Stability of Real Solutions to Nonlinear Equations and Its Applications

We study the stability of solutions to nonlinear equations in finite-dimensional spaces. Namely, we consider an equation of the form $F(x)=\overline {y}$ in the neighborhood of a given solution $\overline {x}$. For this equation we present sufficient conditions under which the equation $F(x)+g(x)=y$ has a solution close to $\overline {x}$ for all $y$ close to $\overline {y}$ and for all continuous perturbations $g$ with sufficiently small uniform norm. The results are formulated in terms of $\lambda $-truncations and contain applications to necessary optimality conditions for a conditional optimization problem with equality-type constraints. We show that these results on $\lambda $-truncations are also meaningful in the case of degeneracy of the linear operator $F'(\overline {x})$.