On the Fitzpatrick transform of a monotone bifunction

Abstract

A new definition of monotone bifunctions is given, which is a slight generalization of the original definition given by Blum and Oettli, but which is better suited for relating monotone bifunctions to monotone operators. In this new definition, the Fitzpatrick transform of a maximal monotone bifunction is introduced so as to correspond exactly to the Fitzpatrick function of a maximal monotone operator in case the bifunction is constructed starting from the operator. Whenever the monotone bifunction is lower semicontinuous and convex with respect to its second variable, the Fitzpatrick transform permits to obtain results on its maximal monotonicity.

Keywords:

• monotone bifunction
• equilibrium problem
• Fitzpatrick function
• maximal monotone operator

AMS Subject Classifications::

• 47H05
• 47H04
• 90C33