Pairs of Monotone Operators

This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent. Introduction We continue with the notation and the numbering of [4]. For the moment, we shall assume that E is reflexive; if we not making this assumption, we shall say specifically that E is a general Banach space. Let S1, S2 : E → 2E be maximal monotone. The main result of this note is the “Six Set Theorem”, Theorem 44(c), in which we prove that int(D(S1)−D(S2)) is identical with five other sets. We deduce from this in Theorem 44(d) and Theorem 44(e) that int(D(S1)−D(S2)) is always convex, and that any point surrounded by co(D(S1)−D(S2)) is always an interior point of D(S1)−D(S2). (See Definition 38 for the technical meaning of “surrounded”. It is easily seen that co(D(S1)−D(S2)) = coD(S1)− coD(S2).) We next deduce the “Nine Set Theorem”, Theorem 45(a), from the Six Set Theorem. In the Nine Set Theorem, we prove the identity of D(S1)−D(S2) with eight other sets if D(S1) − D(S2) is sufficiently fat. We deduce from this that D(S1)−D(S2) is then also convex. These results parallel results known for a single maximal monotone operator in a general Banach space. In Remark 46, we give comparisons of these two series of results. The following eight “constraint qualifications” discussed in [4] are known to guarantee the maximal monotonicity of S1 + S2: D(S1)−D(S2) is absorbing, (0.2) coD(S1)− coD(S2) is absorbing, (0.3) domχS1 − domχS2 is absorbing, (0.4) ⋃ λ>0 λ(D(S1)−D(S2)) = lin(D(S1)−D(S2)), (0.5) coD(S1)− coD(S2) is a neighborhood of 0 in lin(D(S1)−D(S2)), (0.6) Received by the editors December 10, 1996. 1991 Mathematics Subject Classification. Primary 47H05; Secondary 46B10.