On the solution of generalized equations and variational inequalities

Abstract

Uko and Argyros provided in [18] a Kantorovich–type theorem on the existence and uniqueness of the solution of a generalized equation of the form f(u)+g(u) ∋ 0, where f is a Fr´echet–differentiable function, and g is a maximal monotone operator defined on a Hilbert space. The sufficient convergence conditions are weaker than the corresponding ones given in the literature for the Kantorovich theorem on a Hilbert space. However, the convergence was shown to be only linear.

In this study, we show under the same conditions, the quadratic instead of the linear convergenve of the generalized Newton iteration involved.