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Revisiting Estimates for Domains of Invertibility of Diffeomorphisms

Abstract

This paper aims at revisiting estimates for domains of invertibility of diffeomorphisms. In contrast to previous results, we establish the estimates without the assumption of the underlying function being a diffeomorphism from its domain onto its range, and show that the arguments can be easily exploited to derive an analogous result in the framework of finite-dimensional spaces. Furthermore, the established results provide more estimates of the balls on which diffeomorphisms take place. In addition, we use these results to derive several variants of global inverse function theorem, which extend the known ones. The results presented in the paper are also applied to validate the existence of solutions for nonlinear equations.

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Notes

  1. The [4, formula (46)] should be (24), and the [4, formula (47)] should be

    $$\begin{aligned} \left\| \left[ \mathcal {H}(x,y)\right] ^{-1}\right\| \le \omega (\Vert (x,y)-(x_0,y_0)\Vert ).\end{aligned}$$

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Correspondence to Jinhai Chen.

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Communicated by Viorel Barbu.

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Chen, J., Pesch, H.J. Revisiting Estimates for Domains of Invertibility of Diffeomorphisms. J Optim Theory Appl 165, 1–13 (2015). https://doi.org/10.1007/s10957-014-0632-5

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  • DOI: https://doi.org/10.1007/s10957-014-0632-5

Keywords

  • Diffeomorphism
  • Domain of invertibility
  • Continuation property
  • Existence of solutions

Mathematical Subject Classification

  • 46T20
  • 57R50
  • 47J05
  • 26B10
  • 26B05