Applications

There is a strong connection between the implicit function theorem and the theory of differential equations. This is true even from the historical point of view, for Pi-card’s iterative proof of the existence theorem for ordinary differential equations inspired Goursat to give an iterative proof of the implicit function theorem (see Goursat [Go 03]). In the mid-twentieth century, John Nash pioneered the use of a sophisticated form of the implicit function theorem in the study of partial differential equations. We will discuss Nash’s work in Section 6.4. In this section, we limit our attention to ordinary (rather than partial) differential equations because the technical details are then so much simpler. Our plan is first to show how a theorem on the existence of solutions to ordinary differential equations can be used to prove the implicit function theorem. Then we will go the other way by using a form of the implicit function theorem to prove an existence theorem for differential equations.