We show that there is a solution
f
f
of the equation
\[
f
(
x
)
=
f
(
a
)
+
(
R
L
)
∫
a
x
(
f
H
+
f
G
)
f(x) = f(a) + (RL)\int _a^x {(fH + fG)}
\]
such that
f
(
p
)
=
0
f(p) = 0
and
f
(
q
)
≠
0
f(q) \ne 0
for some pair
p
,
q
∈
[
a
,
b
]
p,q \in [a,b]
iff there is a number
t
∈
[
a
,
b
]
t \in [a,b]
such that one of
1
−
H
(
t
−
,
t
)
,
1
−
H
(
t
,
t
+
)
,
1
+
G
(
t
−
,
t
)
1 - H({t^ - },t),1 - H(t,{t^ + }),1 + G({t^ - },t)
or
1
+
G
(
t
,
t
+
)
1 + G(t,{t^ + })
is zero or a right divisor of zero, where
f
,
G
f,G
and
H
H
are functions of bounded variation with ranges in a normed ring
N
N
. Furthermore, if
N
N
is a field, then for each discontinuity of
H
H
on
[
a
,
b
]
[a,b]
there exists
λ
∈
N
\lambda \in N
and a finite set of linearly independent nonzero solutions on
[
a
,
b
]
[a,b]
of the equation
f
(
x
)
=
f
(
a
)
+
(
R
L
)
∫
a
x
(
f
H
+
f
G
)
λ
f(x) = f(a) + (RL)\int _a^x {(fH + fG)\lambda }
such that if
f
f
is a solution and has bounded variation on
[
a
,
b
]
[a,b]
, then
f
f
is a linear combination of this set of solutions. Product integrals are used extensively in the proofs.