Skip to main content
SpringerLink
Log in

Revisiting Estimates for Domains of Invertibility of Diffeomorphisms

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

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

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

References

  1. Rheinboldt, W.C.: Local mapping relations and global implicit function theorems. Trans. Am. Math. Soc. 138, 183–198 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  3. Plastock, R.: Homeomorphisms between Banach spaces. Trans. Am. Math. Soc. 200, 169–183 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chen, J.: Estimating invertible domains of diffeomorphisms. J. Optim. Theory Appl. 154, 818–829 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mustafa, O.G., Rogovchenko, Y.V.: Estimates for domains of local invertibility of diffeomorphisms. Proc. Am. Math. Soc. 135, 69–75 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Sotomayor, J.: Inversion of smooth mappings. Z. Angew. Math. Phys. 41, 306–310 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cristea, M.: A note on global inversion theorems and applications to differential equations. Nonlinear Anal. 5, 1155–1161 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rădulescu, M., Rădulescu, S.: Global inversion theorems and applications to differential equations. Nonlinear Anal. 4, 951–965 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rădulescu, M., Rădulescu, S.: Local inversion theorems without assuming continuous differentiability. J. Math. Anal. Appl. 138, 581–590 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  10. Freund, R.M., Mizuno, S.: Interior point methods: current status and future directions. In: Frenk, H., et al. (eds.) High Performance Optimization, pp. 441–466. Kluwer Acad. Publ., Dordrecht (2000)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Rochester, Rochester, NY, 14642, USA

    Jinhai Chen

  2. University of Bayreuth, 95440, Bayreuth, Germany

    Hans Josef Pesch

Authors
  1. Jinhai Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Josef Pesch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinhai Chen.

Additional information

Communicated by Viorel Barbu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Mathematical Subject Classification

Search

Navigation