Asymptotic integration of functional differential systems with oscillatory decreasing coefficients:

Asymptotic integration of functional differential systems with oscillatory decreasing coefficients:

a center manifold approach

Pavel Nesterov

Yaroslavl State University, Sovetskaya str. 14, Yaroslavl, 150003, Russia Received 25 November 2015, appeared 2 June 2016

Communicated by Mihály Pituk

Abstract. In this paper we study the asymptotic integration problem in the neighbor- hood of infinity for a certain class of linear functional differential systems. We propose a method for the construction of the asymptotics of solutions in the critical case. Using the ideas of the center manifold theory we show the existence of the so called critical manifold that is positively invariant for trajectories of the initial system. We establish that the dynamics of solutions lying on this manifold defines the asymptotics for all solutions. We illustrate the proposed method with an example of the construction of the asymptotics for solutions of a certain scalar delay differential equation.

Keywords: asymptotic integration, functional differential systems, center manifold, method of averaging, Levinson’s theorem, oscillatory decreasing coefficients.

2010 Mathematics Subject Classification: 34K06, 34K19, 34K25, 34C29.

1 Introduction

We study the asymptotic integration problem for the functional differential system

˙

x= B0xt+G(t,xt) (1.1)

as t → _∞. Here x ∈ _C^m, x_t(θ) = x(t+θ) (−h ≤ θ ≤ 0) denotes the element of C_h, where C_h ≡C [−h, 0],C^m

is the set of all continuous functions defined on[−h, 0]and acting toC^m. Further, B₀is a bounded linear functional acting fromC_h toC^m andG(t,x_t)has the form

G(t,xt) =B(t,xt) +R(t,xt). (1.2) We assume that B(t,·)andR(t,·)are linear bounded functionals fromC_hto C^m such that for each ϕ∈ C_hfunctionsB(·,ϕ)andR(·,ϕ)are Lebesgue measurable fort ≥t₀and, moreover,

|R(t,ϕ)| ≤γ(t)kϕk_Ch, γ(t)∈ L₁[t₀,∞) kϕk_Ch =sup_{−h≤θ≤0}|ϕ(θ)|. (1.3)

BEmail: nesterov.pn@gmail.com

The structure of the functionalB(t,·)will be defined later. We note only that for each ϕ∈C_h functionB(·,ϕ)has, in general, an oscillatory decreasing form ast →_∞.

This paper continues our studies of the asymptotic integration problem for Eq. (1.1) that we began in [25]. In the mentioned paper the case of the zero operator B₀ was discussed. In the case of nonzero operatorB₀ the asymptotic integration method, developed in [25], cannot be applied. Since the case of nonzero operator B0 is the situation of general position, it is of significant importance to provide the method for constructing the asymptotics in this case.

We emphasize that the method, we propose below, is not an adaptation or extension of the corresponding method from [25]. However, both methods in the final step use the so-called method of averaging together with the known asymptotic theorems. This will be discussed in details in Section5of this paper.

Functional differential systems of the form (1.1) with linear and nonlinear functionals on the right-hand side, considered as perturbations of linear autonomous system

˙

x= B₀x_t, (1.4)

were studied by many authors (see, e.g., [1–4,9,11,13,14,27–29]). We also remark that the first asymptotic theorems for scalar delay differential equations were proposed by R. Bellman and K. L. Cooke [8] (see also [19, Chapter 9] for a brief survey).

Throughout the paper we study Eq. (1.1) under the condition that the characteristic equa- tion

det∆(λ) =0, ∆(λ) =λI−B0(e^λθI), (1.5) hasNroots (with account of their multiplicities)λ₁, . . . ,λ_N with zero real parts and all other roots have negative real parts. This makes possible to use the ideas of the center manifold theory (see, e.g., [2,5,6,10]) for asymptotic integration of Eq. (1.1). The paper is devoted to the adaptation of this technique for the considered asymptotic integration problem. Particularly, a significant role will be also played by the averaging method proposed in [23] for the asymp- totic integration of the ordinary differential systems with oscillatory decreasing coefficients.

This paper is organized as follows. In Section 2 we give some notations and facts from the theory of functional differential systems needed for the sequel. In Section3 we propose an algorithm for an approximate construction of the so called critical manifold inC_h that is positively invariant for sufficiently larget for trajectories xt(θ)of Eq. (1.1). It turns out that the dynamics of solutions of Eq. (1.1) lying on this manifold defines the asymptotics for all its solutions. The main theorems describing the properties of critical manifold are established in Section4. Finally, in Section5we study the asymptotic integration problem ast →_∞for the system on critical manifold. In this section we also use the developed technique to construct the asymptotics ast→_∞for solutions of the scalar delay differential equation

˙

x =−^π

2x(t−1) + ^asinωt

t^ρ x(t), (1.6)

where a,ω ∈ _R\{0} and ρ > 0. The proof of Theorem 3.2 from Section 3, concerning the solvability of certain algebraic problems, is given in Appendix.

2 Preliminaries

The facts and notations given in this section may be found in [19] (see also [20,22]).

We say that function x(t)with values in C^m satisfies (1.1) for t ≥ T if x(t)is continuous on [T−h,∞), absolutely continuous on[T,∞)and (1.1) holds almost everywhere on[T,∞). Under the above conditions, for each ϕ∈C_h and eachT≥ t₀ there is a uniquex(t)satisfying (1.1) forT≥ t₀ withx_T = ϕ. We will call the functionx(t)the solution of Eq. (1.1) with initial value xT = ϕ.

It is known that linear autonomous equation (1.4) generates in C_h for t ≥ 0 a strongly continuous semigroup T(t): C_h → C_h. The solution operator T(t) of Eq. (1.4) is defined by T(t)ϕ = x_t^ϕ(θ), where ϕ ∈ C_h and x_t^ϕ(θ) is a unique solution of (1.4) with initial value x₀^ϕ(θ) = ϕ. The infinitesimal generator A of this semigroup is defined by Aϕ = ϕ⁰(θ) for ϕ∈ D(A). The domain of A

D(_A) =ϕ∈ C_h

ϕ⁰(θ)∈C_h, ϕ⁰(₀) =_B0ϕ is dense inC_h. The following equalities hold:

d

dtT(t)ϕ=T(t)Aϕ= AT(t)ϕ, ϕ∈D(A). (2.1) In the sequel we will use the Riesz representation ofB₀:

B0ϕ=

Z ₀

−hdη(θ)ϕ(θ)_{, (2.2)}

whereη(θ)is(m×m)-matrix function of bounded variation on[−h, 0]. Using (2.2), we obtain the following expressions for matrix∆(λ)from (1.5) and its derivatives:

∆(λ) =λI−

Z ₀

−hdη(θ)e^λθ, ∆⁰(λ) = I−

Z ₀

−hθdη(θ)e^λθ,

∆^(j)(λ) =−

Z ₀

−hθ^jdη(θ)e^λθ, j≥2. (2.3) We can associate with (1.4) the transposed equation

˙ y=−

Z ₀

−hy(t−θ)dη(θ), (2.4)

where y(t) is an m-dimensional complex row vector. The phase space for (2.4) is C⁰_h ≡ C [0,h],C^m∗

, whereC^m∗ is the space of m-dimensional row vectors. Forψ ∈ C⁰_h and ϕ∈ C_h we define the bilinear form

ψ(ξ),ϕ(θ) =ψ(0)ϕ(0)−

Z ₀

−h

Z _θ

0 ψ(ξ−θ)dη(θ)ϕ(ξ)dξ. (2.5) If

Λ=λ_i ∈_C|det∆(λ_i) =0, i=1, . . . ,N , (2.6) then we can decompose C_h into a direct sum

C_h= P_Λ⊕Q_Λ. (2.7)

Here P_Λ is the generalized eigenspace associated with Λ andQ_Λ is the complementary sub- space of C_h such thatT(t)Q_Λ ⊆ Q_Λ, t ≥ 0. Let Φ(θ) be the (m×N)-matrix function whose columns are the generalized eigenfunctionsϕ₁(θ), . . . ,ϕ_N(θ)of Acorresponding to the eigen- values from Λ. Thus, the columns of Φ(θ) form the basis of P_Λ. Moreover, let Ψ(ξ) _be

(N×m)-matrix whose rows ψ₁(ξ), . . . ,ψ_N(ξ)form the basis of the generalized eigenspaceP_Λ^T of the transposed equation (2.4) associated with Λ. We can choose matrices Φ(θ) and Ψ(ξ) such that

Ψ(ξ),Φ(θ) = ψ_i(ξ),ϕ_j(θ)

1≤i,j≤N = I. (2.8)

SinceΦ(θ)is the basis ofP_Λ and AP_Λ ⊆ P_Λ, there exists (N×N)-matrix D, whose spectrum isΛ, such that AΦ(θ) =_Φ(θ)D. From (2.1) and the definition of A, we deduce that

Φ(θ) =_Φ(0)e^Dθ, T(t)_Φ(θ) =_Φ(θ)e^Dt=_Φ(0)e^D(t+θ), (2.9) where−h≤θ ≤0 andt ≥0. Analogously, for matrixΨ(ξ)we have

Ψ(ξ) =e^−DξΨ(0), (2.10)

where 0≤ ξ ≤h. MatricesΦ(0)andΨ(0)are chosen in the following way. Since the columns of matrix Φ(θ) are the generalized eigenfunctions of A, they should belong to D(A). This implies that

Φ⁰(0) =_Φ(0)D=B₀Φ=

Z ₀

−hdη(θ)_Φ(0)e^Dθ. (2.11) The same reasoning, using (2.4) and (2.10), yields

Ψ⁰(0) =−_DΨ(0) =−

Z ₀

−he^DθΨ(0)dη(θ). (2.12) Finally, the spacesP_Λ andQ_Λ from decomposition (2.7) ofC_hmay be defined as follows:

P_Λ =ϕ∈ C_h| ϕ(θ) =_Φ(θ)a, a∈_C^N ,

Q_Λ =ϕ∈ C_h|(_Ψ,ϕ) =0 . (2.13)

Let xt(θ) be the solution of (1.1) for t ≥ t0 with initial value xt₀ = ϕ. The following variation-of-constants formula holds (see [20]):

xt(θ) =T(t−t0)ϕ+

Z _t

t0

dK(t,s)G(s,xs)ds, t≥ t0. (2.14) Here the kernelK(t,·):[t0,t]→C_h is given by

K(t,s)(θ) =

Z _s

t₀ X(t+θ−α)dα, (2.15)

where X(t) is the fundamental matrix of (1.4), i.e., the unique matrix solution of (1.4) with initial condition

X0(θ) =

(I, θ =0,

0, −h≤θ <_0. ^(2.16)

We can write (2.14) formally as (see [19]) x_t(θ) =T(t−t₀)ϕ+

Z _t

t₀ T(t−s)X₀(θ)G(s,x_s)ds, t≥t₀, (2.17) whereT(t−s)X₀(θ) =X(t+θ−s)_.

We decompose now solutionx_t(θ)of Eq. (1.1) with initial valuex_t0 = ϕaccording to (2.7).

By (2.14), we have

xt(θ) =x_t^PΛ+x^Q_t^Λ, ϕ(θ) =ϕ^PΛ+ϕ^QΛ, (2.18) x^P_t^Λ(θ) =T(t−t₀)ϕ^PΛ+

Z _t

t0

T(t−s)X₀^PΛ(θ)G(s,xs)ds, (2.19) x^Q_t ^Λ(θ) =T(t−t₀)ϕ^QΛ+

Z _t

t₀ d

K(t,s)^QΛG(s,x_s)ds, (2.20) wheret ≥t0and

X^P₀^Λ(θ) =_Φ(θ)_Ψ(0), K(t,s)^QΛ =K(t,s)−_Φ(θ) _Ψ,K(t,s). (2.21) If we make decomposition (2.18) in (2.17), we obtain formulas analogous to (2.19), (2.20). The only difference is that (2.20) should be replaced by formula

x^Q_t ^Λ(θ) =T(t−t₀)ϕ^QΛ+

Z _t

t0

T(t−s)X₀^QΛ(θ)G(s,x_s)ds, (2.22) whereX₀^QΛ =X₀(θ)−X₀^PΛ(θ). It is sometimes more appropriate to use (2.22) instead of (2.20).

We should only keep in mind that to attain the necessary mathematical strictness we need to replace integrands of the form T(t−s)X₀(θ)(_{. . .})ds and T(t−s)X₀^QΛ(_{. . .})ds in the obtained formulas by integrandsdK(t,s)(θ)(. . .)andd

K(t,s)^QΛ(θ)(. . .)respectively. Let

x^P_t^Λ(θ) =_Φ(θ)u(t)_, u(t)∈_C^N, (2.23) thenu(t) = (_Ψ,xt)and, moreover, functionu(t)is the solution of ordinary differential system

˙

u=Du+_Ψ(0)G(t,x_t), t≥t₀ (2.24) with initial conditionu(t₀) = (_Ψ,ϕ).

Assume thatΛis defined by (2.6) and suppose that it coincides with the set

λ∈_C|det∆(λ) =0, Reλ>β (2.25) for some β∈ R. Then for anyε > 0 there exists constant M = M(ε) such that the following inequalities hold:

kT(t)ϕ^QΛk_Ch ≤ Me^(β+ε)tkϕ^QΛk_Ch, t≥0, ϕ∈ C_h, (2.26) kT(t)X^Q₀^Λk_Ch ≤ Me^(β+ε)t, t≥0, (2.27)

Z _t

t₀ d

K(t,s)^QΛG(s,x_s)ds Ch

≤ M Z _t

t₀ e^{(β+ε)(t−s)}

G(s,x_s)ds, t≥t₀. (2.28) Note that the matrix T(t)X^Q₀^Λ on the left-hand side of inequality (2.27) belongs C_h only for t ≥ h. Nevertheless, we can use the norm of C_h on the left-hand of (2.27) since T(t)X₀^QΛ is bounded in θ∈ [−h, 0]_fort∈[_0,h]_.

3 Approximate construction of the critical manifold

We begin this section by clarifying the form of the operatorB(t,ϕ)in (1.2). Suppose that B(t,ϕ) =

∑

v_i(t)B_i(t,ϕ) +

∑

1≤i1≤i2≤n

v_i1(t)v_i2(t)B_i1i2(t,ϕ) +· · ·+

∑

1≤i₁≤···≤i_k≤n

v_i1(t)· · ·v_ik(t)B_i1...ik(t,ϕ). (3.1) HereB_i1...il(t,·)are bounded linear functionals acting fromC_h toC^m. We assume that

B_i1...il(t,ϕ) =

∑

Γ⁽_j^i1...il)(t)`⁽_j^i1...il)(ϕ), ϕ∈C_h. (3.2)

In the formula above,`⁽_j^i1...il)(ϕ)are bounded linear functionals acting fromC_h to C^m that do not depend ontandΓ⁽_j^i1...il)(t)are some matrices whose entries are trigonometric polynomials, i.e.,

Γ⁽_j^i1...il)(t) =

∑

β⁽_sj^i1...il)e^iωst, (3.3) where β⁽_sj^i1...il) are constant complex (m×m)-matrices and ωs are real numbers. Finally, v₁(t), . . . ,vn(t)are absolutely continuous functions acting from[t₀,∞)toCsuch that

1⁰. v₁(t)→0,v₂(t)→0, . . . ,vn(t)→0 ast →_∞;

2⁰. ˙v₁(t), ˙v₂(t), . . . , ˙v_n(t)∈ L₁[t₀,∞);

3⁰. There existsk∈_Nsuch thatv_i1(t)v_i2(t)· · ·v_ik+1(t)∈ L₁[t0,∞)for any sequence 1≤i₁≤ i₂ ≤ · · · ≤i_k+1≤ n.

We now define the setΛby formula (2.6). Assume that the following hypotheses hold.

H₁. Reλ=0 for allλ∈_Λ;

H2. The setΛcoincides with set (2.25) for some β<_0.

Note that if hypothesesH1,H2do not hold for Eq. (1.1) with operatorG(t,x_t)having form (1.2), (1.3), (3.1), then we can make the change of variablex(t) =y(t)e^dt, where

d=sup

Reλ|det∆(λ) =0 .

The transformed system will have the same structure as the initial one (with another func- tionals having the same properties) and, moreover, hypotheses H₁, H₂ will hold for it. We remark that the verification of hypothesesH₁, H₂ for a certain Eq. (1.1) is not a trivial prob- lem. This problem is typical, say, for bifurcation theory. As a rule, various algebraic methods, methods from complex analysis and the methods to study location of the operator spectrum are used to establish the validity of hypothesesH₁,H₂ or, at least, to find the quantityd (see, e.g., [8, Chapters 12, 13]). Thus, it is a distinct and a serious problem and, therefore, it is not studied here. Finally, we note that hypothesisH₂ ensures that the critical manifold, defining below, possesses the property of global attraction, i.e., it attracts all the solutions of Eq. (1.1).

We decompose nowC_hbyΛinto direct sum (2.7).

Definition 3.1. A setW(t)⊂C_h(linear space) is said to be a critical (or center-like) manifold of Eq. (1.1) fort ≥t∗ ≥t0if the following conditions hold.

1. There exists an (m×N)-matrix function H(t,θ) which is continuous in t ≥ t∗ and θ ∈ [−h, 0]with columns belonging toQ_Λ fort ≥t∗such thatkH(t,·)k_Ch →0 ast →_{∞, where}

kH(t,·)k_Ch = sup

−h≤θ≤0

|H(t,θ)|

and| · | is some matrix norm;

2. Fort ≥t∗, the setW(t)has the form

W(t) =ⁿϕ(θ)∈C_h |ϕ(θ) =_Φ(θ)u+H(t,θ)u, u∈_C^No, (3.4) whereΦ(θ)is a basis for a generalized eigenspaceP_Λ from (2.7);

3. The setW(t)is positively invariant for trajectories of Eq. (1.1) fort≥t∗, i.e., if xT ∈ W(_T)_, T ≥t∗, thenxt∈ W(t)fort ≥T.

Assume that a critical manifold W(t) of Eq. (1.1) exists for sufficiently large t (the corre- sponding theorem will be proved in the next section). We propose the method for construction of a certain matrix that is an approximation in some sense for the matrix H(t,θ)from (3.4).

An algorithm we describe below has much in common with an approximation scheme of a center manifold for nonlinear functional differential systems (see, e.g., [2]).

Letx(t)be the solution of Eq. (1.1) with initial value att =T ≥t∗≥ t0. Then fort+θ ≥T we have the following equalities:

d

dtx_t(θ) =





 d

dθxt(θ), −h≤θ <0, B₀x_t+G(t,x_t), θ=0.

(3.5) Suppose that at the initial moment t = T the vector functionx_T(θ)belongs toW(T). Due to the positively invariance of W(t)we obtain that

x_t(θ) =_Φ(θ)u(t) +H(t,θ)u(t), t ≥T, u(t)∈_C^N. (3.6) We remark that formula (3.6) is, actually, decomposition (2.18). Consequently, by (2.24), func- tionu(t)satisfies the ordinary differential system

˙

u=^hD+_Ψ(0)G t,Φ(θ) +H(t,θ)ⁱu, t≥ T. (3.7) This system will be referred to as a projection of Eq. (1.1) on critical manifoldW(t)or, simply, as asystem on critical manifold. We substitute (3.6) in (3.5). This gives for t+θ ≥ T

Φ(θ) +H(t,θ)u˙(t) + ^∂H

∂t u=





 h∂Φ

∂θ + ^∂H

∂θ i

u, −h≤θ <0,

B₀Φ+B₀H+G t,Φ(θ) +H(t,θ)u, θ=0.

We then use (3.7) for ˙uand also (2.9), (2.11). We conclude that

Φ(θ)_Ψ(0)G t,Φ(θ) +H(t,θ)+H(t,θ) D+_Ψ(0)G t,Φ(θ) +H(t,θ)+^∂H

∂t

=







∂H

∂θ, −h≤ θ<_0,

B₀H+G t,Φ(θ) +H(t,θ), θ =0.

(3.8)

Therefore, if a critical manifold W(t) exists for t ≥ t∗ then for all (t,θ) such that t+θ ≥ t∗ matrix H(t,θ) should satisfy Eq. (3.8). Particularly, since the solution x(t) (xt∗ = ϕ ∈ W(t∗)) is absolutely continuous fort≥ t∗ it follows from (3.6) that matrixH(t,θ)is absolutely continuous intfort≥t∗−θand inθ forθ≥t∗−t.

We will try now to satisfy Eq. (3.8) up to terms ˆR(t,θ) such that kR^ˆ(t,·)k_Ch ∈ L₁[t0,∞). Namely, let

Hˆ(t,θ) =

∑

v_i(t)H_i(t,θ) +

∑

1≤i₁≤i₂≤n

v_i1(t)v_i2(t)H_i1i2(t,θ) +· · ·+

∑

1≤i1≤···≤ik≤n

v_i1(t)· · ·v_ik(t)H_i1...ik(t,θ). (3.9) Here the entries of (m×N)-matrices H_i1...il(t,θ) to be found are trigonometric polynomi- als in t and continuously differentiable in θ ∈ [−h, 0]. The natural number k is defined by property 3⁰ of functionsv₁(t), . . . ,v_n(t). Moreover, we assume that the columns of matrices H_i1...il(t,θ)belong to Q_Λ for allt∈ _R. We substitute (3.9) for H(t,θ)in (3.8) and collect terms corresponding to factorsv_i1(t)· · ·v_il(t)(l≤k). We obtain the following equations for matrices H_i1...il(t,θ):

F_i^PΛ

1...i_l(t,θ) +H_i1...il(t,θ)D+F_i^QΛ

1...i_l(t,θ) + ^∂Hi1...il

∂t =







∂H_i1...il

∂θ , −h≤θ <0, B₀H_i1...il+G_i1...il(t), θ =0.

(3.10) Here we also used formulas (1.2), (3.1). These formulas yield, in particular, that to solve Eq. (3.10) we need to compute matricesH_j1...js(t,θ)withs <l. Then F_i^PΛ

1...i_l(t,θ)andF_i^QΛ

1...i_l(t,θ) in Eq. (3.10) are some well-defined matrices that include matricesH_j1...js(t,θ) (s < l)defined in the earlier steps. By (1.2), (3.1), (3.2) and constraints imposed on matrices H_j1...js(t,θ) (s < l) we conclude that the entries of matrices F_i^PΛ

1...i_l(t,θ) _and F_i^QΛ

1...i_l(t,θ) are trigonometric polynomials int. Moreover, the columns of matrix F_i^PΛ

1...il(t,θ)belong to P_Λ and the columns of matrix F_i^QΛ

1...il(t,θ)belong to Q_Λ for all t ∈ R. Further, G_i1...il(t)is a certain matrix, whose entries are trigonometric polynomials:

G_i1...il(t) =

∑

j

G⁽_j^i1...il)e^iωjt, (3.11) where G⁽_j^i1...il) are constant (m×N)-matrices and ω_j are real numbers. It follows from (3.8) that matricesF_i^PΛ

1...il(t,θ)andF_i^QΛ

1...il(t,θ)have the following form:

F_i^PΛ

1...il(t,θ) =_Φ(θ)_Ψ(0)G_i1...il(t) =

∑

j

P_j^(i1...il)(θ)e^iωjt, (3.12) P_j^(i1...il)(θ) =_Φ(θ)_Ψ(0)G⁽_j^i1...il), (3.13) F_i^QΛ

1...il(t,θ) =

∑

j

Q⁽_j^i1...il)(θ)e^iωjt, (3.14) whereP_j^(i1...il)(θ)_andQ⁽_j^i1...il)(θ)are certain matrices continuously differentiable on−h≤θ≤0.

We seek solution of Eq. (3.10) in the form H_i1...il(t,θ) =

∑

j

β⁽_j^i1...il)(θ)e^iωjt, (3.15)

where β⁽_j^i1...il)(θ)are some continuously differentiable on −h≤ θ ≤ 0 matrices to be defined.

We substitute (3.11)–(3.15) in (3.10) and match the coefficients of the corresponding exponen- tials. Omitting for the sake of brevity the dependence of matrices on the indices set (i₁. . .i_l) and also on the indexj, we obtain the following functional boundary value problem for matrix β(θ) = β⁽_j^i1...il)(θ):





 dβ

dθ = β(θ)(D+iω_jI) +P(θ) +Q(θ), −h≤θ <0, β(0)(D+iω_jI) +P(0) +Q(0) =B0β+G,

(3.16)

where P(θ) = P_j^(i1...il)(θ), Q(θ) = Q⁽_j^i1...il)(θ)and G = G⁽_j^i1...il). Note that since solutionβ(θ) of (3.16) should be continuous on −h ≤ θ ≤ 0 we need the solve the first equation of this problem with initial condition β(₀)that is defined from the second equation. We also should take into account that the columns of β(θ) belong to Q_Λ. Consequently, due to (2.13), we should solve problem (3.16) together with additional condition

Ψ(ξ)_,β(θ) =_{0. (3.17)}

We assume that matrixD, whose spectrum isΛ, has Jordan canonical form

D=diag(D⁽¹⁾, . . . ,D^(l)), D⁽ⁱ⁾ =















λ⁽ⁱ⁾ 1 0 . . . 0 0 λ⁽ⁱ⁾ 1 . . . 0 . . . .

0 . . . 0 λ⁽ⁱ⁾ 1 0 . . . 0 λ⁽ⁱ⁾















, (3.18)

whereλ⁽ⁱ⁾ ∈_Λ,D⁽ⁱ⁾is(N_i×N_i)-matrix and N₁+· · ·+N_l = N. We write matricesβ(θ),P(θ), Q(θ)andGin the following form:

β(θ) =β⁽¹⁾(θ), . . . ,β^(l)(θ), P(θ) =P⁽¹⁾(θ), . . . ,P^(l)(θ), Q(θ) =Q⁽¹⁾(θ), . . . ,Q^(l)(θ), G=G⁽¹⁾, . . . ,G^(l)

. (3.19)

Hereβ⁽ⁱ⁾(θ),P⁽ⁱ⁾(θ),Q⁽ⁱ⁾(θ),G⁽ⁱ⁾ are(m×N_i)-matrices and square brackets[·, . . . ,·]stand for the matrix whose columns are vectors pointed inside the brackets and located in the natural order from left to right. Then we can rewrite (3.16), (3.17) in the form of l independent subsystems













 dβ dθ

(i)

= β⁽ⁱ⁾(θ)(D⁽ⁱ⁾+iω_jI) +P⁽ⁱ⁾(θ) +Q⁽ⁱ⁾(θ), −h ≤θ <0, β⁽ⁱ⁾(0)(D⁽ⁱ⁾+iω_jI) +P⁽ⁱ⁾(0) +Q⁽ⁱ⁾(0) =B₀β⁽ⁱ⁾+G⁽ⁱ⁾,

Ψ(ξ)_,β⁽ⁱ⁾(θ)=_0,

(3.20)

where I is the identity matrix of the order N_i andi=1, . . . ,l. Let β⁽ⁱ⁾(θ) =z⁽₁ⁱ⁾(θ)_{, . . . ,}z⁽_Nⁱ⁾

i(θ)_, P⁽ⁱ⁾(θ) =p₁⁽ⁱ⁾(θ)_{, . . . ,}p⁽_Nⁱ⁾

i(θ)_, Q⁽ⁱ⁾(θ) =q₁⁽ⁱ⁾(θ), . . . ,q⁽_Nⁱ⁾

i(θ), G⁽ⁱ⁾ =g⁽₁ⁱ⁾, . . . ,g⁽_Nⁱ⁾

i

,

(3.21)

wherez⁽_sⁱ⁾(θ), p⁽_sⁱ⁾(θ),q⁽_sⁱ⁾(θ), g⁽_sⁱ⁾(s=1, . . . ,N_i) arem-dimensional column vectors. Finally, let Z˜⁽ⁱ⁾(θ) =col z⁽₁ⁱ⁾(θ), . . . ,z⁽_Nⁱ⁾

i(θ), P˜⁽ⁱ⁾(θ) =col p⁽₁ⁱ⁾(θ), . . . ,p⁽_Nⁱ⁾

i(θ), Q˜⁽ⁱ⁾(θ) =_col q⁽₁ⁱ⁾(θ)_{, . . . ,}q⁽_Nⁱ⁾

i(θ)_, G^˜(i)=_col g⁽₁ⁱ⁾, . . . ,g⁽_Nⁱ⁾

i

, B₀Z˜⁽ⁱ⁾=col B₀z⁽₁ⁱ⁾, . . . ,B₀z⁽_Nⁱ⁾

i

(3.22)

denote mN_i-dimensional column vectors composed from the vectors pointed in col(·, . . . ,·) located from the top downward in the natural order. Using these notations we rewrite (3.20) as follows:













 dZ˜

dθ

(i)

=_A⁽ⁱ⁾Z^˜(i)(θ) +P^˜(i)(θ) +Q^˜(i)(θ)_, −h≤θ <_0, A⁽ⁱ⁾Z˜⁽ⁱ⁾(0) +P^˜(i)(0) +Q^˜(i)(0) =B₀Z˜⁽ⁱ⁾+G⁽ⁱ⁾,

Ψ(ξ)_,z⁽_sⁱ⁾(θ)=_0, s =_{1, . . . ,}N_i.

(3.23)

Here the (mN_i×mN_i)-matrixA⁽ⁱ⁾is defined by formula

A⁽ⁱ⁾ =

















µ⁽ⁱ⁾I 0 . . . 0 I µ⁽ⁱ⁾I 0 . . . 0 0 . .. ... ... ... ... . .. I µ⁽ⁱ⁾I 0 0 . . . 0 I µ⁽ⁱ⁾I

















, µ⁽ⁱ⁾=λ⁽ⁱ⁾+iω_j, (3.24)

where I is the identity matrix of the orderm.

Solving the first equation in (3.23), we obtain Z˜⁽ⁱ⁾(θ) =e^A(i)θZ˜⁽ⁱ⁾(0) +

Z _θ

0 e^A(i)(θ−s) P˜⁽ⁱ⁾(s) +Q^˜(i)(s)ds, −h≤θ <0. (3.25) We substitute this expression into the second equation in (3.23). This results in the following linear algebraic equation for vector ˜Z⁽ⁱ⁾(0):

A⁽ⁱ⁾−B₀e^A(i)θZ˜⁽ⁱ⁾(0) =B_{0 Z θ}

0 e^A(i)(θ−s)

P˜⁽ⁱ⁾(s) +Q^˜(i)(s)ds

+G⁽ⁱ⁾−P^˜(i)(0)−Q^˜(i)(0).

Here we apply the functionalB₀to the columns of (mN_i×mN_i)-matrixe^A(i)θ in the same way as in (3.22) taking into account that

e^A(i)θ =



















I 0 . . . 0

θI I 0 . . . 0

... . .. . .. ... ...

θ^Ni−2

(Ni−2)!I . . . θI I 0

θ^Ni−1

(Ni−1)!I _(N^θNi−2

i−2)!I . . . θI I

















 e^µ(i)θ.

We recall now formulas (2.2), (2.3), notations (3.22) and the third equation in (3.23). We obtain the following algebraic problems for vectorsz⁽₁ⁱ⁾(0), . . . ,z⁽_Nⁱ⁾

i(0)(i=1, . . . ,l).

P1:











∆(µ⁽ⁱ⁾)z⁽₁ⁱ⁾(0) =B_{0 Z θ}

0 e^{µ(i)(θ−s)} p⁽₁ⁱ⁾(s) +q⁽₁ⁱ⁾(s)ds

+g⁽₁ⁱ⁾−p⁽₁ⁱ⁾(0)−q⁽₁ⁱ⁾(0), Ψ(ξ),e^µ(i)θI

z⁽₁ⁱ⁾(0) =−

Ψ(ξ), Z _θ

0 e^{µ(i)(θ−s)} p⁽₁ⁱ⁾(s) +q⁽₁ⁱ⁾(s)ds

;

(3.26)

P₂:







































∆⁰(µ⁽ⁱ⁾)z⁽₁ⁱ⁾(0) +_∆(µ⁽ⁱ⁾)z⁽₂ⁱ⁾(0)

= B₀ Z _θ

0

(θ−s)e^{µ(i)(θ−s)} p⁽₁ⁱ⁾(s) +q⁽₁ⁱ⁾(s)+e^{µ(i)(θ−s)} p⁽₂ⁱ⁾(s) +q⁽₂ⁱ⁾(s)ds

+g⁽₂ⁱ⁾−p⁽₂ⁱ⁾(0)−q⁽₂ⁱ⁾(0), Ψ(ξ),θe^µ(i)θI

z⁽₁ⁱ⁾(0) + _Ψ(ξ),e^µ(i)θI z⁽₂ⁱ⁾(0)

= −

Ψ(ξ), Z _θ

0

(θ−s)e^{µ(i)(θ−s)} p⁽₁ⁱ⁾(s) +q₁⁽ⁱ⁾(s)ds

−

Ψ(ξ), Z _θ

0 e^{µa(i)(θ−s)} p⁽₂ⁱ⁾(s) +q⁽₂ⁱ⁾(s)ds

;

(3.27)

...

P_Ni:























































∆^(Ni−1)(µ⁽ⁱ⁾)

(N_i−1)! z⁽₁ⁱ⁾(₀) +· · ·+_∆⁰(µ⁽ⁱ⁾)_z⁽_Nⁱ⁾

i−1(₀) +_∆(µ⁽ⁱ⁾)_z⁽_Nⁱ⁾

i(₀)

= B_{0 Z θ}

0

(θ−s)^Ni−1

(N_i−1)! e^{µ(i)(θ−s)} p⁽₁ⁱ⁾(s) +q⁽₁ⁱ⁾(s) +· · ·+e^{µ(i)(θ−s)} p⁽_Nⁱ⁾

i(s) +q⁽_Nⁱ⁾

i(s)ds

+g⁽_Nⁱ⁾

i −p⁽_Nⁱ⁾

i(0)−q⁽_Nⁱ⁾

i(0), Ψ(ξ), θ^Ni−1

(N_i−1)!e^µ(i)θI

z⁽₁ⁱ⁾(0) +· · ·+ _Ψ(ξ),θe^µ(i)θI z⁽_Nⁱ⁾

i−1(0) + _Ψ(ξ),e^µ(i)θI

z⁽_Nⁱ⁾_i(0)

= −_Ψ(ξ), Z _θ

0

(θ−s)^Ni−1

(N_i−1)! e^{µ(i)(θ−s)} p⁽₁ⁱ⁾(s) +q⁽₁ⁱ⁾(s)ds

− · · · −_Ψ(ξ), Z _θ

0 e^{µ(i)(θ−s)} p⁽_Nⁱ⁾

i(s) +q⁽_Nⁱ⁾

i(s)ds .

(3.28)

We can now formulate the main result of this section.

Theorem 3.2. System(3.23)has a unique solutionZ˜⁽ⁱ⁾(θ)(i=1, . . . ,l) that is continuously differen- tiable on −h ≤ θ ≤ 0. This solution is defined by formula(3.25), where the components of the initial vector Z˜⁽ⁱ⁾(0)are unique solutions of problemsP₁, . . . ,P_Ni.

We note that, since ˜P⁽ⁱ⁾(θ) and ˜Q⁽ⁱ⁾(θ)are smooth on −h ≤ θ ≤ 0, the continuous differ- entiability of ˜Z⁽ⁱ⁾(θ)follows immediately from (3.25). Moreover, solution ˜Z⁽ⁱ⁾(θ)is infinitely differentiable on −h ≤ θ ≤ 0 because the entries of matrix Φ(θ)(and, therefore, the compo- nents of vectors ˜P⁽ⁱ⁾(θ), ˜Q⁽ⁱ⁾(θ)as well) are infinitely differentiable. The proof of Theorem3.2 is given in the Appendix.