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].
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