Applications of $\lambda$-truncations to the study of local and global solvability of nonlinear equations

In this paper, we consider the equation $F(x)=y$ in a neighbourhood of a given point $\bar{x}$, where $F$ is a given continuous mapping between finite-dimensional real spaces. We study a class of polynomial mappings and show that these polynomials satisfy certain regularity assumptions. We show that if a $\lambda$-truncation of $F$ at $\bar{x}$ belongs to the considered class of polynomial mappings then for every y close to $F(\bar{x})$ there exists a solution to the equation $F(x) = y$ that is close to $\bar{x}$. For polynomial mappings satisfying the regularity conditions we study their stability to bounded continuous perturbations.