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This paper treats linear
quasi-differential operators of the form
![∑n (j) (∑n (j) ( (∑n (j))′ )′)′
L [y] = p0jy − p1jy − ⋅⋅⋅− pmjy ⋅⋅⋅ ,
j=0 j=0 j=0](a030x.png)
based on an integrable (m + 1) × (n + 1) matrix function [pij], (i = 0,⋯,m;j = 0,⋯,n),
about which suitable regularity assumptions are made. Results obtained by Reid
(Trans. Amer. Math. Soc. Vol. 85 (1957), pp. 446–461) are extended to operators
of the type considered here.
A generalized Green’s function for the system {L[y] = 0,y ∈𝒟} is defined, where
𝒟 is a linear subspace of the domain of L. Resolvent and deterministic properties of
this function are presented, together with the relationship of such a generalized
Green’s function to the generalized Green’s function for the associated adjoint
system.
For a large class of two-point boundary problems in which the boundary
conditions involve the characteristic parameter linearly it is shown that there exists a
simultaneous canonical representation of the boundary conditions for a given problem
and those of its adjoint; in particular, in the self-adjoint case this canonical
representation has the form of boundary conditions and transversality conditions for
a variational problem. Finally, these results are applied to a two-point boundary
problem involving a differential operator of the type considered in the paper of Reid
above.
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