A New Proof of the Maximal Monotonicity of the Sum Using the Fitzpatrick Function

Abstract

A classical result of Rockafellar [6] states that the sum of two maximal monotone multifunctions on a reflexive Banach space is maximal monotone when the interior of the domain of one of them intersects the domain of the other. There are several proofs of this important result. The original proof of Rockafellar [6] uses some results of Browder [1]; put together, the proof is quite involved. The proof furnished in the recent book by Simons [7] uses minimax theorems. We give another (short) proof of this result in reflexive spaces using the Fitzpatrick function associated to a monotone multifunction and a result on the conjugate of the sum of two convex functions. We recall first some notation and results related to convex analysis. For this propose, consider a separated locally convex space E and E∗ its topological dual; we get so the dual system (E, E∗, 〈·, ·〉), where 〈x, x∗〉 := x∗(x) for x ∈ E and x∗ ∈ E∗. We endow E∗ with the weak-star topology w∗ := σ(E∗, E), and so the topological dual of E∗ is identified with E. As usual, having a subset A of E, we use the notation intA, cl A or A, co A, coA and aff A for the interior, closure, convex hull, closed convex hull, and the affine hull of A, respectively; moreover, A and A denote the core (algebraic interior) and the intrinsic core of A, while A is A when aff A is closed and A is the empty set otherwise. The domain, the epigraph and the conjugate of f : E → R are introduced by