On Characterizations of Metric Regularity of Multi-Valued Maps

We provide a new proof along the lines of the recent book of A. Ioffe [Variational Analysis of Regular Mappings: Theory and Applications, Springer Monographs in Mathematics (2017)] of a result of H. Frankowska [Some inverse mapping theorems, Ann. Inst. H. Poincaré, Anal. Non Linéaire 7 (1990) 183–234] showing that metric regularity of a multi-valued map can be characterized by regularity of its contingent variation – a notion extending contingent derivative.