# Boundary Value Problems for Delay Differential Systems

- A. Boichuk
^{1, 2}Email author, - J. Diblík
^{3, 4}, - D. Khusainov
^{5}and - M. Růžičková
^{1}

**2010**:593834

**DOI: **10.1155/2010/593834

© A. Boichuk et al. 2010

**Received: **16 January 2010

**Accepted: **12 May 2010

**Published: **9 June 2010

## Abstract

Conditions are derived of the existence of solutions of linear Fredholm's boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay, assuming that these solutions satisfy the initial and boundary conditions. Utilizing a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay. As an example of application of the results derived, the problem of bifurcation of solutions of boundary-value problems for systems of ordinary differential equations with a small parameter and with a finite number of measurable delays of argument is considered.

## 1. Introduction

We will investigate (1.5) assuming that the operator maps a Banach space of absolutely continuous functions into a Banach space of function integrable on with the degree ; the operator maps the space into the space . Transformations of (1.3), (1.4) make it possible to add the initial vector function , to nonhomogeneity, thus generating an additive and homogeneous operation not depending on , and without the classical assumption regarding the continuous connection of solution with the initial function at .

A solution of differential system (1.5) is defined as an -dimensional column vector function , absolutely continuous on with a derivative in a Banach space of functions integrable on with the degree satisfying (1.5) almost everywhere on . Throughout this paper we understand the notion of a solution of a differential system and the corresponding boundary value problem in the sense of the above definition.

where if , and is the null matrix. A fundamental matrix for the homogeneous (1.5) has the form , .

A serious disadvantage of this approach, when investigating the above-formulated problem, is the necessity to find the Cauchy matrix
[5, 6]. It exists but, as a rule, can only be found numerically. Therefore, it is important to find systems of differential equations with delay such that this problem can be solved directly. Below, we consider the case of a system with what is called a single delay [7]. In this case, the problem of how to construct the Cauchy matrix is solved *analytically* thanks to a delayed matrix exponential, as defined below.

## 2. A Delayed Matrix Exponential

where is the greatest integer function. The delayed matrix exponential equals a unit matrix on and represents a fundamental matrix of a homogeneous system with a single delay.

The following results (proved in [7] and being a consequence of (1.7) with as well) hold.

- (A)The solution of a homogeneous system (2.3) with a single delay satisfying the initial condition (2.4) where is an arbitrary continuously differentiable vector function can be represented in the form(2.11)

- (B)A particular solution of a nonhomogeneous system (2.1) with a single delay satisfying the zero initial condition if can be represented in the form(2.12)

- (C)A solution of a Cauchy problem of a nonhomogeneous system with a single delay (2.1) satisfying a constant initial condition(2.13)

## 3. Main Results

Now we will consider a general Fredholm boundary value problem for system (3.1).

### 3.1. Fredholm Boundary Value Problem

Using the results in [8, 9], it is easy to derive statements for a general boundary value problem if the number of boundary conditions does not coincide with the number of unknowns in a differential system with a single delay.

where
is an
-dimensional constant vector column, and
is a linear vector functional. It is well known that, for functional differential equations, such problems are of Fredholm's type (see, e.g., [1, 9]). We will derive the necessary and sufficient conditions and a representation (in an *explicit analytical* form) of the solutions
of the boundary value problem (3.11), (3.12).

we will denote by an -dimensional matrix constructed from linearly independent columns of the matrix .

is a generalized Green matrix, corresponding to the boundary value problem (3.11), (3.12), and the Cauchy matrix has the form of (3.6). Therefore, the following theorem holds (see [10]).

Theorem 3.1.

Nonhomogeneous problem (3.11), (3.12) is solvable if and only if and satisfy linearly independent conditions (3.21). In that case, this problem has an -dimensional family of linearly independent solutions represented in an explicit analytical form (3.23).

The case of implies the inequality . If , the boundary value problem is overdetermined, the number of boundary conditions is more than the number of unknowns, and Theorem 3.1 has the following corollary.

Corollary 3.2.

The case of is interesting as well. Then the inequality , holds. If the boundary value problem is not fully defined. In this case, Theorem 3.1 has the following corollary.

Corollary 3.3.

Finally, combining both particular cases mentioned in Corollaries 3.2 and 3.3, we get a noncritical case.

Corollary 3.4.

is a related Green matrix, corresponding to the problem (3.11), (3.12).

## 4. Perturbed Boundary Value Problems

being a particular case of (4.3) for , does not have a solution. In such a case, according to Theorem 3.1, the solvability criterion (3.21) does not hold for problem (4.10). Thus, we arrive at the following question.

*Is it possible to make the problem ( 4.10 ) solvable by means of linear perturbations and, if this is possible, then of what kind should the perturbations*
*and the delays*
*, *
*be for the boundary value problem ( 4.3 ) to be solvable?*

constructed by using the coefficients of the problem (4.3).

Using the Vishik and Lyusternik method [11] and the theory of generalized inverse operators [9], we can find bifurcation conditions. Below we formulate a statement (proved using [8] and [9, page 177]) which partially answers the above problem. Unlike an earlier result [9], this one is derived in an *explicit analytical* form. We remind that the notion of a solution of a boundary value problem was specified in part 1.

Theorem 4.1.

converging for fixed , where is an appropriate constant characterizing the domain of the convergence of the series (4.18), and are suitable coefficients.

Remark 4.2.

Coefficients , , in (4.18) can be determined. The procedure describing the method of their deriving is a crucial part of the proof of Theorem 4.1 where we give their form as well.

Proof.

which determines .

By Theorem 3.1, the homogeneous boundary value problem (4.19) has an -parametric family of solutions where the -dimensional column vector can be determined from the solvability condition of the problem for .

which determines .

Here, is an -dimensional constant vector, which is determined at the next step from the solvability condition of the boundary value problem for

and .

The convergence of the series (4.18) can be proved by traditional methods of majorization [9, 11].

In the case , the condition (4.17) is equivalent with , and problem (4.13), (4.14) has a unique solution.

Example 4.3.

## Declarations

### Acknowledgments

The authors highly appreciate the work of the anonymous referee whose comments and suggestions helped them greatly to improve the quality of the paper in many aspects. The first author was supported by Grant 1/0771/08 of the Grant Agency of Slovak Republic (VEGA) and Project APVV-0700-07 of Slovak Research and Development Agency. The second author was supported by Grant 201/08/0469 of Czech Grant Agency and by the Council of Czech Government MSM 0021630503, MSM 0021630519, and MSM 0021630529. The third author was supported by Project M/34-2008 of Ukrainian Ministry of Education. The fourth author was supported by Grant 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and project APVV-0700-07 of Slovak Research and Development Agency.

## Authors’ Affiliations

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