Convergence of Inexact Newton Methods for Generalized Equations1

For solving the generalized equation f(x) + F (x) 3 0, where f is a smooth function and F is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by (f(xk) + Df(xk)(xk+1 − xk) + F (xk+1)) ∩Rk(xk, xk+1) 6 = ∅, where Df is the derivative of f and the sequence of mappings Rk represents the inexactness. We show how regularity properties of the mappings f + F and Rk are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.