Variational Sum of Monotone Operators

The sum of (nonlinear) maximal monotone operators is reconsidered from the Yosida approximation and graph-convergence point of view. This leads to a new concept, called variational sum, which coincides with the classical (pointwise) sum when the classical sum happens to be maximal monotone. In the case of subdiierentials of convex lower semicontinuous proper functions, the variational sum is equal to the subdiierential of the sum of the functions. A general feature of the variational sum is to involve not only the values of the two operators at the given point but also their values at nearby points. 1. The variational sum, an introduction In this introduction, we present the main lines of the new concept of \variational sum" of maximal monotone operators. The deenitions and tools required to make this notion precise are introduced in the next sections. Let H be a real Hilbert space. For an operator A : H ! ! H, we note: