Continuous Selections of Solutions for Locally Lipschitzian Equations

Aram V. Arutyunov (), Alexey F. Izmailov () and Sergey E. Zhukovskiy ()
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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

Journal of Optimization Theory and Applications, 2020, vol. 185, issue 3, No 1, 679-699

Abstract: 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.

Keywords: Nonlinear equation; Locally Lipschitzian mapping; Clarke’s generalized Jacobian; Inverse function theorem; Continuous selection of solutions; Hadamard’s theorem; 47J05; 47J07; 49J52; 49J53; 58C15 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-020-01674-1

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