We consider the equation $F(x)+\Phi(x)=y.$ Here $F: \mathbb{R}^n \to \mathbb{R}^m$ is a nonlinear smooth mapping, $x$ is unknown, $\Phi$ is a continuous mapping, $y$ is a vector. Using $\lambda$-truncations we obtain conditions for the equation to have a solution $x(y,\Phi)$ close to the given point $\bar x$. The perturbation $\Phi$ is assumed to be sufficiently small around $\bar x$ in the uniform convergence metric, and the perturbation $y$ is assumed to be close to $F(\bar x)$. We derive a priori estimates of the solution $x(y,\Phi).$
Keywords: stability of a mapping at a point, inverse function, $\lambda$-truncation
Received April 20, 2025
Revised May 14, 2025
Accepted May 19, 2025
Funding Agency: This work was supported by Russian Science Foundation, project 24-21-00012,
https://rscf.ru/project/24-21-00012/.
Aram V. Arutyunov, Dr. Phys.-Math. Sci., V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, 117997 Russia, е-mail: arutyunov@cs.msu.ru
Sergey E. Zhukovskiy, Dr. Phys.-Math. Sci., V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, 117997 Russia, e-mail: s-e-zhuk@yandex.ru
REFERENCES
1. Arutyunov A.V. Existence of real solutions of nonlinear equations without a priori normality assumptions. Math. Notes, 2021, vol. 109, no. 1, pp. 3–14. https://doi.org/10.1134/S0001434621010016
2. Arutyunov A.V., Zhukovskiy S.E. Solvability of nonlinear degenerate equations and estimates for inverse functions. Sb. Math., 2025, vol. 216, no. 1, pp. 1–24. https://doi.org/10.4213/sm10060e
3. Arutyunov A.V., Zhukovskiy S.E. Stability of real solutions to nonlinear equations and its applications. Proc. Steklov Inst. Math., 2023, vol. 323, no. 1, pp. 1–11. https://doi.org/10.1134/S0081543823050012
4. Kozlov V.V. On real solutions of systems of equations. Funct. Anal. Its Appl., 2017, vol. 51, no. 4, pp. 306–309. https://doi.org/10.1007/s10688-017-0197-9
5. Chebotarev N.G. Newton’s polygon and its role in modern development of mathematics. In: Isaak N’yuton (1643–1727). Sbornik Statey k Trokhsotletiyu so Dnya Rozhdeniya [Isaac Newton (1643–1727). Collection of articles for the tercentenary of his birth], ed. by Academician S. I. Vavilov, Akad. Nauk SSSR, 1943, pp. 99–126.
6. Bartle R.G., Graves L.M. Mappings between function spaces. Trans. Amer. Math. Soc., 1952, vol. 72, no. 3, pp. 400–413. https://doi.org/10.2307/1990709
7. Dontchev A.L., Rockafellar R.T. Implicit functions and solution mappings. NY, Springer, 2009, 376 p. https://doi.org/10.1007/978-0-387-87821-8
8. Arutyunov A.V., Zhukovskiy S.E. Applications of λ-truncations to the study of local and global solvability of nonlinear equations. Eurasian Math. J., 2024, vol. 15, no. 1, pp. 23–33.
https://doi.org/10.32523/2077-9879-2024-15-1-23-33
9. Arutyunov A.V., Zhukovskiy S.E. On a class of regular λ-truncations and its application to inverse functions. Mat. Notes, 2025, vol. 117, no. 5, pp. 637–652 (in Russian).
Cite this article as: Aram V.Arutyunov, Sergey E.Zhukovskiy. On stability of smooth nonlinear mappings at a given point. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 30–37.
