As an example of application of Theorem 3.1, we consider the problem of bifurcation from point of solutions , satisfying, for a.e. , systems of ordinary differential equations

where is constant matrix, is an matrix, , consisting of matrices , , having entries in , is a small parameter, delays are measurable on , , , , , , and satisfying the initial and boundary conditions

where , is a given vector function with components in , and is a linear vector functional. Using denotations (1.3), (1.4), and (1.6), it is easy to show that the perturbed nonhomogeneous linear boundary value problem (4.1), (4.2) can be rewritten as

In (4.3) we specify as a single delay defined by formula ;

is an -dimensional column vector, and is an -dimensional column vector given by

It is easy to see that . The operator maps the space into the space

that is, . Using denotation (1.3) for the operator , we have the following representation:

where

is the characteristic function of the set

Assume that nonhomogeneities and are such that the shortened boundary value problem

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?*

We can answer this question with the help of the -matrix

where

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.

Consider system

where is constant matrix, is an matrix, , consisting of matrices , , having entries in , is a small parameter, delays are measurable on , , , , , with the initial and boundary conditions

where , is a given vector function with components in , and is a linear vector functional, and assume that

(satisfying ) and are such that the shortened problem

does not have a solution. If

then the boundary value problem (4.13), (4.14) has a set of linearly independent solutions in the form of the series

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.

Substitute (4.18) into (4.3) and equate the terms that are multiplied by the same powers of . For , we obtain the homogeneous boundary value problem

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 .

For , we get the boundary value problem

which determines .

It follows from Theorem 3.1 that the solvability criterion (3.21) for problem (4.20) has the form

from which we receive, with respect to , an algebraic system

The right-hand side of (4.22) is nonzero only in the case that the shortened problem does not have a solution. The system (4.22) is solvable for arbitrary and if the condition (4.17) is satisfied [9, page 79]. In this case, system (4.22) becomes resolvable with respect to up to an arbitrary constant vector from the null-space of matrix and

This solution can be rewritten in the form

where

and is an -dimensional matrix whose columns are a complete set of linearly independent columns of the -dimensional matrix with

So, for the solutions of the problem (3.14), we have the following formulas:

Assuming that (3.24) and (4.17) hold, the boundary value problem (4.20) has the -parametric family of solutions

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

For , we get the boundary value problem

which determines . The solvability criterion for the problem (4.29) has the form (in computations below we need a composition of operators and the order of operations is following the inner operator which acts to matrices and vector function having an argument denoted by and the outer operator which acts to matrices having an argument denoted by

or, equivalently, the form

Assuming that (4.17) holds, the algebraic system (4.31) has the following family of solutions:

where

So, for the -parametric family of solutions of the problem (4.20), we have the following formula:

where

Again, assuming that (4.17) holds, the boundary value problem (4.29) has the -parametric family of solutions

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

The solvability criterion for the problem (4.37) has the form

or, equivalently, the form

Under condition (4.17), the last equation has the -parametric family of solutions

where

So, for the coefficient , we have the following formula:

where

Continuing this process, by assuming that (4.17) holds, it follows by induction that the coefficients of the series (4.18) can be determined, from the relevant boundary value problems as follows:

where

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.

Consider the linear boundary value problem for the delay differential equation

where, as in the above, and are measurable functions. Using the symbols and (see (1.3), (1.4), (1.6), and (4.7)), we arrive at the following operator system:

where is an matrix , and

Under the condition that the generating boundary value problem has no solution, we consider the simplest case of . Using the delayed matrix exponential (2.5), it is easy to see that, in this case, is a normal fundamental matrix for the homogeneous unperturbed system , and

Then the matrix has the form

If , problem (4.46) has a unique solution with the properties

Let, say, where , , then

or, equivalently,

Now the matrix turns into

and the boundary value problem (4.46) is uniquely solvable if