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Inversion is Continuous in a Banach Space

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Inversion of Nonsmooth Maps between Banach Spaces

Set-Valued and Variational Analysis volume 27,pages 921–947 (2019)Cite this article

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Abstract

We study the invertibility nonsmooth maps between infinite-dimensional Banach spaces. To this end, we introduce an analogue of the notion of pseudo-Jacobian matrix of Jeyakumar and Luc in this infinite-dimensional setting. Using this, we obtain several inversion results. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.

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Acknowledgments

We would like to thank an anonymous referee for pointing out a mistake in a previous version of this paper, and also for suggesting to consider the regularity of Nemytskii operators.

Author information

Authors and Affiliations

  1. Instituto de Matemática Interdisciplinar (IMI) and Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Jesús A. Jaramillo

  2. Departamento de Matemáticas, Universidad de Castilla-La Mancha, Escuela de Ingenieros Industriales, 02071, Albacete, Spain

    Sebastián Lajara

  3. Departamento de Matemática Aplicada, Ciencia e Ingeniería de los Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos, Calle Tulipán, s/n, 28933, Móstoles, Spain

    Óscar Madiedo

Corresponding author

Correspondence to Jesús A. Jaramillo.

Additional information

Research supported in part by MICINN under Project MTM2015-65825-P (Spain). The research of the second author is also supported by MICINN under Project MTM2014-54182-P (Spain) and by Fundació n Séneca (Agencia de Ciencia y Tecnología de la Región de Murcia) under Project 19275/PI/14. The research of the third author is also supported by IMI (Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid), REDIum (Red de Institutos Universitarios de Matemáticas de España) and IMACI (Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha).

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Jaramillo, J.A., Lajara, S. & Madiedo, Ó. Inversion of Nonsmooth Maps between Banach Spaces. Set-Valued Var. Anal 27, 921–947 (2019). https://doi.org/10.1007/s11228-018-0499-y

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  • DOI : https://doi.org/10.1007/s11228-018-0499-y

Keywords

  • Global invertibility
  • Nonsmooth analysis
  • Pseudo-Jacobian
  • Hadamard integral condition

Mathematics Subject Classification (2010)

  • 49J52
  • 49J53
  • 46G05

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Source: https://link.springer.com/article/10.1007/s11228-018-0499-y

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