Scholarlink

Abstract <p> We consider the equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(F(x)+\Phi(x)=y\)</EquationSource> </InlineEquation>. Here <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(F \colon\, \mathbb{R}^n \to \mathbb{R}^m\)</EquationSource> </InlineEquation> is a nonlinear smooth mapping, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(x\)</EquationSource> </InlineEquation> is the unknown, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Phi\)</EquationSource> </InlineEquation> is a continuous mapping, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(y\)</EquationSource> </InlineEquation> is a vector. Using <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\lambda\)</EquationSource> </InlineEquation>-truncations we obtain conditions for the equation to have a solution <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(x(y,\Phi)\)</EquationSource> </InlineEquation> close to a given point <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\overline x\)</EquationSource> </InlineEquation>. The perturbation <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\Phi\)</EquationSource> </InlineEquation> is assumed to be sufficiently small in a given neighborhood of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\overline x\)</EquationSource> </InlineEquation> in the uniform metric, and the perturbation <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(y\)</EquationSource> </InlineEquation> is assumed to be close to <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(F(\overline x)\)</EquationSource> </InlineEquation>. We also derive a priori estimates for the solution <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(x(y,\Phi)\)</EquationSource> </InlineEquation>. </p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On Stability of Smooth Nonlinear Mappings at a Given Point

  • 期刊论文

  • 发表日期：

    2025年8月1日

查看全文
  • A. V. Arutyunov,
  • S. E. Zhukovskiy

摘要

Abstract

We consider the equation \(F(x)+\Phi(x)=y\) . Here \(F \colon\, \mathbb{R}^n \to \mathbb{R}^m\) is a nonlinear smooth mapping, \(x\) is the unknown, \(\Phi\) is a continuous mapping, and \(y\) is a vector. Using \(\lambda\) -truncations we obtain conditions for the equation to have a solution \(x(y,\Phi)\) close to a given point \(\overline x\) . The perturbation \(\Phi\) is assumed to be sufficiently small in a given neighborhood of \(\overline x\) in the uniform metric, and the perturbation \(y\) is assumed to be close to \(F(\overline x)\) . We also derive a priori estimates for the solution \(x(y,\Phi)\) .