On Existence and Uniqueness Results for Nonsmooth Implicit Differential Equations

The classical implicit function theorem gives conditions that the function is Frechet differentiable and the derivative is surjective. In this short article they are generalized to conditions of Lipschitz and monotone type. The newly obtained implicit function theorems are used to derive two sets of sufficient conditions for the existence and uniqueness of solutions to the initial value problems of nonsmooth implicit differential equations.