Journal of Convex Analysis 20 (2013), No. 1, 001--012
Copyright Heldermann Verlag 2013
Pseudomonotone Diagonal Subdifferential Operators
Marco Castellani
Dept. of Systems and Institutions for the Economy, University of L'Aquila, Via Giovanni Falcone 25, 67100 L'Aquila, Italy
marco.castellani@univaq.it
Massimiliano Giuli
Dept. of Systems and Institutions for the Economy, University of L'Aquila, Via Giovanni Falcone 25, 67100 L'Aquila, Italy
massimiliano.giuli@univaq.it
[Abstract-pdf]
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.
Keywords: Equilibrium problem, pseudomonotone bifunction, pseudomonotone operator, diagonal subdifferential.
MSC: 91B50, 47H05
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