The problem is to show that (1)
V
(
t
,
x
)
=
S
(
t
,
∫
0
t
H
(
t
,
s
,
x
(
s
)
)
d
s
)
V(t,x) = S(t, \int _0^t H(t, s, x(s)) \, ds )
has a solution, where
V
V
defines a contraction,
V
~
\tilde V
, and
S
S
defines a compact map,
S
~
\tilde S
. A fixed point of
P
φ
=
S
~
φ
+
(
I
−
V
~
)
φ
P \varphi = \tilde S \varphi + (I - \tilde V) \varphi
would solve the problem. Such equations arise naturally in the search for a solution of
f
(
t
,
x
)
=
0
f(t, x) = 0
where
f
(
0
,
0
)
=
0
f(0,0) = 0
, but
∂
f
(
0
,
0
)
/
∂
x
=
0
\partial f(0,0) / \partial x = 0
so that the standard conditions of the implicit function theorem fail. Now
P
φ
=
S
~
φ
+
(
I
−
V
~
)
φ
P \varphi = \tilde S \varphi + ( I - \tilde V) \varphi
would be in the form for a classical fixed point theorem of Krasnoselskii if
I
−
V
~
I - \tilde V
were a contraction. But
I
−
V
~
I - \tilde V
fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that
I
−
V
~
I - \tilde V
has enough properties that an extension of Krasnoselskii’s theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.