On Solvability of Equations Defined by Continuous and Smooth Regular Mappings

Abstract

We consider equations defined by continuous mappings acting between finite-dimensional real vector spaces. It is assumed that the mappings are differentiable in the first variable. A regularity condition for this type of equations is obtained. It is shown that the regularity assumption implies the existence of solutions to the considered equations. Systems of two equations defined by continuous mappings acting between finite-dimensional real vector spaces are considered. It is assumed that the first mapping is differentiable in the first variable and the second mapping is differentiable in the second variable. A regularity condition for this type of systems is obtained. It is shown that the regularity assumption implies the existence of solutions to the considered system. The proofs of the main results of the paper are based on Brouwer's fixed point theorem and global implicit function theorem.