Implicit Functions and Solution Mappings pp 61-130 | Cite as

# Implicit Function Theorems for Variational Problems

## Abstract

Solutions mappings in the classical setting of the implicit function theorem concern problems in the form of parameterized equations. The concept can go far beyond that, however. In any situation where some kind of problem in *x* depends on a parameter *p*, there is the mapping *S* that assigns to each *p* the corresponding set of solutions *x*. The same questions then arise about the extent to which a localization of *S* around a pair \((\bar{p},\bar{x})\) in its graph yields a function *s* which might be continuous or differentiable, and so forth.

This chapter moves into that much wider territory in replacing equation-solving problems by more complicated problems termed “generalized equations.” Such problems arise variationally in constrained optimization, models of equilibrium, and many other areas. An important feature, in contrast to ordinary equations, is that functions obtained implicitly from their solution mappings typically lack differentiability, but often exhibit Lipschitz continuity and sometimes combine that with the existence of one-sided directional derivatives.