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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-monotonicity when the relative parameters are different and prove that-quasimonotonicity implies-quasimonotonicity as well as-pseudomonotonicity implies-pseudomonotonicity. Then, under local Lipschitzianity assumption, we establish first-order criteria for-monotonicity,-pseudomonotonicity, and-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.

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