Pseudomonotone Diagonal Subdifferential Operators

Let

be an equilibrium bifunction defined on the product space

\mathbb{X}\times\mathbb{X} X × X

, where

\mathbb{X} X

is a Banach space. If

f f

is locally Lipschitz with respect to the second variable, for every

x\in\mathbb{X} x ∈ X

we define

T_f(x) T f ( x )

as the Clarke subdifferential of

f(x,\cdot) f ( x , ⋅ )

evaluated at

x x

. This multivalued operator plays a fundamental role for the reformulation of equilibrium problems as variational inequality ones. We analyze additional conditions on

f f

which ensure the

D D

-maximal pseudomonotonicity and the cyclically pseudomonotonicity of

T_f T f

. Such results have consequences in terms of the characterization of the set of solutions of a subclass of pseudomonotone equilibrium problems.