# Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity

Borwein, Jonathan M. and Zhu, Qiji J. (1996) Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity. SIAM Journal on Control and Optimization, 34 (5). pp. 1568-1591. ISSN 0363-0129

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In Gateaux or Bornologically differentiable spaces there are two natural natural generalizations of the concept of a Fr\' echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish a fuzzy sum rule for it in a smooth Banach space. This rule is applied to obtain existence and comparison results for viscosity solutions of the Hamilton-Jacobi-Bellman equation in $\beta$-smooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustriates the flexibility of viscosity subderivatives as a tool for analysis.