On the maximality of sums of nonlinear monotone operators

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