Some remarks on the solvability of a nonlinear equations and existence of the fixed points

Abstract

A theorem showing how to locally determine the image of some subset under a continuous mapping acting in a Banach space is obtained, as well as a solvability theorem. Furthermore, some results about the completeness of the image of a continuous mapping are presented, and a solvability theorem is proved for the boundary value problem for one class of fully nonlinear ordinary differential equations.