Convergence of inexact Newton methods for generalized equations

For solving the generalized equation , where is a smooth function and is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by (f(x(k)) + Df(x(k))(x(k+1) - x(k)) + F(x(k+1) )) boolean AND R-k(x(k), x(k+1) ) not equal empty set, where is the derivative of and the sequence of mappings represents the inexactness. We show how regularity properties of the mappings and 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.