Solvability of nonlinear degenerate equations and estimates for inverse functions

For a continuous map $F$ from a finite-dimensional real space to another such space the question of the solvability of the nonlinear equation of the form $F(x)=y$ is investigated for $y$ close to a fixed value $F(\overline x)$. To do this, the concept of $\lambda$-truncation of the map $F$ in a neighbourhood of the point $\overline x$ is introduced and examined. A theorem on the uniqueness of a $\lambda$-truncation is proved. The regularity condition is introduced for $\lambda$-truncations; it is shown to be sufficient for the solvability of the equation in question. A priori estimates for the solution are obtained. Bibliography: 16 titles.