A unified approach to boundary value problems on compact intervals

Franklin Takashi Iha

Vol. 54 (1974), No. 1, 145–156

DOI: 10.2140/pjm.1974.54.145

Abstract

Let L be a formal differential operator of order n and consider L as an operator from Cⁿ([a,b]) ⊂ L²([a,b]) into L²([a,b]). Let {η₁,⋯,η_l} be a set of linear functionals defined on Cⁿ([a,b]) with the property that each η_jT,j = 1,⋯,l, is continuous, where T is a continuous right inverse of L. Let M be the set of all f ∈ Cⁿ([a,b]) such that η_j(f) = 0, 1 ≦ j ≦ l, and N be the set of all f ∈ M such that Lf = 0. It is shown that the inverse of L from L(M), the image of M under L, into 1ψ ∩ N^⊥ is a compact operator and can be represented as an integral operator. In particular, if l = n and {η_f} is linearly independent, the inverse of L maps C([a,b]) onto M and it is compact. The Hilbert-Schmidt expansion theorem is generalized to these inverse operators when L is self-adjoint on M.

Mathematical Subject Classification 2000

Primary: 34B05

Secondary: 47E05

Milestones

Received: 8 May 1973

Revised: 29 October 1973

Published: 1 September 1974