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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Generalized gradients and applications


Author: Frank H. Clarke
Journal: Trans. Amer. Math. Soc. 205 (1975), 247-262
MSC: Primary 26A51; Secondary 53C70
DOI: https://doi.org/10.1090/S0002-9947-1975-0367131-6
MathSciNet review: 0367131
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Abstract: A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets. It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis. A theorem is proved concerning the differentiability properties of a function of the form $ \max \{ g(x,u):u \in U\} $. This result unifies and extends some theorems of Danskin and others. The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0367131-6
Keywords: Nondifferentiable functions, Lipschitz, generalized gradients, max functions, tangent cones, directional derivatives, flow-invariant sets
Article copyright: © Copyright 1975 American Mathematical Society