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