The lack of self-adjointness in three-point boundary value problems

John William Neuberger

Vol. 18 (1966), No. 1, 165–168

DOI: 10.2140/pjm.1966.18.165

Abstract

Suppose that a < c < b, C_[a,b] is the set of all real-valued continuous functions on [a,b], each of p and q is in C_[a,b], p(x) > 0 for all x in [a,b] and each of P, Q and S is a real 2 ×2 matrix. The assumption is made that the only member f of C_[a,b] so that (pf′)′− qf = 0 and

is the zero function. It follows that there is a real-valued continuous function K₁₂ on [a,b] × [a,b] such that if g is in C_[a,b], then the only element f of C_[a,b] so that (pf′)′− qf = g and (Δ) holds is given by

In this note it is shown that if in addition it is specified that Q is not the zero 2 ×2 matrix, then K₁₂ is not symmetric, i.e., it is not true that K₁₂(x,t) = K₁₂(t,x) for all x, t in [a,b].

Mathematical Subject Classification

Primary: 34.36

Milestones

Received: 1 February 1965

Published: 1 July 1966