Integral equations, implicit functions, and fixed points

Abstract

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.