Generalized equations, variational inequalities and a weak Kantorovich theorem

Abstract

We prove a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form \(f(u)+g(u)\owns 0\) where f is a Frechet-differentiable function and g is a maximal monotone operator defined on a Hilbert space. The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems we obtain novel results for these problems as well. Our approach to the solution of a generalized equation is iterative, and the solution is obtained as the limit of the solutions of partially linearized generalized Newton subproblems of the type \(Az+g(z)\owns b\) where A is a linear operator.