Nonlinear maximal monotone operators in Banach space

Such a set G is said to be maximal monotone if it is maximal among monotone sets in the sense of inclusion, and a mapping T is said to be maximal monotone if its graph G(T) is a maximal monotone set. If D(T) is convex and T is hemicontinuous from D(T) to X* (i.e. T is continuous from each line segment of D(T) to the weak* topology of X*), then T is maximal monotone if and only if it is maximal in the sense of operator inclusion in the family of monotone mappings from X to X*. For operators T with D(T) = X , the basic result obtained independently by BROWNER [5] and MINTY [31] states that for a reflexive space X, if T is a hemicontinuous monotone operator from D(T) = X to X* and if in addition, T is coercive, i.e.

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