Implicit function theorems for nonregular mappings in Banach spaces. Exit from singularity

摘要

We consider the equation F(x, y) = 0, where F : X × Y→ Z is a smooth mapping,and X, Y and Z are Banach spaces. In the case when F(x~* , y~* ) = 0 and the mapping F is regular at (x~*, y~*), i.e., when F'_y (x~* , y~*), the derivative of F with respect to y, is invertible, the classical implicit function theorem guarantees the existence of a smooth mapping Φ defined on a neighborhood of x~* such that F (x , Φ(x)) = 0 and Φ(x~*) = y~*. We are interested in the case when the mapping F is nonregular and the classical implicit function theorem is not applicable. We present generalizations of the implicit function theorem for this case. The results are illustrated by some examples, including differential equations.