Relatively Monotone and Relatively Generalized Monotone Maps in Nonsmooth Setting and Relative r-Monotonicity

The aim of this article is twofold. First, we consider relatively monotone, relatively pseudomonotone, and relatively quasimonotone maps in nonsmooth setting. For continuous maps, we describe (α,β)-monotonicity when the relative parameters are different and prove that (α,β)-quasimonotonicity implies (α,α)-quasimonotonicity as well as (α,β)-pseudomonotonicity implies (α,α)-pseudomonotonicity. Then, under local Lipschitzianity assumption, we establish first-order criteria for (α,α)-monotonicity, (α,α)-pseudomonotonicity, and (α,α)-quasimonotonicity. Second, we introduce and study the new notion of relatively r-monotone operators. We provide characterizations and properties. We give in particular criteria of first order for relative r-monotonicity, in smooth and nonsmooth locally Lipschitz settings, when the relative parameters are equal.