Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential inclusions. The present paper contains a study on solvability and error bounds for generalized equations of the form F (x) ⊆ C, where F is a given set-valued mapping and C is a closed, convex cone. A property called metric C-increase, matching the metric behaviour of F with the partial order associated with C, is singled out, which ensures solution existence and error bound estimates in terms of problem data. Applications to the exact penalization of optimization problems with constraint systems, defined by the above class of generalized equations, and to the existence of ideal efficient solutions in vector optimization are proposed.

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In this paper, a systematic study of the strong metric subregularity property of mappings is carried out by means of a variational tool, called steepest displacement rate. With the aid of this tool, a simple characterization of strong metric subregularity for multifunctions acting in metric spaces is formulated. The resulting criterion is shown to be useful for establishing stability properties of the strong metric subregularity in the presence of perturbations, as well as for deriving various conditions, enabling to detect such a property in the case of nonsmooth mappings. Some of these conditions, involving several nonsmooth analysis constructions, are then applied in studying the isolated calmness property of the solution mapping to parameterized generalized equations.

Some properties, connected with recent generalizations of the classic notion of Lipschitz continuity for multifunctions, are investigated with reference to variational systems, that is to solution maps associated to parametrized generalized equations. The latter ones are a convenient framework to address several questions, mainly related to the stability and sensitivity analysis, arising in mathematical programming, optimal control, equilibrium and variational inequality theory. Global and local criteria for metric regularity and Lipschitz-likeness of variational systems are obtained. Some applications to the exact penalization of mathematical programs with equilibrium constraints and to the Lipschitzian stability of fixed points for multivalued contractions are then considered.