We consider approximations of an arbitrarymap F: X â Y between Banach spaces X and Y by an affine operator A: X â Y in the Lipschitz metric: the difference F - A has to be Lipschitz continuous with a small constant eopen > 0. In the case Y = â we show that if F can be affinely eopen-approximated on any straight line in X, then it can be globally 2eopen-approximated by an affine operator on X. The constant 2eopen 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. PÃ¡les in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed. Â© 2013 Pleiades Publishing, Ltd.
|Titolo:||Lipschitz stability of operators in Banach spaces|
|Data di pubblicazione:||2013|
|Appare nelle tipologie:||1.1 Articolo in rivista|