Equations on monotone graphs

Abstract

This paper studies the local analysis of equations on a product U × U of Banach spaces, whose variables lie in a subset having the special property that it is locally Lipschitz-homeomorphic to an open subset of U. A prominent example, to which we devote most of the paper, is a system of equations whose variables lie in the graph of a maximal monotone operator. This general formulation covers many specific problems of interest, and our objective is to understand the local behavior of solutions of such equations when they depend on parameters. We analyze this local behavior in stages, the first stage being to apply a tailored implicit-function theorem to an abstract formulation of the problem, thereby producing a set of results applicable to any particular problem instance. The second stage is to specialize the analysis to a Hilbert space H, with the subset mentioned above being the graph of a maximal monotone operator on H. This makes the results of the first stage applicable to many variational problems of practical importance. We then develop in detail the analytical steps to apply these results to finite-dimensional variational conditions with constraints of generalized nonlinear-programming type. The conditions thus identified generalize the strong second-order sufficient condition and linear-independence constraint qualification of nonlinear programming. A detailed example brings out some of the issues involved in practical implementation of this method. It also shows that aspects of representation (problem formulation) can strongly influence the feasibility of local analysis. This sensitivity to representation does not seem to be well known.