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Restrictive metric regularity and generalized differential calculus in Banach spaces

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  • Boris S. Mordukhovich
  • Bingwu Wang

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.

Suggested Citation

  • Boris S. Mordukhovich & Bingwu Wang, 2004. "Restrictive metric regularity and generalized differential calculus in Banach spaces," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-28, January.
    • Handle: RePEc:hin:jijmms:683907
    DOI: 10.1155/S0161171204405183
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    Cited by:

    1. Huynh Van Ngai & Michel Théra, 2015. "Directional Metric Regularity of Multifunctions," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 969-991, October.
    2. Aram V. Arutyunov & Sergey E. Zhukovskiy, 2023. "Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 112-142, October.
    3. Bingwu Wang & Xinmin Yang & Pujun Long, 2024. "Generalized Sequential Normal Compactness and Weak Differentiabilities," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1309-1324, November.
    4. Huynh Van Ngai & Phan Nhat Tinh, 2015. "Metric Subregularity of Multifunctions: First and Second Order Infinitesimal Characterizations," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 703-724, March.
    5. Aram V. Arutyunov & Alexey F. Izmailov, 2006. "Directional Stability Theorem and Directional Metric Regularity," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 526-543, August.

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