An implicit-function theorem for a class of monotone generalized equations

Implicit-function theorems for generalized equations play an important role in many applications, especially in the stability and sensitivity analysis of variational inequalities and optimization problems and in the convergence analysis of numerical algorithms solving such problems. We refer for instance to Fiacco [7] and Ioffe and Tihomirov [9] for applications of the classical implicit-function theorem in this context. Further results and some extensions of the classical implicit-function theorem can be found in Fiacco [8]. In [15] Robinson proved an implicit-function theorem for a class of generalized equations which he called strongly regular. This result has been widely used in the stability and sensitivity analysis of optimization and optimal control problems (see e.g. Robinson [15, 16], Alt [1, 2], Ito-Kunisch [10], Malanowski [13]) and in the convergence analysis of algorithms solving optimization problems and variational inequalities (see e.g. Robinson [16], Alt [1, 2]). In a recent paper [17], Robinson could extend his implicit-function theorem to a class of nonsmooth functions. In [11, 12] Kassay and Kolumban derived implicit-function theorems for a class of generalized equations defined by a monotone set-valued mapping. They have shown that from these theorems the classical implicit function theorem and Browder's surjectivity theorem can be easily derived. They also presented applications to variational inequalities. The aim of the present paper is to further develop the rather general implicitfunction theorem of Kassay and Kolumban [12] in view of applications to variational inequalities and a class of generalized equations defined by nonsmooth functions.