ON CONVEXITY OF PREIMAGES OF MONOTONE OPERATORS

In this paper we first study the relationship between local and global Minty-Browder monotone operators and then we show that these operators have generally convex preimages. Our results allow to show that positive semidefinitedness on the complement of a discrete set of the differential operator implies the Minty-Browder monotonicity of the operator itself. We also show that complex functions of one complex variable are Minty-Browder monotone under suitable conditions. Finally, we obtain some injectivity/univalency theorems that generalize some well-known results.