Previous examples of non-type (D) maximal monotone operators were restricted to ℓ 1 , L 1 , and Banach spaces containing isometric copies of these spaces. This fact led to the conjecture that non-type (D) operators were restricted to this class of Banach spaces. We present a linear non-type (D) operator in c 0 . keywords: maximal monotone, type (D), Banach space, extension, bidual. 1 Introduction Let U, V arbitrary sets. A point-to-set (or multivalued) operator T : U ⇒ V is a map T : U → P(V ), where P(V ) is the power set of V . Given T : U ⇒ V , the graph of T is the set Gr(T ) := {(u, v) ∈ U × V | v ∈ T (u)}, the domain and the range of T are, respectively, dom(T ) := {u ∈ U | T (u) = ∅}, R(T ) := {v ∈ V | ∃u ∈ U, v ∈ T (u)} and the inverse of T is the point-to-set operator T −1 : V ⇒ U,A point-to-set operator T : U ⇒ V is called point-to-point if for every u ∈ dom(T ), T (u) has only one element. Trivially, a point-to-point operator is injective if, and only if, its inverse is also point-to-point.

We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martinez-Legaz and Svaiter. We first recover several properties similar to the monotone polar, including a characterization in terms of normal cones. Next, we use it to analyze certain aspects of maximal (in the sense of graph inclusion) quasimonotonicity, and its relation to the notion of maximal quasimonotonicity introduced by Aussel and Eberhard. Furthermore, we study the connections between quasimonotonicity and Minty Variational Inequality Problems.

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In this paper, we deal with three aspects of p-monotone operators. First we study p-monotone operators with a unique maximal extension (called pre-maximal), and with convex graph. We then deal with linear operators, and provide characterizations of p-monotonicity and maximal p-monotonicity. Finally we show that the Brezis-Browder theorem preserves p-monotonicity in reflexive Banach spaces.

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