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)=y in the neighborhood of a given solution x . For this equation we present sufficient conditions under which the equation F(x)+g(x)=y has a solution close to x for all y close to y and for all continuous perturbations g with sufficiently small uniform norm. The results are formulated in terms of λ -truncations and contain applications to necessary optimality conditions for a constrained optimization problem with equality-type constraints. We show that these results on λ -truncations are also meaningful in the case of degeneracy of the linear operator F'(x) .