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Let E be a real Banach space, E∗Open image in new window be the dual space of E, E∗∗Open image in new window be the dual space of E∗Open image in new window. Let T:D(T)⊆E∗∗→2E∗Open image in new window be a monotone type mapping. In this paper, first, we introduce the special case when T is the weak* sub-differential ∂∗ϕOpen image in new window of a convex function ϕ and obtain a surjective result for the mapping ∂∗(ϕ+ϵ∥⋅∥2)Open image in new window, where ϵ>0Open image in new window. Second, we show the existence of solutions of the variational inequality problems for strictly quasi-monotone operators and semi-monotone operators. Finally, we construct a degree theory for mappings of the class (S+)Open image in new window and then construct a generalized degree for the weak* sub-differential of a convex function.

Monotone type operators in nonreflexive Banach spaces