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 (x). To do this, the concept of -truncation of the map F in a neighbourhood of the point x is introduced and examined. A theorem on the uniqueness of a -truncation is proved. The regularity condition is introduced for -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.