In the present paper we shall examine linear boundary value problems for linear equations of order N 2 2 in a right invertible operator. Particular cases with rather strong assumptions on the existence of resolving operators have been considered by BERG [l], [2], TASCHE [l] and D. PRZEWORSKA-ROLEWICZ [I] (c.f. also related results of DIMOVSKI [l], DIMOVSKI and PETROVA [l], JODAR [l], KORNACKI [l], MIELOSZYK [l], MILMAN [l]). A construction of resolving operators of linear boundary value problems for a wide class of initial operators and equations with scalar and stationary coefticients was given recently by the second author in [4]. This construction has been modified and applied by the first author to boundary value problems of a much more general form (KARWOWSKI, [l]).
In the present paper we shall construct the so-called GREEN operator for an arbitrary power DN of a given right invertible operator D. Having already constructed this operator, we can solve in an explicit form boundary value problems of various types for the operator DN. The results obtained here will be applied in the next papers to boundary value problems for polynomials with scalar, stationary and operator coefficients by a reduction of these problems to "integral" equations. i.e. equations in right inverses. Through the paper, for the sake of brevity, the results of the authors will be cited with their initials. Also all references to the book [2] of the second author will be denoted by "AA'.