Structure Theory for Maximally Monotone Operators with Points of Continuity

Abstract

In this paper, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the norm-toweak * closedness and of property (Q) for these operators (as recently proven by Voisei). Various applications and limiting examples are given.

Sum Theorems for Maximally Monotone Operators of Type (Fpv)

“…By the Banach-Alaoglu theorem (see [29,Theorem 3.15]), there exist a weak * convergent subnet, (c * β ) β∈J of (c * k ) k∈N such that c * β w* c * ∞ ∈ X * . Borwein and Yao [12,Corollary 4.1] showed that (λ n a n , c * ∞ ) ∈ gra B, which contradicts our assumption that λ n a n dom B.…”

Faces and Support Functions for the Values of Maximal Monotone Operators

“…Conversely, if x ∈ D(A) and A is locally bounded at x, then x ∈ int D(A) (see [14,Theorem 1.14] or [8,Theorem 3.11.15]). Moreover, if [15,Theorem 4.1]), i.e., there exists i 0 ∈ I and M > 0 such that…”

“…Hence, A • (x + t n w n ), w n ≥ x * , w n for every x * ∈ Ax and so (15) holds. Taking n → ∞ in (15), by the lower semicontinuity of σ Ax , we get…”

“…Since x + v ∈ int(D(A)), A is locally bounded around x + v, i.e., there exist r , M > 0 such that Ay = ∅ and Ay ⊂ MB * for all y ∈ x + v + 4r B. According to [15,Lemma 4.1], there exists K > 0 such that…”

“…x * . Since A is maximal monotone and int(D(A)) = ∅, by [15,Theorem 4.1], the net {Ã(x + t i w i )} i∈I is eventually bounded, and by [15,Corollary 4.1], x * ∈ Ax. It follows from the monotonicity of A, for every ξ ∈ Ax, we have…”