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

cited. 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.

On the other hand, there can exist a nonoscillating solution of the system (1) whenever ∈ R and is even. For instance, if = 2 and = ( 0 1 −1 0 ), then (1) has the form with ∈ R 2 , 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 ≡ 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][2][3][4][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.
We will denote Θ and the × zero and identity matrix, respectively.
From now on, we assume the property of empty sum and empty product; that is, for any function and matrix function , whether they are defined or not for indicated argument.
We will need a property of multinomial coefficients described in the next lemma.
Proof. If = 2, then the statement follows from the property of binomial coefficients: Let the statement be true for − 1. Next, we use the property of multinomial coefficient with inductive hypothesis to derive Clearly, from (16), we get . . .
Applying the case = 2 (property of binomial coefficient) and (16), we get Putting (18) and (19) in (17), we obtain that the statement holds for and the proof is complete.
Define the functions X for any ∈ R. We will need functions X 2 , Y 2 : R → (R ) for > 0 and × complex matrix (cf. [16]) defined as with the propertieṡ for any ∈ R, considering the one-sided derivatives at − , 0. Some of properties of functions X are concluded in Lemma 4, but to prove it we will need the next lemma.
(45) 6 Abstract and Applied Analysis Next, for any fixed ∈ {1, . . . , } we split the second sum to = 1 and ≥ 2, that is, and use the equality So we obtain Now, we add and subtract to the right-hand side of (50) to geẗ Denoting # the number of elements of the set , we split the last two terms of the right-hand side of the latter equality with respect to 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.

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