Factorization of the linear differential operator
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RESEARCH
Open Access
Factorization of the linear differential operator Klara R Janglajew1* and Kim G Valeev2 *
Correspondence: [email protected] 1 Institute of Mathematics, University of Białystok, Białystok, Poland Full list of author information is available at the end of the article
Abstract 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 Keywords: linear differential equations; differential operator; factorization; integral manifold of solutions
1 Introduction 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: [–]. 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 [–]). 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, [–] and the references given therein. In this paper, we focus on an nth order linear differential operator of the form Ln (t, D) := IDn +
n–
Ak (t)Dk ,
k=
Dk :=
dk , dt k
()
where we assume that Ak (t) (k = , , . . . , n – ) are m × m real-valued matrices with the entries being continuous and bounded functions on R and that I is the m × m identity matrix. Our purpose here is to give a proof that () can be presented as Ln (t, D) = Lp (t, D)Lq (t, D), where p + q = n and Lp (t, D) = IDp +
p– k=
Bk (t)Dk ,
Lq (t, D) = IDq +
q–
Ck (t)Dk .
()
k=
© 2013 Janglajew and Valeev; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Janglajew and Valeev Advances in Difference Equations 2013, 2013:237 http://www.advancesindifferenceequations.com/content/2013/1/237
Page 2 of 12
We give the necessary and sufficient conditions for factorization of the above operator Ln into the product of lower order factors Lq and Lp . 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 Let us consider the linear differential equation of order n, formed by acting the operator () on a vector function Z: n
Ln (t, D)Z(t) = ID +
n–
Ak (t)D
k
Z(t) = ,
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