Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Abstract

Nonhomogeneous system of linear differential equations of second order with multiple different delays and pairwise permutable matrices defining the linear parts is considered. Solution of corresponding initial value problem is represented using matrix polynomials.

1. Introduction

Motivated by delayed exponential representing a solution of a system of differential or difference equations with one or multiple fixed or variable delays [1–6], which has many applications in theory of controllability, asymptotic properties, boundary-value problems, and so forth [3–5, 7–15], we extended representation of a solution of a system of differential equations of second order with delay [1] 𝑥̈(𝑡)=−𝐵²𝑥(𝑡−𝜏) (1) to the case of two delays 𝑥̈(𝑡)=−𝐵²₁𝑥𝑡−𝜏₁−𝐵²₂𝑥𝑡−𝜏₂, (2) where the linear parts were given by permutable matrices [16]. Equations (1), (2), and the below-stated (11) with 𝑓≡0 are generalizations of the scalar equation 𝑥̈(𝑡)=−𝑏²𝑥(𝑡) (3) representing linear oscillator, to 𝑁-dimensional space with one or multiple fixed delays. Clearly, each solution of the latter equation is oscillating whenever 0≠𝑏∈ℝ. Analogically, (1) with 𝑥∈ℝ^𝑁 can have at least one oscillating solution whenever 𝑁 is odd. Indeed, if 𝐵 is 𝑁×𝑁 matrix, 𝑁≥3 is odd, and 𝐵 has a simple real nonzero eigenvalue 𝜆, then there exists a regular matrix 𝑆 such that 𝑆⁻¹𝐵𝑆=𝐽=𝐽^0𝜆0̃ where 𝐽̃ is (𝑁−1)×(𝑁−1) matrix. On letting 𝑥=𝑆𝑦, one gets 𝑦̈=−𝐽²𝑦(𝑡−𝜏) (4) or rewrites as the system 𝑦̈₁=−𝜆²𝑦₁(),𝑦̈𝑡−𝜏₂=−𝐽²̃𝑦₂(𝑡−𝜏), (5) where 𝑦=(𝑦₁,𝑦₂)∈ℝ×ℝ^𝑁−1. Note that the first column 𝑣 of 𝑆 is the eigenvector of 𝐵 corresponding to 𝜆. Clearly, if solution 𝑦₁ of (5) is oscillating, then solution 𝑦 of (4) is oscillating in the first coordinate whenever its initial condition satisfies {𝑦(𝑡)∣𝑡∈[−𝜏,0]}⊂ℝ×{0}^𝑁−1. Consequently, solution 𝑥 of (1) is oscillating in span{𝑣} whenever {𝑥(𝑡)∣𝑡∈[−𝜏,0]}⊂span{𝑣}. Taking 𝑦₁(𝑡)=𝑒^𝜇𝑡, one obtains characteristic equation 𝜇²=−𝜆²𝑒^−𝜇𝜏 of (5), which has solutions 𝜇_1,2=𝛼±𝚤𝛽∈ℂ with 𝛽≠0. Thus, 𝑦₁ is oscillating.

On the other hand, there can exist a nonoscillating solution of the system (1) whenever 𝑥∈ℝ^𝑁 and 𝑁 is even. For instance, if 𝑁=2 and 𝐵=⁰¹₋₁₀, then (1) has the form 𝑥̈(𝑡)=𝑥(𝑡−𝜏) (6) with 𝑥∈ℝ², which, obviously, does not have an oscillating solution satisfying nonoscillating initial condition. Similarly, it can be shown that system with odd dimension can possess a nonoscillating solution satisfying an appropriate initial condition.

For simplicity, we call the generalizations (1), (2), and (11) with 𝑓≡0, of scalar equation (3), oscillating although their solutions do not always have to be oscillating. Nevertheless, at the end of this paper, in Corollary 8 we state the representation of a solution of more general system (86) without squares of matrices.

We note that the delayed matrix exponential from [1–5] as well as the representation of a solution of second-order differential equations derived in [1, 16] and in this paper can lead to new results in nonlinear boundary value problems for impulsive functional differential equations considered in [17] or stochastic delayed differential equations from [18].

So, in the present paper, we extend our result from [16] to three and more delays by the assumption of pairwise permutable matrices defining linear parts. By such an assumption, we are able to construct matrix functions solving homogeneous system of differential equations of second order with any number of fixed delays, and, consequently, we use these functions to represent a solution of the corresponding nonhomogeneous initial value problem. As will be shown in the next sections, extending from two to more delays brings many technical difficulties, for example, the use of multinomial coefficients. Naturally, the results of the present paper hold with one or two different delays as well. However, these cases can by studied in a simpler way, which was already done in [1, 16]. Thus, we focus our attention on the case of three and more different delays.

First, we recall our result from [16].

Theorem 1. Let 𝜏₁,𝜏₂>0, 𝜏∶=max{𝜏₁,𝜏₂}, and 𝜑∈𝐶¹([−𝜏,0],ℝ^𝑁). Let 𝐵₁,𝐵₂ be 𝑁×𝑁 permutable matrices; that is, 𝐵₁𝐵₂=𝐵₂𝐵₁, and let 𝑓∶[0,∞)→ℝ^𝑁 be a given function. Solution 𝑥(𝑡) of 𝑥̈(𝑡)=−𝐵²₁𝑥𝑡−𝜏₁−𝐵²₂𝑥𝑡−𝜏₂+𝑓(𝑡) (7) satisfying initial condition 𝑥(𝑡)(𝑡),𝑥̇(𝑡)=𝜑̇(𝑡),=𝜑−𝜏≤𝑡≤0 (8) has the form 𝑥(𝑡)=𝜑(𝑡)𝒳(𝑡)𝜑(0)(𝑡)𝜑̇(0),−𝜏≤𝑡<0,+𝒴−𝐵²₁⁰_−𝜏1𝒴𝑡−𝜏₁−𝑠𝜑(𝑠)𝑑𝑠−𝐵²₂⁰_−𝜏2𝒴𝑡−𝜏₂𝜑(𝑠)+−𝑠𝑑𝑠_𝑡0𝒴(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,0≤𝑡, (9) where 𝒳(𝑡)=𝒳^{𝐵21,𝐵22}_𝜏1,𝜏2(𝑡)∶=_{𝑖,𝑗≥0𝑖𝜏1+𝑗𝜏2≤𝑡}(−1)^𝑖+𝑗×𝑖𝐵𝑖+𝑗_12𝑖𝐵₂^2𝑗𝑡−𝑖𝜏₁−𝑗𝜏₂^2(𝑖+𝑗)2!,𝒴(𝑡)𝑖+𝑗=𝒴^{𝐵21,𝐵22}_𝜏1,𝜏2(𝑡)∶=_{𝑖,𝑗≥0𝑖𝜏1+𝑗𝜏2≤𝑡}(−1)^𝑖+𝑗×𝑖𝐵𝑖+𝑗_12𝑖𝐵₂^2𝑗𝑡−𝑖𝜏₁−𝑗𝜏₂^{2(𝑖+𝑗)+1}2!.𝑖+𝑗+1 (10)

We will denote Θ and 𝐸 the 𝑁×𝑁 zero and identity matrix, respectively.

2. Systems with Multiple Delays

In this section, we derive the representation of a solution of 𝑥̈(𝑡)=−𝐵²₁𝑥𝑡−𝜏₁−⋯−𝐵²_𝑛𝑥𝑡−𝜏_𝑛+𝑓(𝑡) (11) satisfying the initial condition (8), where 𝑛≥3, 𝜏₁,…,𝜏_𝑛>0, 𝜏∶=max_{𝑖=1,…,𝑛}𝜏_𝑖, 𝐵₁,…,𝐵_𝑛 are 𝑁×𝑁 pairwise permutable matrices; that is, 𝐵_𝑖𝐵_𝑗=𝐵_𝑗𝐵_𝑖 for each 𝑖,𝑗∈{1,…,𝑛}, 𝜑∈𝐶¹([−𝜏,0],ℝ^𝑁), and 𝑓∶[0,∞)→ℝ^𝑁 are given functions. The solution 𝑥(𝑡) will be represented using matrix functions analogical to (10) and will be stated in Section 3. We note that the same problems with 𝑛=1,2 were studied in [1, 16].

From now on, we assume the property of empty sum and empty product; that is, _𝑖∈∅𝑓(𝑖)=0,_𝑖∈∅𝐹(𝑖)=Θ,_𝑖∈∅𝑓(𝑖)=1,_𝑖∈∅𝐹(𝑖)=𝐸 (12) for any function 𝑓 and matrix function 𝐹, whether they are defined or not for indicated argument.

We recall that (𝑗₁,…,𝑗_𝑛)! is a multinomial coefficient [19] given by 𝑗₁,…,𝑗_𝑛𝑗!=¹+⋯+𝑗_𝑛!𝑗₁!⋯𝑗_𝑛!. (13) Note that if 𝑛=2, then (𝑗₁,𝑗₂)=^𝑗1+𝑗2_𝑗1 and (20) coincides with (10).

We will need a property of multinomial coefficients described in the next lemma.

Lemma 2. Let 𝑛≥2 be fixed. Then 𝑖₁,𝑖₂,…,𝑖_𝑛!=𝑖₁−1,𝑖₂,…,𝑖_𝑛!+𝑖₁,𝑖₂−1,𝑖₃,…,𝑖_𝑛!+𝑖₁,…,𝑖_𝑛−1,𝑖_𝑛−1! (14) for any 𝑖₁,…,𝑖_𝑛≥1.

Proof. If 𝑛=2, then the statement follows from the property of binomial coefficients: 𝑖₁,𝑖₂𝑖!=¹+𝑖₂𝑖₁=𝑖₁+𝑖₂𝑖−1₁+𝑖−1¹+𝑖₂𝑖−1₁=𝑖₁−1,𝑖₂!+𝑖₁,𝑖₂−1!. (15) Let the statement be true for 𝑛−1. Next, we use the property of multinomial coefficient 𝑖₁,𝑖₂,𝑖₃,…,𝑖_𝑛!=𝑖₁+𝑖₂,𝑖₃,…,𝑖_𝑛!𝑖₁,𝑖₂! (16) with inductive hypothesis to derive 𝑖₁,𝑖₂,𝑖₃,…,𝑖_𝑛!=𝑖₁+𝑖₂−1,𝑖₃,…,𝑖_𝑛!+𝑖₁+𝑖₂,𝑖₃−1,…,𝑖_𝑛!+⋯+𝑖₁+𝑖₂,𝑖₃,…,𝑖_𝑛−1!𝑖₁,𝑖₂!. (17) Clearly, from (16), we get 𝑖₁+𝑖₂,𝑖₃−1,…,𝑖_𝑛!𝑖₁,𝑖₂!=𝑖₁,𝑖₂,𝑖₃−1,…,𝑖_𝑛⋮𝑖!,₁+𝑖₂,𝑖₃,…,𝑖_𝑛−1!𝑖₁,𝑖₂!=𝑖₁,𝑖₂,𝑖₃,…,𝑖_𝑛−1!. (18) Applying the case 𝑛=2 (property of binomial coefficient) and (16), we get 𝑖₁+𝑖₂−1,𝑖₃,…,𝑖_𝑛!𝑖₁,𝑖₂!=𝑖₁+𝑖₂−1,𝑖₃,…,𝑖_𝑛!𝑖₁−1,𝑖₂!+𝑖₁,𝑖₂!=𝑖−1₁−1,𝑖₂,𝑖₃,…,𝑖_𝑛!+𝑖₁,𝑖₂−1,𝑖₃,…,𝑖_𝑛!. (19) Putting (18) and (19) in (17), we obtain that the statement holds for 𝑛 and the proof is complete.

In further work, we write ({𝑗∣𝑗∈𝑀})! for the multinomial coefficient of elements of the finite set 𝑀, and (𝑖,{𝑗∣𝑗∈𝑀})! for the multinomial coefficient of 𝑖 and elements of the finite set 𝑀; for example, if 𝑀={1,2}, then (𝑎,{𝑗∣𝑗∈𝑀})!=(𝑎,1,2)!. For the completeness, we define ({𝑗∣𝑗∈∅})!∶=1.

Define the functions 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛},𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}∶ℝ→𝐿(ℝ^𝑁) as 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)∶=_{𝑗1,…,𝑗𝑛𝑗≥01𝜏1+⋯+𝑗𝑛𝜏𝑛≤𝑡}(−1)^{𝑗1+⋯+𝑗𝑛}𝑗₁,…,𝑗_𝑛!×^𝑛_𝑖=1𝐵^2𝑗𝑖_𝑖𝑡−𝑗₁𝜏₁−⋯−𝑗_𝑛𝜏_𝑛^{2(𝑗1+⋯+𝑗𝑛)}2𝑗₁+⋯+𝑗_𝑛!,𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)∶=_{𝑗1,…,𝑗𝑛𝑗≥01𝜏1+⋯+𝑗𝑛𝜏𝑛≤𝑡}(−1)^{𝑗1+⋯+𝑗𝑛}𝑗₁,…,𝑗_𝑛!×^𝑛_𝑖=1𝐵^2𝑗𝑖_𝑖𝑡−𝑗₁𝜏₁−⋯−𝑗_𝑛𝜏_𝑛^{2(𝑗1+⋯+𝑗𝑛)+1}2𝑗₁+⋯+𝑗_𝑛+1! (20) for any 𝑡∈ℝ.

We will need functions 𝒳^𝐵2_𝜏,𝒴^𝐵2_𝜏∶ℝ→𝐿(ℝ^𝑁) for 𝜏>0 and 𝑁×𝑁 complex matrix 𝐵 (cf. [16]) defined as 𝒳^𝐵2_𝜏(𝑡)∶=_{𝑖≥0𝑖𝜏≤𝑡}(−1)^𝑖𝐵^2𝑖(𝑡−𝑖𝜏)^2𝑖()!,𝒴2𝑖_𝐵2𝜏(𝑡)∶=_{𝑖≥0𝑖𝜏≤𝑡}(−1)^𝑖𝐵^2𝑖(𝑡−𝑖𝜏)^2𝑖+1(2𝑖+1)! (21) with the properties 𝒳̇^𝐵2_𝜏(𝑡)=−𝐵²𝒴^𝐵2_𝜏(𝑡−𝜏),𝒳̈^𝐵2_𝜏(𝑡)=−𝐵²𝒳^𝐵2_𝜏(),𝒴̇𝑡−𝜏_𝐵2𝜏(𝑡)=𝒳^𝐵2_𝜏(𝑡),𝒴̈^𝐵2_𝜏(𝑡)=−𝐵²𝒴^𝐵2_𝜏(𝑡−𝜏) (22) for any 𝑡∈ℝ, considering the one-sided derivatives at −𝜏, 0.

Some of properties of functions 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛} and 𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛} are concluded in Lemma 4, but to prove it we will need the next lemma.

Lemma 3. Let 𝑛≥1 and 𝜏₁,…,𝜏_𝑛>0. Let 𝐵₁,…,𝐵_𝑛 be 𝑁×𝑁 pairwise permutable matrices, that is, 𝐵_𝑖𝐵_𝑗=𝐵_𝑗𝐵_𝑖 for each 𝑖,𝑗∈{1,…,𝑛}. Then for any 𝑡∈ℝ, 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=_{{}𝑀⊂1,…,𝑛}𝑆_𝑀(𝑡),𝒴^{𝐵21,…𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=_{{}𝑀⊂1,…,𝑛}𝑆_𝑀(𝑡), (23) where the sums are taken over all subsets of {1,…,𝑛} including the trivial ones, and 𝑆_𝑀(𝑡)∶=_{𝑗𝑖∑≥1,𝑖∈𝑀𝑖∈𝑀𝑗𝑖𝜏𝑖≤𝑡}(−1)^{∑𝑖∈𝑀𝑗𝑖}𝑗_𝑖∣𝑖∈𝑀!×_𝑖∈𝑀𝐵^2𝑗𝑖_𝑖𝑡−∑_𝑖∈𝑀𝑗_𝑖𝜏_𝑖^{2∑𝑖∈𝑀𝑗𝑖}2∑_𝑖∈𝑀𝑗_𝑖!, (24)𝑆_𝑀(𝑡)∶=_{𝑗𝑖∑≥1,𝑖∈𝑀𝑖∈𝑀𝑗𝑖𝜏𝑖≤𝑡}(−1)^{∑𝑖∈𝑀𝑗𝑖}𝑗_𝑖∣𝑖∈𝑀!×_𝑖∈𝑀𝐵^2𝑗𝑖_𝑖𝑡−∑_𝑖∈𝑀𝑗_𝑖𝜏_𝑖^{2∑𝑖∈𝑀𝑗𝑖+1}2∑_𝑖∈𝑀𝑗_𝑖!.+1 (25)

Proof. Denote ℕ₀, ℕ the set of all nonnegative, positive integers, respectively; that is, ℕ₀={0}∪ℕ. Thus, we have the trivial identity ℕ₀×⋯×ℕ_0𝑛={0}×ℕ₀×⋯×ℕ_0𝑛−1∪ℕ×ℕ₀×⋯×ℕ_0𝑛−1=⋯=_{𝑀1,…,𝑀𝑛∈0}}{{,ℕ}𝑀₁×⋯×𝑀_𝑛. (26)
Analogically, for any 𝑡∈ℝ each 𝑛-tuple 𝑗₁,…,𝑗_𝑛≥0 such that ∑^𝑛_𝑖=1𝑗_𝑖𝜏_𝑖≤𝑡 can be divided in two distinct sets of 𝑖-s so that 𝑗_𝑖≥1 if 𝑖∈𝑀⊂{1,…,𝑛} and 𝑗_𝑖=0 if 𝑖∈{1,…,𝑛}⧵𝑀. That is, 𝑀 denotes the set of all indices 𝑖 such that 𝑗_𝑖=0. Moreover, ∑^𝑛_𝑖=1𝑗_𝑖𝜏_𝑖=∑_𝑖∈𝑀𝑗_𝑖𝜏_𝑖. Accordingly, we can write 𝑗₁,…,𝑗_𝑛∈ℕ^𝑛₀∣^𝑛_𝑖=1𝑗_𝑖𝜏_𝑖=≤𝑡_{{}𝑀⊂1,…,𝑛}𝑗₁,…,𝑗_𝑛∈ℕ^𝑛₀∣𝑗_𝑖=0∀𝑖∉𝑀,_𝑖∈𝑀𝑗_𝑖𝜏_𝑖≤𝑡, (27) where the union is taken over all subsets of {1,…,𝑛} including the trivial ones. So, in the view of definition (20), the statement for 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛} follows.
Statement for 𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛} can be proved in a similar way.

Lemma 4. Let 𝑛≥3 and 𝜏₁,…,𝜏_𝑛>0. Let 𝐵₁,…,𝐵_𝑛 be 𝑁×𝑁 pairwise permutable matrices; that is, 𝐵_𝑖𝐵_𝑗=𝐵_𝑗𝐵_𝑖 for each 𝑖,𝑗∈{1,…,𝑛}. Then the following holds for any 𝑡∈ℝ:(1)if 𝐵_𝑖=Θ for some 𝑖∈{1,…,𝑛}, then 𝒳^{𝐵21,…,𝐵2𝑖−1,𝐵2𝑖,𝐵2𝑖+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,…,𝜏𝑛}(𝑡)=𝒳^{𝐵21,…,𝐵2𝑖−1,𝐵2𝑖+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑖−1,𝜏𝑖+1,…,𝜏𝑛}(𝑡), (28)(2)if 𝜏_𝑖=𝜏_𝑘 for 𝑖<𝑘, 𝑖,𝑘∈{1,…,𝑛}, then 𝒳^{𝐵21,…,𝐵2𝑖−1,𝐵2𝑖,𝐵2𝑖+1,…,𝐵2𝑘−1,𝐵2𝑘,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,…,𝜏𝑘−1,𝜏𝑘,𝜏𝑘+1,…,𝜏𝑛}(𝑡)=𝒳^{𝐵21,…,𝐵2𝑖−1,𝐵2𝑖+𝐵2𝑘,𝐵2𝑖+1,…,𝐵2𝑘−1,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,…,𝜏𝑘−1,𝜏𝑘+1,…,𝜏𝑛}(𝑡), (29)(3)for any bijective mapping 𝜎∶{1,…,𝑛}→{1,…,𝑛} we get 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒳^{𝐵2𝜎(1),…,𝐵2𝜎(𝑛)}_{𝜏𝜎(1),…,𝜏𝜎(𝑛)}(𝑡), (30)(4)taking the one-sided derivatives at 0,𝜏₁,…,𝜏_𝑛, then 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=−𝐵²₁𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏₁−⋯−𝐵²_𝑛𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏_𝑛, (31)(5)considering the one-sided derivatives at 0 (they both equal Θ), then 𝒴̇^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡). (32)Statements (1)–(4) hold with 𝒴 instead of 𝒳.

Proof. Statement (1) follows easily from definition of 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}, because Θ^2𝑖=𝐸 if 𝑖=0 and Θ^2𝑖=Θ whenever 𝑖>0. Next, if 𝜏_𝑖=𝜏_𝑘, then _{𝑗1,…,𝑗𝑛𝑗≥01𝜏1+⋯+𝑗𝑛𝜏𝑛≤𝑡}𝐹𝑗₁,…,𝑗_𝑛=_{𝑗1,…,𝑗𝑖−1,𝑙,𝑗𝑖+1,…,𝑗𝑘−1,𝑗𝑘+1,…,𝑗𝑛𝑗≥01𝜏1+⋯+𝑗𝑖−1𝜏𝑖−1+𝑙𝜏𝑖+𝑗𝑖+1𝜏𝑖+1+⋯+𝑗𝑘−1𝜏𝑘−1+𝑗𝑘+1𝜏𝑘+1+⋯+𝑗𝑛𝜏𝑛≤𝑡𝑗𝑖,𝑗𝑘𝑗≥0𝑖+𝑗𝑘=𝑙}𝐹𝑗₁,…,𝑗_𝑛 (33) for any matrix function 𝐹. Thus, using the property of multinomial coefficient (see (16)) 𝑗₁,…,𝑗_𝑛!=𝑗₁,…,𝑗_𝑖−1,𝑗_𝑖+𝑗_𝑘,𝑗_𝑖+1,…,𝑗_𝑘−1,𝑗_𝑘+1,…,𝑗_𝑛!𝑗_𝑖,𝑗_𝑘!, (34) for (2), we obtain 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=_{𝑗1,…,𝑗𝑖−1,𝑙,𝑗𝑖+1,…,𝑗𝑘−1,𝑗𝑘+1,…,𝑗𝑛𝑗≥01𝜏1+⋯+𝑗𝑖−1𝜏𝑖−1+𝑙𝜏𝑖+𝑗𝑖+1𝜏𝑖+1+⋯+𝑗𝑘−1𝜏𝑘−1+𝑗𝑘+1𝜏𝑘+1+⋯+𝑗𝑛𝜏𝑛≤𝑡}(−1)^{∑𝑗𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘𝑠+𝑙}×𝑗₁,…,𝑗_𝑖−1,𝑙,𝑗_𝑖+1,…,𝑗_𝑘−1,𝑗_𝑘+1,…,𝑗_𝑛!×_{𝑗𝑖,𝑗𝑘𝑗≥0𝑖+𝑗𝑘=𝑙}𝑗_𝑖,𝑗_𝑘!𝐵^2𝑗𝑖_𝑖𝐵^2𝑗𝑘_𝑘×_{{}𝑠∈1,…,𝑛𝑠≠𝑖,𝑘}𝐵^2𝑗𝑠_𝑠𝑡−∑_{𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘}𝑗_𝑠𝜏_𝑠−𝑙𝜏_𝑖^{𝑗2(∑𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘𝑠+𝑙)}2∑_{𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘}𝑗_𝑠!=+𝑙_{𝑗1,…,𝑗𝑖−1,𝑙,𝑗𝑖+1,…,𝑗𝑘−1,𝑗𝑘+1,…,𝑗𝑛𝑗≥01𝜏1+⋯+𝑗𝑖−1𝜏𝑖−1+𝑙𝜏𝑖+𝑗𝑖+1𝜏𝑖+1+⋯+𝑗𝑘−1𝜏𝑘−1+𝑗𝑘+1𝜏𝑘+1+⋯+𝑗𝑛𝜏𝑛≤𝑡}(−1)^{∑𝑗𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘𝑠+𝑙}×𝑗₁,…,𝑗_𝑖−1,𝑙,𝑗_𝑖+1,…,𝑗_𝑘−1,𝑗_𝑘+1,…,𝑗_𝑛!𝐵²_𝑖+𝐵²_𝑘^𝑙×_{{}𝑠∈1,…,𝑛𝑠≠𝑖,𝑘}𝐵^2𝑗𝑠_𝑠𝑡−∑_{𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘}𝑗_𝑠𝜏_𝑠−𝑙𝜏_𝑖^{𝑗2(∑𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘𝑠+𝑙)}2∑_{𝑠∈{1,…,𝑛}𝑠≠𝑖,𝑘}𝑗_𝑠+𝑙!=𝒳^{𝐵21,…,𝐵2𝑖−1,𝐵2𝑖+𝐵2𝑘,𝐵2𝑖+1,…,𝐵2𝑘−1,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,…,𝜏𝑘−1,𝜏𝑘+1,…,𝜏𝑛}(𝑡). (35) Property (3) is trivial.
Now, we prove the statement (4). If 𝜏∶=𝜏₁=⋯=𝜏_𝑛, then 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒳̈^{𝐵21+⋯+𝐵2𝑛}_𝜏(𝑡)=−𝐵²₁+⋯+𝐵²_𝑛𝒳^{𝐵21+⋯+𝐵2𝑛}_𝜏(𝑡−𝜏)=−𝐵²₁𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏₁−⋯−𝐵²_𝑛𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏_𝑛 (36) by (2) and from the property of 𝒳^{𝐵21+⋯+𝐵2𝑛}_𝜏(𝑡) (see (22)).
Hence, without any loss of generality, we assume that 𝜏_𝑖≠𝜏_𝑗 for each 𝑖≠𝑗, 𝑖,𝑗∈{1,…,𝑛} (in the other case, we collect matrices as stated in (2)). Note the case 𝑛=2 was proved in [16, Lemma 2.3.] Now, assume that 𝒳^{𝐵21,…,𝐵2𝑛−1}_{𝜏1,…,𝜏𝑛−1}(𝑡) solves 𝑥̈(𝑡)=−𝐵²₁𝑥𝑡−𝜏₁−⋯−𝐵²_𝑛−1𝑥𝑡−𝜏_𝑛−1, (37) that is, that the statement is fulfilled for 𝑛−1 different delays.
Let 𝜏_𝑘∶=max_{𝑖=1,…,𝑛}𝜏_𝑖. If 𝑡<𝜏_𝑘, then 𝑡−𝜏_𝑘<0, that is, 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏_𝑘=Θ, (38) and from definition (20) it holds 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒳^{𝐵21,…,𝐵2𝑘−1,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑘−1,𝜏𝑘+1,…,𝜏𝑛}(𝑡) (39) for such 𝑡. Consequently, 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒳̈^{𝐵21,…,𝐵2𝑘−1,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑘−1,𝜏𝑘+1,…,𝜏𝑛}(𝑡)=−_{𝑖=1,…,𝑛𝑖≠𝑘}𝐵²_𝑖𝒳^{𝐵21,…,𝐵2𝑘−1,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑘−1,𝜏𝑘+1,…,𝜏𝑛}𝑡−𝜏_𝑖=−^𝑛_𝑖=1𝐵²_𝑖𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏_𝑖 (40) by the inductive hypothesis.
Now, let 𝑡≥max_{𝑖=1,…,𝑛}𝜏_𝑖. Applying Lemma 3, we get 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=_{{}𝑀⊂1,…,𝑛}𝑆_𝑀(𝑡) (41) with 𝑆_𝑀(𝑡) given by (24) and the sum taken over all subsets of {1,…,𝑛} including the trivial ones. Note that 𝑆_∅(𝑡)=_{𝑗𝑖≥1,𝑖∈∅0≤𝑡}(−1)⁰𝑗_𝑖()∣𝑖∈∅!𝐸𝑡−0⁰=0!_0≤𝑡𝐸=𝐸𝜒_[0,∞)(𝑡) (42) with a characteristic function 𝜒𝑀 of a set 𝑀 given by 𝜒𝑀(𝑡)=1,𝑡∈𝑀,0,𝑡∉𝑀. (43) Since each 𝑀⊂{1,…,𝑛} is a finite set, Lemma 2 yields 𝑗_𝑖∣𝑖∈𝑀!=_𝑖∈𝑀𝑗_𝑖−1,𝑗_𝑘∣𝑘∈𝑀⧵{𝑖}!. (44) We apply this identity to derive a formula for the second derivative of 𝑆_𝑀 for any ∅≠𝑀⊂{1,…,𝑛}: 𝑆_𝑀^′′(𝑡)=_{𝑗𝑖∑≥1,𝑖∈𝑀𝑖∈𝑀𝑗𝑖𝜏𝑖≤𝑡}(−1)^{∑𝑖∈𝑀𝑗𝑖}𝑗_𝑖!×∣𝑖∈𝑀_𝑖∈𝑀𝐵^2𝑗𝑖_𝑖𝑡−∑_𝑖∈𝑀𝑗_𝑖𝜏_𝑖^{2(∑𝑖∈𝑀𝑗𝑖−1)}2∑_𝑖∈𝑀𝑗_𝑖−1!=_{𝑖∈𝑀𝑗𝑘∑≥1,𝑘∈𝑀𝑘∈𝑀𝑗𝑘𝜏𝑘≤𝑡}(−1)^{∑𝑘∈𝑀𝑗𝑘}𝑗_𝑖−1,𝑗_𝑘∣𝑘∈𝑀⧵{𝑖}!×_𝑘∈𝑀𝐵^2𝑗𝑘_𝑘𝑡−𝜏_𝑖−∑_{{𝑖}𝑘∈𝑀⧵}𝑗_𝑘𝜏_𝑘−𝑗_𝑖−1𝜏_𝑖^{2(∑𝑘∈𝑀𝑗𝑘−1)}2∑_𝑘∈𝑀𝑗_𝑘!.−1 (45) Next, for any fixed 𝑖∈{1,…,𝑛} we split the second sum to 𝑗_𝑖=1 and 𝑗_𝑖≥2, that is, _{𝑗𝑘∑≥1,𝑘∈𝑀𝑘∈𝑀𝑗𝑘𝜏𝑘≤𝑡}𝐹𝑗₁,…,𝑗_𝑖−1,𝑗_𝑖,𝑗_𝑖+1,…,𝑗_𝑛=_{𝑗𝑘{𝑖}∑≥1,𝑘∈𝑀⧵{𝑖}𝑘∈𝑀⧵𝑗𝑘𝜏𝑘≤𝑡−𝜏𝑖}𝐹𝑗₁,…,𝑗_𝑖−1,1,𝑗_𝑖+1,…,𝑗_𝑛+_{𝑗𝑘{𝑖}𝑗≥1,𝑘∈𝑀⧵𝑖∑≥2𝑘∈𝑀𝑗𝑘𝜏𝑘≤𝑡}𝐹𝑗₁,…,𝑗_𝑖−1,𝑗_𝑖,𝑗_𝑖+1,…,𝑗_𝑛, (46) and use the equality _{𝑗𝑘{𝑖}𝑗≥1,𝑘∈𝑀⧵𝑖∑≥2𝑘∈𝑀𝑗𝑘𝜏𝑘≤𝑡}𝐹𝑗₁,…,𝑗_𝑖−1,𝑗_𝑖,𝑗_𝑖+1,…,𝑗_𝑛=_{𝑗𝑘∑≥1,𝑘∈𝑀𝑘∈𝑀𝑗𝑘𝜏𝑘≤𝑡−𝜏𝑖}𝐹𝑗₁,…,𝑗_𝑖−1,𝑗_𝑖+1,𝑗_𝑖+1,…,𝑗_𝑛 (47) since _𝑘∈𝑀𝑗_𝑘𝜏_𝑘≤𝑡⟺_{{𝑖}𝑘∈𝑀⧵}𝑗_𝑘𝜏_𝑘+𝑗_𝑖−1𝜏_𝑖≤𝑡−𝜏_𝑖. (48) So we obtain 𝑆_𝑀^′′(𝑡)=−_𝑖∈𝑀𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖+𝑆_𝑀𝑡−𝜏_𝑖 (49) for each ∅≠𝑀⊂{1,…,𝑛}. Obviously, 𝑆_∅^′′(𝑡)=Θ. Consequently, 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖+𝑆_𝑀𝑡−𝜏_𝑖=−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖. (50) Now, we add and subtract _{{}∅≠𝑀⊂1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖 (51) to the right-hand side of (50) to get 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=−_{}^𝑛_{∅≠𝑀⊂1,…,𝑛𝑖=1}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖+_{{}∅≠𝑀⊂1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖 (52) and apply 𝑀=𝑀⧵{𝑖} whenever 𝑖∉𝑀: 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=−_{}^𝑛_{∅≠𝑀⊂1,…,𝑛𝑖=1}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖+_{{}∅≠𝑀⊂1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖. (53) Denoting #𝑀 the number of elements of the set 𝑀, we split the last two terms of the right-hand side of the latter equality with respect to _{{}∅≠𝑀⊂1,…,𝑛}=_{{}𝑀⊂1,…,𝑛1≤#𝑀≤𝑛−1}+_{{}𝑀⊂1,…,𝑛#𝑀=𝑛}=_{{}𝑀⊂1,…,𝑛#𝑀=1}+_{{}𝑀⊂1,…,𝑛2≤#𝑀≤𝑛}. (54) Hence, we have _{{}∅≠𝑀⊂1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=_{{}𝑀⊂1,…,𝑛1≤#𝑀≤𝑛−1𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖+_{{}𝑀=1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{1}{𝑛𝑀∈{{,…,}}𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{{}𝑀⊂1,…,𝑛2≤#𝑀≤𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖. (55) Now, we show that _{{}𝑀⊂1,…,𝑛1≤#𝑀≤𝑛−1𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=_{{}𝑀⊂1,…,𝑛2≤#𝑀≤𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖. (56) Let 𝑀⊂{1,…,𝑛}, and let 𝑖∉𝑀 be arbitrary and fixed such that 1≤#𝑀≤𝑛−1. Then, clearly, 𝐵_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=𝐵_𝑖𝑆_{(𝑀∪{𝑖})⧵{𝑖}}𝑡−𝜏_𝑖 (57) and 2≤#(𝑀∪{𝑖})≤𝑛, 𝑖∈𝑀∪{𝑖}. Moreover, if 𝑀₁,𝑀₂⊂{1,…,𝑛}, 𝑖∉𝑀_1,2 are such that 𝑀₁≠𝑀₂, 1≤#𝑀_1,2≤𝑛−1, then 𝑀₁∪{𝑖}≠𝑀₂∪{𝑖}.
On the other side, if 𝑀⊂{1,…,𝑛}, 𝑖∈𝑀 are arbitrary and fixed such that 2≤#𝑀≤𝑛, then 𝐵_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=𝐵_𝑖𝑆_{(𝑀⧵{𝑖})⧵{𝑖}}𝑡−𝜏_𝑖 (58) and 1≤#(𝑀⧵{𝑖})≤𝑛−1,𝑖∉𝑀⧵{𝑖}. Furthermore, if 𝑀₁,𝑀₂⊂{1,…,𝑛},𝑖∈𝑀_1,2 are such that 𝑀₁≠𝑀₂, 2≤#𝑀_1,2≤𝑛, then, 𝑀₁⧵{𝑖}≠𝑀₂⧵{𝑖}. In conclusion, there is 1−1 correspondence between the terms on the left-hand side of (56) and the terms on the right-hand side. So (56) is valid.
Putting (56) in (55) we obtain _{{}∅≠𝑀⊂1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{{}∅≠𝑀⊂1,…,𝑛𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=_{{}𝑀=1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖−_{1}{𝑛𝑀∈{{,…,}}𝑖∈𝑀}𝐵²_𝑖𝑆_𝑀{𝑖}𝑡−𝜏_𝑖. (59) Next, by the property of empty sum, we get _{{}𝑀=1,…,𝑛𝑖∉𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=_{{}𝑀=1,…,𝑛}Θ=Θ. (60) Moreover, it holds _{1}{𝑛𝑀∈{{,…,}}𝑖∈𝑀}𝐵²_𝑖𝑆_{𝑀⧵{𝑖}}𝑡−𝜏_𝑖=^𝑛_𝑖=1𝐵²_𝑖𝑆_∅𝑡−𝜏_𝑖=^𝑛_{𝑀=∅𝑖=1}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖. (61) Therefore, putting (60) and (61) in (59) and the result in (53), we obtain 𝒳̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=−_{}^𝑛_{∅≠𝑀⊂1,…,𝑛𝑖=1}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖−^𝑛_{𝑀=∅𝑖=1}𝐵²_𝑖𝑆_𝑀𝑡−𝜏_𝑖=−^𝑛_𝑖=1𝐵²_{𝑖{}𝑀⊂1,…,𝑛}𝑆_𝑀𝑡−𝜏_𝑖=−^𝑛_𝑖=1𝐵²_𝑖𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏_𝑖. (62) Hence, 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡) solves (31) for all 𝑡≥0. Clearly, the same is true for 𝑡<0.
For 𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡), statements (1)–(3) can be proved as for 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡). Next, if 𝜏∶=𝜏₁=⋯=𝜏_𝑛, we apply the point (2) of this lemma and property (22) for 𝒴^{𝐵21+⋯+𝐵2𝑛}_𝜏(𝑡) to see that 𝒴̈^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒴̈^{𝐵21+⋯+𝐵2𝑛}_𝜏(𝑡)=−𝐵²₁+⋯+𝐵²_𝑛𝒴^{𝐵21+⋯+𝐵2𝑛}_𝜏(𝑡−𝜏)=−𝐵²₁𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏₁−⋯−𝐵²_𝑛𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}𝑡−𝜏_𝑛. (63) So, 𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡) is a solution of (31) when all delays are the same.
Again, the case 𝑛=2 with different delays was proved in [16]; thus, we assume that the statement is fulfilled for 𝑛−1, 𝑛≥3 and that 𝜏_𝑖≠𝜏_𝑗 for each 𝑖≠𝑗,𝑖,𝑗∈{1,…,𝑛}. As before, if 𝑡<𝜏_𝑘 and 𝜏_𝑘∶=max_{𝑖=1,…,𝑛}𝜏_𝑖, then 𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=𝒴^{𝐵21,…,𝐵2𝑘−1,𝐵2𝑘+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑘−1,𝜏𝑘+1,…,𝜏𝑛}(𝑡) (64) by definition (20), and the statement follows from the inductive hypothesis. For 𝑡≥max_{𝑖=1,…,𝑛}𝜏_𝑖, we apply Lemma 3 to see that 𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)=_{{}𝑀⊂1,…,𝑛}𝑆_𝑀(𝑡) (65) with 𝑆_𝑀(𝑡) given by (25). This time 𝑆_∅(𝑡)=_{𝑗𝑖≥1,𝑖∈∅0≤𝑡}(−1)⁰𝑗_𝑖()∣𝑖∈∅!𝐸𝑡−0¹=0!_0≤𝑡𝐸𝑡=𝑡𝐸𝜒_[0,∞)(𝑡) (66) and 𝑆_∅^′′(𝑡)=Θ. The rest proceeds analogically to 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡).
The final statement follows directly from definition (20).

Remark 5. Another proof of statements (1)–(3) of the previous lemma can be made with the aid of statement (4) of the same lemma and uses the uniqueness of a solution of the corresponding initial value problem. For instance in statement (1) of the lemma, both 𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡),𝒳^{𝐵21,…,𝐵2𝑖−1,𝐵2𝑖+1,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑖−1,𝜏𝑖+1,…,𝜏𝑛}(𝑡) (67) solve 𝑥̈(𝑡)=−𝐵²₁𝑥𝑡−𝜏₁−⋯−𝐵²_𝑖−1𝑥𝑡−𝜏_𝑖−1−𝐵²_𝑖+1𝑥𝑡−𝜏_𝑖+1−⋯−𝐵²_𝑛𝑥𝑡−𝜏_𝑛 (68) with initial condition 𝑥(𝑡)=𝑥̇(𝑡)Θ,−𝜏≤𝑡<0,𝐸,𝑡=0,=Θ,−𝜏≤𝑡≤0 (69) and 𝜏=max_{𝑖=1,…,𝑛}𝜏_𝑖.

We are ready to state and prove our main result.

3. Main Result

Here we find a solution of the initial value problem (11), (8) in the sense of the next definition.

Definition 6. Let 𝜏₁,…,𝜏_𝑛>0, 𝜏∶=max_{𝑖=1,…,𝑛}𝜏_𝑖, and 𝜑∈𝐶¹([−𝜏,0],ℝ^𝑁), and let 𝐵₁,…,𝐵_𝑛 be 𝑁×𝑁 matrices, and let 𝑓∶[0,∞)→ℝ^𝑁 be a given function. Function 𝑥∶[−𝜏,∞)→ℝ^𝑁 is a solution of (11) and initial condition (8), if 𝑥∈𝐶¹([−𝜏,∞),ℝ^𝑁)∩𝐶²([0,∞),ℝ^𝑁) (taken the second right-hand derivative at 0) satisfies (11) on [0,∞) and condition (8) on [−𝜏,0].

Theorem 7. Let 𝑛≥3, 𝜏₁,…,𝜏_𝑛>0, 𝜏∶=max_{𝑖=1,…,𝑛}𝜏_𝑖, and 𝜑∈𝐶¹([−𝜏,0],ℝ^𝑁), and let 𝐵₁,…,𝐵_𝑛 be 𝑁×𝑁 pairwise permutable matrices; that is, 𝐵_𝑖𝐵_𝑗=𝐵_𝑗𝐵_𝑖 for each 𝑖,𝑗∈{1,…,𝑛}, and let 𝑓∶[0,∞)→ℝ^𝑁 be a given function. Solution 𝑥(𝑡) of (11) satisfying initial condition (8) has the form 𝑥(𝑡)=𝜑(𝑡)𝒳(𝑡)𝜑(0)(𝑡)𝜑̇(0)−,−𝜏≤𝑡<0,+𝒴_𝑛𝑖=1𝐵²_𝑖⁰_−𝜏𝑖𝒴𝑡−𝜏_𝑖−𝑠𝜑(𝑠)𝑑𝑠+^𝑡₀𝒴(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,0≤𝑡, (70) where 𝒳(𝑡)=𝒳^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡) and 𝒴(𝑡)=𝒴^{𝐵21,…,𝐵2𝑛}_{𝜏1,…,𝜏𝑛}(𝑡).

Proof. Obviously, 𝑥(𝑡) satisfies the initial condition on [−𝜏,0), and, from definition (20), 𝑥(0)=𝜑(0). For the derivative, it holds lim_𝑡→0⁻𝑥̇(𝑡)=𝜑̇(0). Moreover, if 0≤𝑡<min_{𝑖=1,…,𝑛}𝜏_𝑖, then 𝑥(𝑡)(0)𝜑̇(0)−=𝜑+𝑡_𝑛𝑖=1𝐵²_𝑖^{𝑡−𝜏𝑖}_−𝜏𝑖𝑡−𝜏_𝑖𝜑(𝑠)+−𝑠𝑑𝑠_𝑡0(𝑡−𝑠)𝑓(𝑠)𝑑𝑠 (71) since 𝒴𝑡−𝜏_𝑖−𝑠=𝑡−𝜏_𝑖−𝑠𝐸,𝑠∈−𝜏_𝑖,𝑡−𝜏_𝑖,Θ,𝑠∈𝑡−𝜏_𝑖,0 (72) for each 𝑖=1,…,𝑛. Thus 𝑥̇(𝑡)=𝜑̇(0)−^𝑛_𝑖=1𝐵²_𝑖^{𝑡−𝜏𝑖}_−𝜏𝑖𝜑(𝑠)𝑑𝑠+^𝑡₀𝑓(𝑠)𝑑𝑠 (73) and lim_𝑡→0⁺𝑥̇(𝑡)=𝜑̇(0). Clearly, 𝑥∈𝐶¹(−𝜏,∞),ℝ^𝑁∩𝐶²(0,∞)⧵𝜏₁,…,𝜏_𝑛,ℝ^𝑁. (74)
We show that, although 𝒳(𝑡) is not 𝐶² at 𝜏₁,…,𝜏_𝑛, function 𝑥(𝑡) is 𝐶² at these points and, therefore, in (0,∞). At once, we prove that 𝑥(𝑡) is a solution of (11).
Assume that 0≤𝑡<min_{𝑖=1,…,𝑛}𝜏_𝑖. Then identities (71) and (73) are valid, and by differentiating (73) for such 𝑡 we get 𝑥̈(𝑡)=−^𝑛_𝑖=1𝐵²_𝑖𝜑𝑡−𝜏_𝑖+𝑓(𝑡)=−^𝑛_𝑖=1𝐵²_𝑖𝑥𝑡−𝜏_𝑖+𝑓(𝑡) (75) since 𝑥(𝑡−𝜏_𝑖)=𝜑(𝑡−𝜏_𝑖) for each 𝑖=1,…,𝑛.
Now, let ∅≠𝑀_1,2⊂{1,…,𝑛} be such that 𝜏_𝑖≤𝑡<𝜏_𝑗 for each 𝑖∈𝑀₁, 𝑗∈𝑀₂. Then 𝒴𝑡−𝜏_𝑗=𝒴−𝑠𝑡−𝜏_𝑗−𝑠,𝑠∈−𝜏_𝑗,𝑡−𝜏_𝑗,Θ,𝑠∈𝑡−𝜏_𝑗,0 (76) whenever 𝑗∈𝑀₂, and (70) becomes 𝑥(𝑡)=𝒳(𝑡)𝜑(0)+𝒴(𝑡)𝜑̇(0)−_𝑖∈𝑀1𝐵²_𝑖⁰_−𝜏𝑖𝒴𝑡−𝜏_𝑖−𝑠𝜑(𝑠)𝑑𝑠−_𝑗∈𝑀2𝐵²_𝑗^{𝑡−𝜏𝑗}_−𝜏𝑗𝒴𝑡−𝜏_𝑗𝜑(𝑠)+−𝑠𝑑𝑠_𝑡0𝒴(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (77) By the point (5) of Lemma 4, we get 𝑥̇(𝑡)=𝒳̇(𝑡)𝜑(0)+𝒴̇(𝑡)𝜑̇(0)−_𝑖∈𝑀1𝐵²_𝑖⁰_−𝜏𝑖𝒴̇𝑡−𝜏_𝑖𝜑(𝑠)−−𝑠𝑑𝑠_𝑗∈𝑀2𝐵²_𝑗^{𝑡−𝜏𝑗}_−𝜏𝑗𝒳𝑡−𝜏_𝑗−𝑠𝜑(𝑠)𝑑𝑠+^𝑡₀𝒳(𝑡−𝑠)𝑓(𝑠)𝑑𝑠, (78) and for the second derivative it holds 𝑥̈(𝑡)=𝒳̈(𝑡)𝜑(0)+𝒴̈(𝑡)𝜑̇(0)−_𝑖∈𝑀1𝐵²_𝑖⁰_−𝜏𝑖𝒴̈𝑡−𝜏_𝑖−𝑠𝜑(𝑠)𝑑𝑠−_𝑗∈𝑀2𝐵²_𝑗𝜑𝑡−𝜏_𝑗+^{𝑡−𝜏𝑗}_−𝜏𝑗𝒴̈𝑡−𝜏_𝑗𝜑(𝑠)(𝑡)+−𝑠𝑑𝑠+𝑓^𝑡₀𝒴̈(𝑡−𝑠)𝑓(𝑠)𝑑𝑠 (79) since 𝒳(0)=𝐸. Now, we apply the property (4) of Lemma 4 together with 𝒳𝑡−𝜏_𝑗=𝒴𝑡−𝜏_𝑗=Θ,∀𝑗∈𝑀₂ (80) to see that both 𝒳 and 𝒴 are solutions of 𝑦̈(𝑡)=−_𝑖∈𝑀1𝐵²_𝑖𝑦𝑡−𝜏_𝑖. (81) Therefore, 𝑥̈(𝑡)=−_𝑘∈𝑀1𝐵²_𝑘𝒳𝑡−𝜏_𝑘𝜑(0)+𝒴𝑡−𝜏_𝑘𝜑̇(0)−_𝑖∈𝑀1𝐵²_𝑖⁰_−𝜏𝑖𝒴𝑡−𝜏_𝑖−𝜏_𝑘−𝑠𝜑(𝑠)𝑑𝑠−_𝑗∈𝑀2𝐵²_𝑗^{𝑡−𝜏𝑗}_−𝜏𝑗𝒴𝑡−𝜏_𝑗−𝜏_𝑘𝜑(𝑠)+−𝑠𝑑𝑠_𝑡0𝒴𝑡−𝜏_𝑘−𝑠𝑓(𝑠)𝑑𝑠−_𝑗∈𝑀2𝐵²_𝑗𝜑𝑡−𝜏_𝑗+𝑓(𝑡)=−_𝑖∈𝑀1𝐵²_𝑖𝑥𝑡−𝜏_𝑖−_𝑗∈𝑀2𝐵²_𝑗𝜑𝑡−𝜏_𝑗+𝑓(𝑡). (82) In fact, this is exactly formula (11) since 𝑥(𝑡−𝜏_𝑗)=𝜑(𝑡−𝜏_𝑗) for each 𝑗∈𝑀₂.
Finally, if max_{𝑖=1,…,𝑛}𝜏_𝑖≤𝑡, we have 𝑥(𝑡)(𝑡)𝜑(0)(𝑡)𝜑̇(0)−=𝒳+𝒴_𝑛𝑖=1𝐵²_𝑖⁰_−𝜏𝑖𝒴𝑡−𝜏_𝑖𝜑(𝑠)+−𝑠𝑑𝑠_𝑡0𝒴(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (83) So, differentiating this formula twice and applying (4) of Lemma 4 result in (11). Hence, one can see that function 𝑥(𝑡) given by (70) really solves (11) and satisfies initial condition (8) and, moreover, that 𝑥∈𝐶²((0,∞),ℝ^𝑁). To see the last one, one has to put 𝜏₁,…,𝜏_𝑛 into the computed derivatives, for example, if 𝜏_𝑘∶=min_{𝑖=1,…,𝑛}𝜏_𝑖<𝜏_𝑖 for each 𝑖=1,…,𝑘−1,𝑘+1,…,𝑛, then by (75) and (82) we get lim_{𝑡→𝜏−𝑘}𝑥̈(𝑡)=−^𝑛_{𝑖=1𝑖≠𝑘}𝐵²_𝑖𝜑𝜏_𝑘−𝜏_𝑖−𝐵²_𝑘𝜑(0)+𝑓𝜏_𝑘=−_𝑗∈𝑀2𝐵²_𝑗𝜑𝜏_𝑘−𝜏_𝑗−𝐵²_𝑘𝑥(0)+𝑓𝜏_𝑘=lim_{𝑡→𝜏+𝑘}𝑥̈(𝑡), (84) where 𝑀₂={1,…,𝑛}⧵{𝑘}.

It is easy to see that defining functions 𝒳^{𝐵1,…,𝐵𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)∶=𝒳^{−𝐵1,…,−𝐵𝑛}_{𝜏1,…,𝜏𝑛}(𝑡),𝒴^{𝐵1,…,𝐵𝑛}_{𝜏1,…,𝜏𝑛}(𝑡)∶=𝒴^{−𝐵1,…,−𝐵𝑛}_{𝜏1,…,𝜏𝑛}(𝑡) (85) leads to the solution of 𝑥̈(𝑡)=𝐵₁𝑥𝑡−𝜏₁+⋯+𝐵_𝑛𝑥𝑡−𝜏_𝑛+𝑓(𝑡) (86) with pairwise permutable matrices 𝐵₁,…,𝐵_𝑛 and initial condition (8). More precisely, we have the following corollary of Theorem 7.

Corollary 8. Let 𝑛≥3, 𝜏₁,…,𝜏_𝑛>0, 𝜏∶=max_{𝑖=1,…,𝑛}𝜏_𝑖, 𝜑∈𝐶¹([−𝜏,0],ℝ^𝑁), and let 𝐵₁,…,𝐵_𝑛 be 𝑁×𝑁 pairwise permutable matrices; that is, 𝐵_𝑖𝐵_𝑗=𝐵_𝑗𝐵_𝑖 for each 𝑖,𝑗∈{1,…,𝑛}, and let 𝑓∶[0,∞)→ℝ^𝑁 be a given function. Solution 𝑥(𝑡) of (86) satisfying initial condition (8) has the form 𝑥(𝑡)=𝜑(𝑡)𝒳(𝑡)𝜑(0)(𝑡)𝜑̇(0)+,−𝜏≤𝑡<0,+𝒴_𝑛𝑖=1𝐵_𝑖⁰_−𝜏𝑖𝒴𝑡−𝜏_𝑖−𝑠𝜑(𝑠)𝑑𝑠+^𝑡₀𝒴(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,0≤𝑡, (87) where 𝒳(𝑡)=𝒳^{𝐵1,…,𝐵𝑛}_{𝜏1,…,𝜏𝑛}(𝑡) and 𝒴(𝑡)=𝒴^{𝐵1,…,𝐵𝑛}_{𝜏1,…,𝜏𝑛}(𝑡).

Proof. The corollary can be proved exactly in the same way as Theorem 7.

Acknowledgments

J. Diblík was supported by the Grant GAČR P201/11/0768. M. Fečkan was supported in part by the Grants VEGA-MS 1/0507/11, VEGA-SAV 2/0029/13, and APVV-0134-10. M. Pospíšil was supported by the Project no. CZ.1.07/2.3.00/30.0005 funded by European Regional Development Fund.