$p$-Regularity Theory. Tangent Cone Description in the Singular Case

Abstract

We present a new proof of the theorem which is one of the main results of the $p$-regularity theory. This gives us a detailed description of the structure of the zero set of a singular nonlinear mapping. We say that $F : X → Y$ is singular at some point $x_0$, where $X$ and $Y$ are Banach spaces, if Im $F′(x_0) ≠ Y$. Otherwise, the mapping $F$ is said to be regular.