Integral equations, implicit functions, and fixed points

The problem is to show that (1) $V(t,x) = S(t, \int _0^t H(t, s, x(s)) \, ds )$ has a solution, where $V$ defines a contraction, $\tilde V$, and $S$ defines a compact map, $\tilde S$. A fixed point of $P \varphi = \tilde S \varphi + (I - \tilde V) \varphi$ would solve the problem. Such equations arise naturally in the search for a solution of $f(t, x) = 0$ where $f(0,0) = 0$, but $\partial f(0,0) / \partial x = 0$ so that the standard conditions of the implicit function theorem fail. Now $P \varphi = \tilde S \varphi + ( I - \tilde V) \varphi$ would be in the form for a classical fixed point theorem of Krasnoselskii if $I - \tilde V$ were a contraction. But $I - \tilde V$ fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that $I - \tilde V$ 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.