Applications of the Duality to Monotone Operators

Abstract

The theory of monotone operators captured the attention of mathematicians not only because of the fineness of the results but also because of the large number of applications, especially in fields like nonlinear analysis, variational inequalities and partial differential equations (see for instance [88, 130]). Different attempts to establish links to the convex analysis have been made (see [90,91,119]), but the most fruitful ones turned out to be the ones based on the so-called Fitzpatrick function discovered by Simons Fitzpatrick in [70]. Neglected for many years until re-popularized in [7, 8, 52, 98, 104–106, 121], this class of functions along with its extensions have given rise to a great number of publications which rediscovered and extended the important results of the theory of monotone operators by using tools from the convex analysis. The investigations we make in this chapter are to be seen belonging to this class of results, whereby, we concentrate ourselves on results based on the conjugate duality theory.