On uniformly \(\Phi\)-convex functions and strongly monotone multifunctions (Q1280023)

In the classical convex analysis there are close connections between convex functions and the monotony of the subdifferential mapping. In the present paper, the author extends these results to spaces without linear structure. In detail, he discusses \(\Phi\)-convex and uniformly \(\Phi\)-convex functions on a metric space using an arbitrary family \(\Phi\) of functions. Regarding the \(\Phi\)-subdifferential of such functions, he introduces different monotony properties of multifunctions: monotony, cyclic monotony, strong monotony, cyclic strong monotony, maximal monotony, maximal strong monotony and maximal cyclic strong monotony. In Theorem 1 it is shown that the \(\Phi\)-subdifferential of a uniformly \(\Phi\)-convex function is cyclic strongly monotone. After an assertion about maximal (cyclic) strongly monotone multifunctions, in Theorem 3 it is shown that in Banach spaces -- taken \(\Phi\) the dual space (i.e, the family of linear continuous functions) -- the subdifferential is even maximal cyclic strongly monotone. The question arises about the converse result. In Lemma 4, a sufficient condition is presented that for a given maximal cyclic strongly monotone multifunction \(\Gamma(x)\) there is a uniformly \(\Phi\)-convex function \(f\) such that \(\partial^\Phi f(x)= \Gamma(x)\). Finally, in the last assertions this sufficient condition is discussed for special types of linear spaces.