Implicit Function Theorem in a Neighborhood of an Abnormal Point

Recieved: 05/23/2023

Accepted: 05/30/2023

Published: 09/20/2023

Keywords: implicit function, abnormal point, nonlinear equation, real solution, truncation, regularity in direction

To cite this article

Arutyunov A.V., Salikhova K. I. Implicit Function Theorem in a Neighborhood of an Abnormal Point. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2023. N 3, p.3-9 https://doi.org/10.55959/MSU/0137–0782–15–2023–47–3–3–9.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Arutyunov A.V. (V. A. Trapeznikov Institute of Management Problems of the Russian Academy of Sciences)
Salikhova K. I. (Lomonosov Moscow State University (MSU))
(V. A. Trapeznikov Institute of Management Problems of the Russian Academy of Sciences)

Abstract

We study the existence of an implicit function, defined by an equation G(x, σ) = 0, in a neighborhood of an abnormal point (x0, σ0). We prove that if some λ-truncation of the mapping F(x) = G(x, σ0) is regular in a certain direction, then the sought implicit function exists.