Fréchet quasidifferential calculus with applications to metric regularity of continuous maps

Abstract

A strong variant for the notion of quasidifferentiability of functions is considered in the light of recent achievements in variational analysis. Several characterization results, some calculus rules and examples are provided. Then a generic result for the corresponding subdifferentiability notion is established in general Banach spaces. Subsequently, such notion is employed in the study of generalized differential criteria for metric regularity. Necessary and sufficient conditions, along with a precise estimation of the norm of metric regularity, are obtained in the case of continuous maps between Banach spaces via a strong slope approach. All these results do not require Asplundity assumptions.

Keywords:

• Fréchet quasidifferentiable functions
• Local calmness
• Ekeland variational principle
• Strong slope
• Metric regularity criteria
• Asplund spaces
• AMS Subject Classifications: 49J52
• 58C15
• 65K10

Acknowledgments

The author wishes to thank Vladimir F. Demyanov for having supplied him with a copy of [30], and the anonymous referee for having contributed to improve the quality of the present article.

Notes

Dedicated to V.F. Demyanov on the occasion of his 65th birthday.

 The varying terminology reflects slightly different starting points having led to the same concept recurring in the literature.