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## Lipschitz stability of operators in Banach spaces

Proceedings of the Steklov Institute of Mathematics. 2013. Vol. 268. No. 1. P. 268-279.
Translator: V. Y. Protasov.

We consider approximations of an arbitrarymap FX → Y between Banach spaces X and Y by an affine operator AX → Y in the Lipschitz metric: the difference F — A has to be Lipschitz continuous with a small constant ɛ > 0. In the case Y = ℝ we show that if F can be affinely ɛ-approximated on any straight line in X, then it can be globally 2ɛ-approximated by an affine operator on X. The constant 2ɛ is sharp. Generalizations of this result to arbitrary dual Banach spaces Y are proved, and optimality of the conditions is shown in examples. As a corollary we obtain a solution to the problem stated by Zs. Páles in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.