Suppose that a < c < b, C_{[a,b]} is
the set of all realvalued 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 realvalued continuous function K_{12} 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 K_{12} is not symmetric, i.e., it is not true that K_{12}(x,t) = K_{12}(t,x)
for all x, t in [a,b].
