On the maximality of sums of nonlinear monotone operators

Abstract

R. T. ROCKAFELLAR(')= {*?+x% I xf e Tx(x), xt e T2(x)}.If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal-some sort of condition is needed, since for example the graph of Tx + T2 can even be empty (as happens when D(Tx) n D(T2)= 0).The problem of determining conditions under which Tx + T2 is maximal turns out to be of fundamental importance in the theory of monotone operators.Results in this direction have been proved by Lescarret [9] and Browder [5], [6], [7].The strongest result which is known at present is : Theorem (Browder [6],[7]).Let X be reflexive, and let Tx and T2 be monotone operators from X to X*. Suppose that Tx is maximal, D(T2) = X, T2 is single-valued