Integral equations, implicit functions, and fixed points

The problem is to show that (1) has a solution, where defines a contraction, , and defines a compact map, . A fixed point of would solve the problem. Such equations arise naturally in the search for a solution of where , but so that the standard conditions of the implicit function theorem fail. Now would be in the form for a classical fixed point theorem of Krasnoselskii if were a contraction. But fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that 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.