Abstract and Applied Analysis

Volume 2013 (2013), Article ID 931493, 10 pages

http://dx.doi.org/10.1155/2013/931493

Research Article

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

Josef Diblík,¹ Michal Fečkan,2,3 and Michal Pospíšil⁴

¹Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic

²Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia

³Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia

⁴Centre for Research and Utilization of Renewable Energy, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 616 00 Brno, Czech Republic

Received 13 January 2013; Accepted 19 April 2013

Academic Editor: Jaan Janno

Copyright © 2013 Josef Diblík et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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]

to the case of two delays

where the linear parts were given by permutable matrices [16]. Equations (1), (2), and the below-stated (11) with are generalizations of the scalar equation

representing linear oscillator, to -dimensional space with one or multiple fixed delays. Clearly, each solution of the latter equation is oscillating whenever . Analogically, (1) with can have at least one oscillating solution whenever is odd. Indeed, if is matrix, is odd, and has a simple real nonzero eigenvalue , then there exists a regular matrix such that where is matrix. On letting , one gets

or rewrites as the system

where . Note that the first column of is the eigenvector of corresponding to . Clearly, if solution of (5) is oscillating, then solution of (4) is oscillating in the first coordinate whenever its initial condition satisfies . Consequently, solution of (1) is oscillating in whenever . Taking , one obtains characteristic equation of (5), which has solutions with . Thus, is oscillating.

On the other hand, there can exist a nonoscillating solution of the system (1) whenever and is even. For instance, if and , then (1) has the form

with , 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 , 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 , , and . Let be permutable matrices; that is, , and let be a given function. Solution of

satisfying initial condition

has the form

where

We will denote and the zero and identity matrix, respectively.

2. Systems with Multiple Delays

In this section, we derive the representation of a solution of

satisfying the initial condition (8), where , , , are pairwise permutable matrices; that is, for each , , and are given functions. The solution will be represented using matrix functions analogical to (10) and will be stated in Section 3. We note that the same problems with were studied in [1, 16].

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 recall that is a multinomial coefficient [19] given by

Note that if , then and (20) coincides with (10).

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

Lemma 2. Let be fixed. Then

for any .

Proof. If , then the statement follows from the property of binomial coefficients:

Let the statement be true for . Next, we use the property of multinomial coefficient

with inductive hypothesis to derive

Clearly, from (16), we get

Applying the case (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.

In further work, we write for the multinomial coefficient of elements of the finite set , and for the multinomial coefficient of and elements of the finite set ; for example, if , then . For the completeness, we define .

Define the functions as

for any .

We will need functions for and complex matrix (cf. [16]) defined as

with the properties

for any , considering the one-sided derivatives at , .

Some of properties of functions and are concluded in Lemma 4, but to prove it we will need the next lemma.

Lemma 3. Let and . Let be pairwise permutable matrices, that is, for each . Then for any ,

where the sums are taken over all subsets of including the trivial ones, and

Proof. Denote , the set of all nonnegative, positive integers, respectively; that is, . Thus, we have the trivial identity

Analogically, for any each -tuple such that can be divided in two distinct sets of -s so that if and if . That is, denotes the set of all indices such that . Moreover, . Accordingly, we can write

where the union is taken over all subsets of including the trivial ones. So, in the view of definition (20), the statement for follows.

Statement for can be proved in a similar way.

Lemma 4. Let and . Let be pairwise permutable matrices; that is, for each . Then the following holds for any :

(1)if for some , then

(2)if for , , then

(3)for any bijective mapping we get

(4)taking the one-sided derivatives at , then

(5)considering the one-sided derivatives at 0 (they both equal ), then

Statements (1)–(4) hold with instead of .

Proof. Statement (1) follows easily from definition of , because if and whenever . Next, if , then

for any matrix function . Thus, using the property of multinomial coefficient (see (16))

for (2), we obtain

Property (3) is trivial.

Now, we prove the statement (4). If , then

by (2) and from the property of (see (22)).

Hence, without any loss of generality, we assume that for each , (in the other case, we collect matrices as stated in (2)). Note the case was proved in [16, Lemma 2.3.] Now, assume that solves

that is, that the statement is fulfilled for different delays.

Let . If , then , that is,

and from definition (20) it holds

for such . Consequently,

by the inductive hypothesis.

Now, let . Applying Lemma 3, we get

with given by (24) and the sum taken over all subsets of including the trivial ones. Note that

with a characteristic function of a set given by

Since each is a finite set, Lemma 2 yields

We apply this identity to derive a formula for the second derivative of for any :

Next, for any fixed we split the second sum to and , that is,

and use the equality

since

So we obtain

for each . Obviously, . Consequently,

Now, we add and subtract

to the right-hand side of (50) to get

and apply whenever :

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

Hence, we have

Now, we show that

Let , and let be arbitrary and fixed such that . Then, clearly,

and , . Moreover, if , are such that , , then .

On the other side, if , are arbitrary and fixed such that , then

and . Furthermore, if are such that , , then, . In conclusion, there is 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

Next, by the property of empty sum, we get

Moreover, it holds

Therefore, putting (60) and (61) in (59) and the result in (53), we obtain

Hence, solves (31) for all . Clearly, the same is true for .

For , statements (1)–(3) can be proved as for . Next, if , we apply the point (2) of this lemma and property (22) for to see that

So, is a solution of (31) when all delays are the same.

Again, the case with different delays was proved in [16]; thus, we assume that the statement is fulfilled for , and that for each . As before, if and , then

by definition (20), and the statement follows from the inductive hypothesis. For , we apply Lemma 3 to see that

with given by (25). This time

and . The rest proceeds analogically to .

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

solve

with initial condition

and .

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 , , and , and let be matrices, and let be a given function. Function is a solution of (11) and initial condition (8), if (taken the second right-hand derivative at 0) satisfies (11) on and condition (8) on .

Theorem 7. Let , , , and , and let be pairwise permutable matrices; that is, for each , and let be a given function. Solution of (11) satisfying initial condition (8) has the form

where and .

Proof. Obviously, satisfies the initial condition on , and, from definition (20), . For the derivative, it holds . Moreover, if , then

since

for each . Thus

and . Clearly,

We show that, although is not at , function is at these points and, therefore, in . At once, we prove that is a solution of (11).

Assume that . Then identities (71) and (73) are valid, and by differentiating (73) for such we get

since for each .

Now, let be such that for each , . Then

whenever , and (70) becomes

By the point (5) of Lemma 4, we get

and for the second derivative it holds

since . Now, we apply the property (4) of Lemma 4 together with

to see that both and are solutions of

Therefore,

In fact, this is exactly formula (11) since for each .

Finally, if , we have

So, differentiating this formula twice and applying (4) of Lemma 4 result in (11). Hence, one can see that function given by (70) really solves (11) and satisfies initial condition (8) and, moreover, that . To see the last one, one has to put into the computed derivatives, for example, if for each , then by (75) and (82) we get

where .

It is easy to see that defining functions

leads to the solution of

with pairwise permutable matrices and initial condition (8). More precisely, we have the following corollary of Theorem 7.

Corollary 8. Let , , , , and let be pairwise permutable matrices; that is, for each , and let be a given function. Solution of (86) satisfying initial condition (8) has the form

where and .

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.

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