The theorem on existence of singular solutions to nonlinear equations

The aim of this paper is to present some applications of $p$-regularity theory to investigations of nonlinear multivalued mappings. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case when the linear part is singular at the considered initial point. We formulate conditions for existence of solutions of equation $F(x)=0$ when first $p-1$ derivatives of $F$ are singular.