periodic functions as well.


1.
Introduction. Neutral functional differential equations (NFDE) play an important role in the theory of functional differential equations (FDE). Its importance as well as the requirement of more refined methods due to theoretical complications, like the non-eventually compactness of the solution operator opposed to the delay case, the theory of NFDE has become an independent tread. Besides its theoretical interest, these equations have some importance in applications (for instance see the book of Kolmanovskii and Myshkis [6]). Many results concerning the theory of NFDE were given in the book by Hale and Verduyn Lunel [5].
One of the subject of intensive investigation during the past decades is the asymptotic behaviour of the solutions of first order linear delay equations or neutral delay differential equations, motivated by the pioneer works by Driver [1] and Driver, Sasser and Slater [2]. Hale and Verduyn Lunel [5,Sec. 8.3] show an application of Floquet theory for delay differential equations in the case where the delays are integer multiples of the common period of the periodic coefficients. Frasson and Verduyn Lunel [4] studied the same equation as an application of their results. Some more recent results have been given by Philos et al. in [7,9,10] and references therein. The paper [10] deals with the asymptotic behaviour of solutions for a first order linear neutral delay differential equation with periodic coefficients and constant delays which are multiples of the common period. Lillo [8] considers the asymptotic distribution of exponential multipliers for periodic differential difference equations for the case in which the delays and the period are rationally related. These works inspired the form of the equation in which we are interested.
Consider the neutral delay differential equation of the form where a j ∈ C, ω > 0, b j (·) are complex-valued continuous functions of a real variable, there is an integer q > 0 such that b j (t + qω) = b j (t) for j = 0, . . . , p, and not all b j are identically zero. In other words, the delays and the common period are rationally related. The change of variablesx(t) = exp − t 0 b 0 (s)ds x(t) transforms the initial value problem (1) into another one of the same form but with b 0 ≡ 0, so we can assume without loss of generality that the NFDE has the form In this work we present results on the asymptotic behaviour of solutions of Equation (2) using spectral properties of the monodromy operator, like the knowledge of a dominant characteristic multiplier of the monodromy operator, following ideas in [4].
This paper is organised in the following way: in Section 2 we present a summary on Floquet Theory for general neutral functional differential equations and spectral properties associated to the solution semigroup following Hale and Verduyn Lunel [5]; in Section 3 we prove our main result, for that computing the resolvent of monodromy operator; in Section 4 we study two particular cases, a simple periodic delay functional differential equation (Section 4.1) and a periodic delay neutral functional differential equation where the coefficient of neutral part is not necessarily constant (Section 4.2).
2. Summary on Floquet theory for periodic NFDE. Let C denote the space of continuous functions from [−h, 0] into the complex space C n , h > 0, with the norm in C given by ϕ = sup |ϕ(u)|, −h u 0, where |ϕ(u)| is any vector norm in C n . Then (C, · ) is a Banach space. Given a function x(t) from [s − h, A] into C n , where A may be infinite, define x t ∈ C, for t s given by x t (u) = x(t + u), −h u 0.
Let σ ∈ R. A complex number ρ is called a characteristic multiplier of the NFDE (3) if ρ is an eigenvalue of finite type of T D,L (σ + ω, σ), that is, an isolated point of the spectrum of T D,L (σ + ω, σ) with finite-dimensional generalised eigenspace. One can show that the characteristic multipliers are independent of σ. For each characteristic multiplier ρ, there is a decomposition of C as C = E σ ⊕ Q σ , where E σ = ker(T D,L (σ + ω, σ) − ρI) k , Q σ = range(T D,L (σ + ω, σ) − ρI) k , for k sufficiently large and E σ has finite dimension, say d. If Φ d is a basis for E σ , then there is a d × d constant matrix B σ where the spectrum of e Bσω equal to ρ and n × d matrix In this sense, there is a Floquet representation on the generalised eigenspace E σ of the character multiplier ρ.
where U (t, σ) is completely continuous for t σ and T D (t, σ) is the solution operator generated by the difference equation D(t)y t = 0.
Using the same type of reasoning as for the autonomous case, if e a D ω is the spectral radius of T D (σ + ω, σ) one proves that any ρ in the spectrum of T D,L (σ + ω, σ) with |ρ| > e a D ω must be a normal eigenvalue and, thus, a characteristic multiplier of Equation (3). Furthermore, there are only a finite number of characteristic multipliers ρ satisfying |ρ| e aω for any constant a > a D . The space C therefore can be decomposed as C = E σ,a ⊕ Q σ,a , where E σ,a and Q σ,a are invariant under T D (σ + ω, σ) and the spectrum of T D (σ + ω, σ)|E σ,a consists only of the multipliers ρ with |ρ| e aω , a > a D , and the spectrum of T D (σ + ω, σ)| Qσ,a lies inside the disk with centre zero and radius < e (a− )ω for some > 0. Therefore, there is a constant K such that Also, there is a constant matrix B σ,a and an ω-periodic matrix C σ,a (t) such that where Φ σ,a is a basis for E σ,a and the only eigenvalues of e Bσ,aω are those characteristic multipliers ρ with |ρ| e aω . Note that if ρ = e λω is simple and dominant (see Definition 2.2), then B σ,a is identical to {ρ}, Φ σ,a = {φ 1 } and C σ,a (t) assumes the simpler form e λt C σ,a (t) = T D,L (t + σ, σ)φ 1 .
Since the characteristic multipliers are independent of the initial time σ, we can consider σ = 0 going forward, therefore the spectral projection onto E 0,a along Q 0,a can be represented using Dunford representation, for ρ ∈ σ(T D,L (ω, 0))\σ(T D (ω, 0)), P ρ = Res z=ρ (zI − T D,L (ω, 0)) −1 , then we can compute spectral projections using residue calculus. In particular, if ρ is a simple characteristic multiplier of T D,L (ω, 0), the spectral projection onto the one-dimensional eigenspace is given by We finish this section presenting results on the asymptotic behaviour of solutions. The proof of the next couple of results are identical to the proof of results in [4], only notice that in the case of RFDE, every element nonzero of the spectrum is an eigenvalue of finite type and always there is a finite number of these eigenvalues which modulus is larger that any value.
If γ is an arbitrary real number such that there is a finite number of characteristic multipliers of T D,L (ω, 0) such that their modulus are larger than e γω , then there are positive constants and N such that for t 0 Motivated by Theorem 2.1 we have the following definition of dominance for characteristic multipliers.
is as Equation (7), then there are positive constants N and so that the large time behaviour of the solution x(t; 0, ϕ) is given by

3.
A class of periodic NFDE. We consider the class of linear periodic NFDE given by where a j ∈ C, ω > 0, b j (·) are complex-valued continuous functions of a real variable, at least one not identically zero, b j (t) = b j (t + qω) for all j = 1, . . . , p where q > 0 is an integer (let q be the least positive integer with such property). We say that x, a continuous complex-valued function defined on the interval [−pω, ∞), is a solution of the NFDE (11) if x(t) + p j=1 a j x(t − jω) is continuously differentiable for t 0 and x satisfies (11) for all t 0.
The computation of the resolvent of Π is given by the following lemma.
We apply Lemma 3.1 and derive the system of ordinary differential equations subjected to the boundary conditions where H(z), M (θ), K(z) and F are as in the statement of Lemma 3.1.
First we treat the case z such that det H(z) = 0. System (29) can be rewritten as the system of ordinary differential equationṡ Let Ω t s (z) the fundamental matrix of the homogeneous systeṁ The variation of constants formula gives us that the solution of (31) is given by where Noticing that d ds Ω θ s (z) = −Ω θ s (z)H −1 M (s), integrating (33) by parts we get (24). In (32) we still have to obtain Ψ (−ω). Using (30) and (32) for θ = 0 we get Finally, noticing that for θ ∈ [−ω, 0] and i = 1 − p, . . . , 0, θ + iω covers the interval [−pω, 0] and ψ(θ + iω) = ψ i (θ) = e T i+p Ψ (θ), from (32) and (34) we get (22). To conclude main part of the proof, we observe that the expression to (zI − Π) −1 ϕ in (22) defines a linear bounded operator for ϕ ∈ C, so we can drop the assumption that ϕ is differentiable, using the fact that the set of differentiable functions is dense in C.
It remains only to prove the statement (25). The formula of the resolvent in (22) is singular when det ∆(z) = 0. It remains to analyse the resolvent when H(z) is singular. Suppose this is the case and that there is a solution Ψ (θ) of (29). Since the first p−1 lines of H are linear independent, one of the last q lines of H is a linear combination of the preceding ones. By the standard algebraic system manipulation operations corresponding to the subtraction of this linear combination from that line, we can derive an algebraic scalar equation involving linear combinations of ψ i (θ), b j i (θ) andφ i (θ) for some indexes i, but necessarily with elementsφ i (θ). Since ϕ →φ i is a unbounded operator, if we could solve the ψ i , the solution would be a unbounded operator, so the resolvent would fail to be a bounded operation and z is in the spectrum of Π.
Before we provide the formula for the spectral projection P µ of Π onto the eigenspace E µ of a simple characteristic multiplier µ, we need the following auxiliary lemma. Lemma 3.3. Let ∆(z) be a n × n square matrix-valued differentiable function of z ∈ C, n 2, and suppose that µ is a simple zero of det ∆(·). Then there are non-null n-column vectors u, v such that adj ∆(µ) = uv T .
From (36) we have rank(adj ∆(µ)) 1 and from (38) and (35) we obtain rank(adj ∆(µ)) 1. Hence rank(adj ∆(µ)) = 1. Letting v T be a non-null row of adj ∆(µ), all other rows are multiple of it and the lemma follows. Now let us prove (35). Suppose by contradiction that dim(ker ∆(µ)) > 1. Then there are at least two linear independent column vectors v 1 , v 2 ∈ ker ∆(z). From {v 1 , v 2 } complete a basis {v 1 , . . . , v n } for C n . Let V be the square matrix whose i th column is v i . Without loss of generality we can assume det V = 1. The columns of ∆(µ)V are given by ∆(µ)v i and at least its first two columns are null.
For z = µ we define the matrix-valued function N (z) obtained from ∆(z)V but multiplying the first column by 1/(z − µ). Then det N (z) = g(z). Since the first column of ∆(z)V is differentiable in z and null for z = µ, the first column of N (z) has a removable singularity at z = µ, so if we define N (µ) = lim z→µ N (z) then N (z) is differentiable and obtain that det N (µ) = g(µ) = 0. A contradiction, since its second column is ∆(µ)v 2 = 0. So we proved the claim (35) and the proof is complete.
Using the Dunford representation for the spectral projection P µ of Π onto a generalised eigenspace E µ , we can compute a formula for the spectral projection of Π using residue calculus.
Proof. Since µ is a simple zero of det(µ) = 0, there is a differentiable function g(z), with g(µ) = d dz [det ∆(z)] z=µ = 0, such that det ∆(z) = (z − µ)g(z). For z = µ in a neighbourhood of µ, we have From (22) and (42), the resolvent of Π has a simple pole at z = µ. From (39) and using Lemma 3.3 we obtain that The factor in (43) between curly brackets evaluates to a scalar function and concentrates all dependence on θ and i, so we can define Φ as in the statement of the theorem. The factor in (43) between round brackets evaluates to a scalar function and concentrates all dependence on ϕ, so that we call it by c(ϕ) as in (41) and (40) follows.
The existence of simple dominant characteristic multiplier µ d allows us to give an explicit formula for the large time behaviour of solutions.
where Φ(·) is the solution through Φ where Φ and c(·) are given in Theorem 3.4.
The proof of this result follows by the previuos Theorem 3.4 and the Corollary 2.3 in Section 2.
4. Case studies. In this section we study some special cases. In first example a class of rather simple periodic retarded FDE is studied and more sharp results are stated. The second example is a class of neutral periodic equation, as we considered in (11). Finally we show by means of an example that techniques of previous sections can be extended to larger classes of periodic equations. We compute the large time behavior of solutions of a neutral functional delay equation where the neutral part is not necessarily constant.

4.1.
A simple class of periodic retarded FDE. We consider the initial value problem (IVP) on delay differential equations of the forṁ where a(t) and b(t) are real continuous 1-periodic functions. By the change of variables we get that (45) is equivalent to the IVṖ where A = 0 −1 a(s)ds. Computing the resolvent as before and using the same notation, we arrive to Setting z = e λ and remarking that the exponential is 2πi-periodic we obtain In order to apply Theorem 3.5 we compute ∆ (µ) = (µ + D)/µ Applying Theorem 3.5, assuming condition (49) we obtain the following asymptotic behavior for the solutions of (47) Undoing the change of variables (46), we obtain the large time behavior for the solutions of the IVP (45) is given by

4.2.
A class of periodic NFDE. Consider the following class of periodic neutral equation d dt where c ∈ C, a(·) and b(·) are 1-periodic functions. Considering the change of variable give by where A = 1 0 a(s)ds. Now, settingc = ce −A andb(t) = e −A (b(t) − ca(t)) which is 1-periodic since b(t) is 1-periodic, we have that find y which satisfies (50) is equivalent to find a solution x of the next equation Therefore, we considerc = −1 andb(t) = cos 2 (2πt) and we study the asymptotic behaviour of solution of the following IVP d dt Using Lemma 3.2 in order to compute the resolvent (zI −Π) −1 of the monodromy operator Π = T (0, ω) and obtain the ODĖ Therefore the fundamental solution is given by Hence, the spectrum of the monodromy operator Π is = 0 or µ = 1}. Now, we analyse the zeros of the characteristic equation Setting z = e λ and remarking that the exponential function is 2πi-periodic we obtain so it suffices to analyse the equation on the Floquet exponents Using a calculus method, we notice that x → f (x) = 1 2(e x −1) is strictly decreasing. Since the identity function is increasing, it follows that there exists two roots, a positive root λ 0 which value is approximately 0, 6034 and the other root which is negative. The characteristic exponent λ 0 is simple. In fact, Hereafter, we are going to prove that the characteristic exponent is dominant. Suppose that λ = a + bi is another root of (54) such that a > 0. Without loss of generality, we can assume that b 0. Taking the real and imaginary part in (54), we obtain the following system 2ae a cos b − 2be a sin b = 2a + 1, (55a) 2ae a sin b + 2be a cos b = 2b.
Making algebraic computations with Equations (55a) and (55b) we obtain the following equation that relate a and b.
Suppose that (H) some µ ∈ C is a simple and dominant root of the characteristic equation (64) and |a(θ)| < |µ| for θ ∈ [0, ω]. In particular, (H) is fulfilled in the restricted cases where (H ) max |a(θ)| < 1 and either i) a is constant and 0 −ω b(s) = 0 or ii) there is a constant α such that b(t) = αȧ(t), when we obtain that Under hypothesis (H), from Theorem 2.1, the asymptotic behavior of solutions x(t) of (61) subjected to the initial condition x 0 = ϕ is given by e −γt |x(t) − c(ϕ)Φ(t)| < N e −δt ϕ .
where γ is such that µ = e γω , δ and N are suitable constants. If (H ) is fulfilled when 1 is the only Floquet multiplier, we can take any δ < ln(max |µ − a(t)|)/ω and N = N (δ) a suitable constant.