On stability of smooth nonlinear mappings at a given point

A. V. Arutyunov, S. E. Zhukovskiy

V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow

Abstract: 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.

UDC: 517.275

MSC: 26B10

Received: 20.04.2025
Revised: 14.05.2025
Accepted: 19.05.2025

DOI: 10.21538/0134-4889-2025-31-2-30-37

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