Stability of Real Solutions to Nonlinear Equations and Its Applications

IF 0.4 4区数学 Q4 MATHEMATICS

Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI:10.1134/s0081543823050012

A. V. Arutyunov, S. E. Zhukovskiy

引用次数: 0

Abstract

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 constrained 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 \(F'(\overline{x})\).

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非线性方程实解的稳定性及其应用

摘要我们研究有限维空间中非线性方程解的稳定性。也就是说，我们考虑在给定解 \(\overline{x}\)的邻域内形式为 \(F(x)=\overline{y}\)的方程。对于这个方程，我们提出了充分条件，在这些条件下，方程 \(F(x)+g(x)=y\) 对于所有接近 \(\overline{x}\) 的 \(y\) 和所有具有足够小的均匀规范的连续扰动 \(g\) 都有一个接近 \(\overline{x}\) 的解。这些结果是用\(\lambda\)-截断来表述的，并且包含了对具有相等类型约束的约束优化问题的必要最优条件的应用。我们证明了这些关于截断的结果在线性算子 \(F'(\overline{x})\)退化的情况下也是有意义的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Proceedings of the Steklov Institute of Mathematics MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

0.90

自引率

20.00%

发文量

审稿时长

4-8 weeks

期刊介绍： Proceedings of the Steklov Institute of Mathematics is a cover-to-cover translation of the Trudy Matematicheskogo Instituta imeni V.A. Steklova of the Russian Academy of Sciences. Each issue ordinarily contains either one book-length article or a collection of articles pertaining to the same topic.