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.
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Acknowledgements
We thank the referees for their useful comments. We are specially grateful for pointing out reference [15] in connection with Theorem 3.1. Research of the first author is supported by the Russian Foundation for Basic Research Grant (Projects No. 18-01-00106 and 19-01-00080). The second author is supported in part by the Russian Foundation for Basic Research Grants 19-51-12003 and 20-01-00106, and by the Volkswagen Foundation. The third author is supported by the Russian Science Foundation (Project No. 20-11-20131).
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Arutyunov, A.V., Izmailov, A.F. & Zhukovskiy, S.E. Continuous Selections of Solutions for Locally Lipschitzian Equations. J Optim Theory Appl 185, 679–699 (2020). https://doi.org/10.1007/s10957-020-01674-1
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DOI: https://doi.org/10.1007/s10957-020-01674-1
Keywords
- Nonlinear equation
- Locally Lipschitzian mapping
- Clarke’s generalized Jacobian
- Inverse function theorem
- Continuous selection of solutions
- Hadamard’s theorem