International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 50, Pages 2653-2680
doi:10.1155/S0161171204405183

Restrictive metric regularity and generalized differential calculus in Banach spaces

Boris S. Mordukhovich1 and Bingwu Wang2

1Department of Mathematics, Wayne State University, Detroit 48202, MI, USA
2Department of Mathematics, Eastern Michigan University, Ypsilanti 48197, MI, USA

Received 20 May 2004

Copyright © 2004 Boris S. Mordukhovich and Bingwu Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider nonlinear mappings f:X→Y between Banach spaces and study the notion of restrictive metric regularity of f around some point x¯, that is, metric regularity of f from X into the metric space E=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case when f is strictly differentiable at x¯ but its strict derivative ∇f(x¯) is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions.