Geodesic
Stability of Real Solutions to Nonlinear Equations and Its Applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Theory of Functions of Several Real Variables and Its Applications, Tome 323 (2023), pp. 5-16.

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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 $F'(\overline {x})$.
Keywords: $\lambda $-truncated mappings, directionally regular $\lambda $-truncation, necessary minimum condition, nonlinear equation, $2$-regularity. 
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A. V. Arutyunov; S. E. Zhukovskiy. Stability of Real Solutions to Nonlinear Equations and Its Applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Theory of Functions of Several Real Variables and Its Applications, Tome 323 (2023), pp. 5-16. http://geodesic.mathdoc.fr/item/TM_2023_323_a0/

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