Journal of Optimization Theory and Applications

, Volume 155, Issue 2, pp 558–571

# Stability of Implicit Multifunctions in Banach Spaces

• N. Q. Huy
• D. S. Kim
• K. V. Ninh
Article

## Abstract

This paper is devoted to present new sufficient conditions for both the metric regularity in the Robinson’s sense and the Lipschitz-like property in the Aubin’s sense of implicit multifunctions in general Banach spaces. The basic tools of our analysis involve the Clarke subdifferential, the Clarke coderivative of set-valued mappings, and the Ekeland variational principle. The metric regularity of implicit multifunction is compared with the Lipschitz-like property.

## Keywords

Implicit multifunction Robinson metric regularity Aubin Lipschitz-like property Clarke subdifferential Clarke coderivative Ekeland variational principle

## Notes

### Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0012780), and was supported by the project “Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas” (Vietnam–USA). The authors are indebted to an anonymous referee and Professor Franco Giannessi for careful reading of the manuscript and many comments which helped to improve the presentation.

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