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Continuous Selections of Solutions for Locally Lipschitzian Equations

Author

Listed:
  • Aram V. Arutyunov

    (V.A. Trapeznikov Institute of Control Sciences of RAS)

  • Alexey F. Izmailov

    (Lomonosov Moscow State University, MSU)

  • Sergey E. Zhukovskiy

    (V.A. Trapeznikov Institute of Control Sciences of RAS)

Abstract

This paper answers in the affirmative the long-standing question of nonlinear analysis, concerning the existence of a continuous single-valued local selection of the right inverse to a locally Lipschitzian mapping. Moreover, we develop a much more general result, providing conditions for the existence of a continuous single-valued selection not only locally, but rather on any given ball centered at the point in question. Finally, by driving the radius of this ball to infinity, we obtain the global inverse function theorem, essentially implying the well-known Hadamard’s theorem on a global homeomorphism for smooth mappings and the more general Pourciau’s theorem for locally Lipschitzian mappings.

Suggested Citation

  • Aram V. Arutyunov & Alexey F. Izmailov & Sergey E. Zhukovskiy, 2020. "Continuous Selections of Solutions for Locally Lipschitzian Equations," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 679-699, June.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:3:d:10.1007_s10957-020-01674-1
    DOI: 10.1007/s10957-020-01674-1
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    References listed on IDEAS

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    1. Hoa T. Bui & Scott B. Lindstrom & Vera Roshchina, 2019. "Variational Analysis Down Under Open Problem Session," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 430-437, July.
    2. DORSCH, Dominik & JONGEN, Hubertus & RUCKMANN, Jan & SHIKHMAN, Vladimir, 2014. "On the local representation of piecewise smooth equations as a Lipschitz manifold," LIDAM Reprints CORE 2656, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Aram V. Arutyunov & Sergey E. Zhukovskiy, 2023. "Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 112-142, October.

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