Does Newton's method for set-valued maps converges uniformly in mild differentiability context?
Alain PiΓ©trus Β· LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones CientΓficas) Β· 2000
In this article, we study the existence of Newton-type sequence for solving the equation y Ο΅ f ( π ) + F( π ) where y is a small parameter, f is a function whose Frechet derivative satisfies a Holder condition of the form II β f ( π k ) - β f (x 2 ) β₯ β€ K β₯ lx 1 - x 2 β₯ d and F is a set-valued map between two Banach spaces X and Y. We prove that the Newton-type method y β f ( π k ) + β f ( π k ) ( π k+1 - π k ) +F( π k+1 ), is locally convergent to a solution of y Ο΅ f ( π ) + F( π ) if the set valued map (f(x* )+ β f (x* )( β - π *)+F( β )) -1 is Aubin continuous at (0, π *) where π * is a solution of 0 Ο΅ f ( π ) + F( π ). Moreover, we show that this convergence is superlinear uniformly in the parameter y and quadratic when d = 1.