Abstract
In this paper, we present the notions of openness and metric regularity for a set-valued map with respect to two fixed sets, proving their equivalence. By using different approaches, we show the stability, with respect to the sum of maps, of the openness property, both in the setting of Banach spaces and of metric spaces. Finally, we infer the regularity of the map solving a generalized parametric equation defined via a parametric map that is, in its turn, perturbed by the sum with another map.
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Acknowledgments
The work of the second author was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0024. The first and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Communicated by Michel Théra.
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Bianchi, M., Kassay, G. & Pini, R. Stability Results of Variational Systems Under Openness with Respect to Fixed Sets. J Optim Theory Appl 164, 92–108 (2015). https://doi.org/10.1007/s10957-014-0560-4
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DOI: https://doi.org/10.1007/s10957-014-0560-4
Keywords
- Linear openness
- Metric regularity
- Sum of maps
- Generalized equation
- Sensitivity analysis
- Fixed point theorem
- Ekeland’s variational principle