Stability of Real Solutions to Nonlinear Equations and Its Applications

Abstract

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 constrained 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 \(F'(\overline{x})\).