On the regularity of selections and on Minty points of generalized monotone set-valued maps

In this paper we deal with set-valued maps T:X⇉X⁎ defined on a Banach space X, that are generalized monotone in the sense of Karamardian. Under various continuity assumptions on T, we investigate the regularity of suitable selections of the set-valued map R++T that shares with T the generalized monotonicity properties. In particular, we show that for every quasimonotone set-valued map T satisfying the Aubin property around (y,y⁎)∈gph(T) with y⁎≠0 there exist locally Lipschitz selections of R++T∖{0}. In the last part some notions of Minty points of T are introduced, and their relationship with zeros as well as effective zeros of T is discussed; a correlation is established between these concepts and the broader context of the generalized monotonicity of the map T.