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Nonlocal Generalized Implicit Function Theorems in Hilbert Spaces

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Abstract

A nonlocal generalized implicit function theorem is obtained for mappings acting in Hilbert spaces. Its proof is based on the theory of ordinary differential equations. Under natural assumptions, we use this theorem to derive a global generalized implicit function theorem as well as a global implicit function theorem and a global inverse function theorem as particular cases of the former. Using the estimates produced, we prove an assertion on the continuation of an implicit function from a given closed set to the entire parameter space.

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Funding

This work was supported by the Russian Foundation for Basic Research, projects nos. 18-01-00106 and 19-01-00080. The results in Sec. 4 were obtained by the authors with the support of the Russian Science Foundation, project no. 20-11-20131.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    A. V. Arutyunov & S. E. Zhukovskiy

  2. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, 127051, Russia

    A. V. Arutyunov

Authors
  1. A. V. Arutyunov
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  2. S. E. Zhukovskiy
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Corresponding authors

Correspondence to A. V. Arutyunov or S. E. Zhukovskiy.

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Translated by V. Potapchouck

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Arutyunov, A.V., Zhukovskiy, S.E. Nonlocal Generalized Implicit Function Theorems in Hilbert Spaces. Diff Equat 56, 1525–1538 (2020). https://doi.org/10.1134/S00122661200120010

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  • DOI: https://doi.org/10.1134/S00122661200120010

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