Convergence of inexact Newton methods for generalized equations

For solving the generalized equation f (x) + F(x) 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)+ D f (xk)(xk+1 − xk)+ F(xk+1)) ∩ Rk(xk, xk+1) = ∅, where D f 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.