Integral equations, implicit functions, and fixed points

The problem is to show that (1) V(t, x) = S(t, integral(0)(t) H(t, s, x(s)) ds) has a solution, where V defines a contraction, (V) over tilde, and S defines a compact map, (S) over tilde. A fixed point of P-phi = (S) over tilde phi + (1 - (V) over tilde)(phi) 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 derivative f(0, 0)/partial derivative x = 0 so that the standard conditions of the implicit function theorem fail. Now P-phi = (S) over tilde(phi) + (I - (V) over tilde)(phi) would be in the form for a classical fixed point theorem of Krasnoselskii if I - (V) over tilde were a contraction. But I - (V) over tilde fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that I - (V) over tilde 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.