Let f f 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.