Abstract:
For nonlinear mappings acting in Banach spaces, we examine inverse and implicit function theorems under various smoothness assumptions. For various regularity (normality) conditions imposed on such mappings, we prove that the corresponding equations have solutions under any sufficiently small (in the norm) completely continuous perturbations. A priori estimates for these solutions are obtained.
This work was supported by the Volkswagen Foundation. The research presented in Section 2 was performed by the second author and supported by the Russian Science Foundation under grant 20-11-20131.
Citation:
A. V. Arutyunov, S. E. Zhukovskiy, “Stable Solvability of Nonlinear Equations under Completely Continuous Perturbations”, Function Spaces, Approximation Theory, and Related Problems of Analysis, Collected papers. In commemoration of the 115th anniversary of Academician Sergei Mikhailovich Nikol'skii, Trudy Mat. Inst. Steklova, 312, Steklov Math. Inst., Moscow, 2021, 7–21; Proc. Steklov Inst. Math., 312 (2021), 1–15
\Bibitem{AruZhu21}
\by A.~V.~Arutyunov, S.~E.~Zhukovskiy
\paper Stable Solvability of Nonlinear Equations under Completely Continuous Perturbations
\inbook Function Spaces, Approximation Theory, and Related Problems of Analysis
\bookinfo Collected papers. In commemoration of the 115th anniversary of Academician Sergei Mikhailovich Nikol'skii
\serial Trudy Mat. Inst. Steklova
\yr 2021
\vol 312
\pages 7--21
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4149}
\crossref{https://doi.org/10.4213/tm4149}
\elib{https://elibrary.ru/item.asp?id=46018167}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2021
\vol 312
\pages 1--15
\crossref{https://doi.org/10.1134/S0081543821010016}
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Linking options:
https://www.mathnet.ru/eng/tm4149
https://doi.org/10.4213/tm4149
https://www.mathnet.ru/eng/tm/v312/p7
This publication is cited in the following 2 articles:
N. S. Borzov, T. V. Zhukovskaya, I. D. Serova, “Obyknovennye differentsialnye uravneniya i differentsialnye uravneniya s zapazdyvaniem: obschie svoistva i osobennosti”, Vestnik rossiiskikh universitetov. Matematika, 28:142 (2023), 137–154
A. V. Arutyunov, S. E. Zhukovskiy, “On implicit function theorem for locally Lipschitz equations”, Math. Program., 198 (2022), 1107–1120