Abstract

In 1950 Graves proved the following theorem: If the function f from a Banach space X into a Banach space Y is strictly differentiable at x0x_0 and the strict derivative ∇f(x0)\nabla f(x_0) is onto, then f is open with linear rate around x0x_0. Under fairly general assumptions, the latter property is equivalent to either the metric regularity or to the Aubin property of the inverse. In this paper, we prove that the Graves theorem is a consequence of the following general result: the openness with linear rate of a locally closed set-valued map F around a point (x0,y0)(x_0, y_0) of its graph is invariant with respect to a perturbation of the form f + F provided that the strict derivative of f at x0x_0 is zero.

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Asen L. Dontchev

Mathematical Reviews, 416 Fourth Street Ann Arbor, MI 48107, U.S.A

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Suggested citation

A. L. Dontchev. “The Graves Theorem Revisited.” Journal of Convex Analysis 3 (1996), No. 1, 45–53. https://doi.org/10.68381/jca03003

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Published by Heldermann Verlag, 1996. Rights now held by Banach Press.