Lipschitz stability of operators in Banach spaces

Abstract

We consider approximations of an arbitrarymap F: X → Y between Banach spaces X and Y by an affine operator A: X → 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. Pales in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.