Vol. 54, No. 1, 1974

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A unified approach to boundary value problems on compact intervals

Franklin Takashi Iha

Vol. 54 (1974), No. 1, 145–156

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,l} be a set of linear functionals defined on Cn([a,b]) with the property that each ηjT,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 ηj(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 {ηf} 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
Received: 8 May 1973
Revised: 29 October 1973
Published: 1 September 1974
Franklin Takashi Iha