Geodesic
Applications of $\lambda$-truncations to the study of local and global solvability of nonlinear equations
Eurasian mathematical journal, Tome 15 (2024) no. 1, pp. 23-33 
A. V. Arutyunov; S. E. Zhukovskiy. Applications of $\lambda$-truncations to the study of local and global solvability of nonlinear equations. Eurasian mathematical journal, Tome 15 (2024) no. 1, pp. 23-33. http://geodesic.mathdoc.fr/item/EMJ_2024_15_1_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Russian Math. Surveys, 67:3 (2012), 403–457 | DOI | MR

[2] A. V. Arutyunov, “Existence of real solutions of nonlinear equations without a priori normality assumptions”, Math. Notes, 109:1 (2021), 3–14 | DOI | MR | Zbl

[3] A. V. Arutyunov, S. E. Zhukovskiy, “On exact penalties for constrained optimization problems in metric spaces”, Eurasian Math. J., 12:4 (2021), 10–20 | DOI | MR | Zbl

[4] A. L. Dontchev, R. T. Rockafellar, Implicit functions and solution mappings, Springer, N. Y., 2009 | MR | Zbl

[5] B. D. Gel'man, V. V. Obukhovskii, “On fixed points of acyclic type multivalued maps”, J. Math. Sci, 225 (2017), 565–574 | DOI | MR

[6] A. F. Izmailov, “Accelerating convergence of a globalized sequential quadratic programming method to critical Lagrange multipliers”, Computational Optimization and Applications, 80 (2021), 943–978 | DOI | MR | Zbl

[7] L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, N.Y., 2001 | MR | Zbl