Pseudomonotone Diagonal Subdifferential Operators

Let $f$ be an equilibrium bifunction defined on the product space $X\times X$, where $X$ is a Banach space. If $f$ is locally Lipschitz with respect to the second variable, for every $x\in X$ we define $T_f(x)$ as the Clarke subdifferential of $f(x,\cdot)$ evaluated at $x$. This multivalued operator plays a fundamental role for the reformulation of equilibrium problems as variational inequality ones. We analyze additional conditions on $f$ which ensure the $D$-maximal pseudomonotonicity and the cyclically pseudomonotonicity of $T_f$. Such results have consequences in terms of the characterization of the set of solutions of a subclass of pseudomonotone equilibrium problems.