Regularity of implicit solution mapping to parametric generalized equation

Abstract

This paper concerns the study of both local and global metric regularity/Lipschitz-like properties concerning the behavior of the implicit solution mapping associated to parametric generalized equation in metric space. We extend some implicit multifunction results to the addition of two multifunctions both depending on parameters. Through the approach of inverse mapping iteration, several results are established regarding the relations between the (partial) metric regularity/Lipschitz-like moduli of multifunctions used as the defining form of the generalized equation and the corresponding implicit solution mapping, the proof of which is completely self-contained. Finally, a local Lyusternik–Graves Theorem is obtained as an application.

COMMUNICATED BY:

• B. Mordukhovich

Keywords:

• Parametric generalized equations
• implicit solution mapping
• (partial) metric regularity
• (partial) Lipschitz-like property

AMS subject classifications:

• 49J53
• 47H04
• 54C60

Acknowledgments

The authors are indebted to the referees for their valuable comments on the original submission.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This author's research was supported by the National Natural Science Foundation of People's Republic of China [grant numbers 11801500 and 11771384] and the Yunnan Provincial Science and Technology Department [grant number 2017FD070]. This author's research was supported by the Yunnnan Provincial Department of Education Research Fund Research Fund [grant number 2019J0040] and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY 201807022).