The Generalized Variational Metric

The variational convergence theory has known these last years a lot of developments. This theory is fundamental to give a global approach of problems which consist in searching a zero of a maximal monotone operator on a real Hilbert space. The notion of perturbation is often used in the resolution of such problems. More and more authors work also with a special or a variable metric on the space to solve this kind of problems. In the present paper, we give a unified context to couple these two approaches: a variational metric between operators, taking into account different metrics on the space.