Vol. 18, No. 1, 1966

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ISSN: 0030-8730
The lack of self-adjointness in three-point boundary value problems

John William Neuberger

Vol. 18 (1966), No. 1, 165–168
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

  [       ]    [        ]    [       ]  [ ]
f(a)         f (c)          f(b)      0
P  p(a)f′(a) + Q p(c)f′(c) + S  p(b)f′(b) =  0         (Δ )

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

      ∫ b
f(x) =   K12(x,t)g(t)dt  for all x in [a,b].
a

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
Authors
John William Neuberger