Vol. 16, No. 2, 1966

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Adjoint quasi-differential operators of Euler type

John Spurgeon Bradley

Vol. 16 (1966), No. 2, 213–237

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

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.

Mathematical Subject Classification
Primary: 34.30
Received: 21 May 1964
Published: 1 February 1966
John Spurgeon Bradley