Factorization of the linear differential operator
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The paper deals with the problem of factorization of a linear differential operator with matrix-valued coefficients into a product of lower order operators of the same type. Necessary and sufficient conditions for the factorization of the considered operator are given. These conditions are obtained by using the integral manifolds approach. Some consequences of the obtained results are also considered.
MSC:34A30, 47A50, 47E05.
Keywordslinear differential equations differential operator factorization integral manifold of solutions
Factorization of differential and difference operators uses analogies between these operators and algebraic polynomials. There is a number of important papers on this subject, of which we only mention a few: [1, 2, 3, 4, 5].
A linear differential (difference) operator L admits factorization if it can be represented as a product of lower order operators of the same type (see [6, 7, 8]). Methods of factorization are exploited in analytic and algebraic approaches to the problem of integration of ordinary differential equations. Many special results are scattered over a large number of research papers; see, for instance, [9, 10, 11, 12] and the references given therein.
where we assume that () are real-valued matrices with the entries being continuous and bounded functions on ℝ and that I is the identity matrix.
We give the necessary and sufficient conditions for factorization of the above operator into the product of lower order factors and . These conditions are connected with the existence of solutions of linear vector differential equations. The results are obtained by the usage of integral manifolds approach in the form elaborated by Valeev in the work .
2 Splitting equations
where for .
Note that the block matrices , , , are , , , matrices, respectively.
We recall here the following definition (see ).
Definition 2.1 The connected subset M of is called the integral manifold of system (4) if implies for all , where and are determined by (4) with , .
where is a real-valued matrix and X, Y satisfy (4) on ℝ, i.e., provided that (6) is satisfied for a certain , then it is valid for all .
i.e., the linear subsystem splits off from the linear system of differential equations (4).
where (; ) are matrices.
The following theorem establishes the existence of an integral manifold of the linear differential equation (3).
Substituting into (3) the derivatives () from (17) and (18), we obtain zero. This proves the theorem. □
3 Existence of the integral manifold of solutions
with a property such that each solution Z of (19) is also a solution of (3) on ℝ. This means that Z has n derivatives on ℝ and matrices , are differentiable up to order p ().
The following result may be proved in much the same way as Theorem 2.2.
Theorem 3.1 If any solution of the linear differential equation (19) with coefficients bounded together with their derivatives up to order p satisfies (3) on ℝ, then the linear system of differential equations (4) has the integral manifold given by (6), where is a matrix.
We substitute an arbitrary but fixed solution Z of (20) into the differential equation (3). For this end, we differentiate (20) p times with respect to t. After each differentiation, we eliminate by (20) and we take into account (10)-(11). In this way we get (17). Similarly, taking into account (12)-(13), we obtain (18). Hence, the existence of solutions of the linear differential equation (19), all solutions of which are the solutions of (3), guarantees the existence of the integral manifold of the form (6) of the linear system (4) provided that all the derivatives up to the order p of matrices () are bounded. This is the desired conclusion. □
Thus, the existence of an integral manifold of the form (6) for the linear system (4) is equivalent to the fact that any solution of (19) satisfies (3) as well.
4 Factorization of the operator
Let the linear vector differential equation (3) be written in the form (4). We assume that any solution of the differential equation (19) is a solution of (3). Then the linear system (4) has the integral manifold of the form (6).
for . Now our main result follows from (33) as the next theorem.
where and are given by (2), if and only if any solution of the linear vector differential equation (19) is a solution of the linear vector differential equation (3) on ℝ.
5 The case of constant coefficients
Let us consider the case when the linear differential equation (3) has constant coefficients. This case is known to be important from the point of view of applications. In this case, from Theorem 4.1 we conclude the following.
From this we have the following corollary.
has the fundamental matrix , which is a solution of the system under condition (34).
Remark 5.4 The main results of this paper were announced in .
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.