Convex Analysis and Monotone Operators: Forward and Backward

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The monotone operators play a central role in many areas of Mathematics and its applications, for instance in Optimization and in the theory of Differential Equations. Many connections between Convex Analysis and the theory of Monotone Operators are known for around 50 years, however they were intensively studied after 2002, when the so-called Fitzpatrick function and its generalizations known as the representative functions were rediscovered. The main scope of this project is to deliver new statements on (combinations of) monotone operators, e.g. concerning maximal monotonicity, by means of the new advances and concepts from Convex Analysis, for instance general regularity conditions. The project is structured into five objectives. Firstly, investigations on representative functions and subdifferential concepts are planned. Afterwards, we intend to extend several results on monotone operators from reflexive to general Banach spaces. Then we will deal with compositions and extensions of monotone operators. Within the framework of the fourth objective investigations on so-called diagonal subdifferential operators are scheduled. Because many interesting problems (e.g. minimization problems, complementarity problems, variational inequalities) can be cast as monotone inclusions, we also plan to deliver splitting type algorithms for finding zeros of combinations of monotone operators and to implement them on concrete applications.