Abstract In this paper we extend some well known properties of monotone and maximal monotone operators to the wider class of e-monotone and maximal e-monotone operators. The main results concern local boundedness of maximal e-monotone operators, maximal 2e-monotonicity of the Clarke–Rockafellar subdifferential $$\partial ^{CR}f$$ ∂ CR f for an e-convex function f, and the characterization of e-monotonicity of an operator T via the behaviour of its e-Fitzpatrick function outside the graph of T.

Abstract In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

A Correction to this paper has been published: 10.1007/s10898-021-01088-x