Monotone Operators

The aim of this work is to present several new results concerning monotone operator theory, applications to injectivity theorems, convex functions and variational inequalities, and some applications of the duality to the theory of maximal monotone operators. We begin with the study under which circumstances the local (generalized) monotonicity property of an operator is equivalent to its global counterpart. We introduce two new concepts, the -monotonicity of operators and the -convexity of real valued functions, that contain in particular several monotonicity/convexity notions from the literature. We dedicate a whole chapter to the study of variational inequalities of Stampacchia and Minty type and we establish several coicidence, respectively fixed point results. The last part this work is devoted to the study of famous sum problems involving two maximal monotone operators. Based on stable strong duality and introducing some generalized infimal convolution formulas, we establish most of the important results obtained so far in the literature related to this problem.