Firmly nonexpansive mappings and maximally monotone operators: correspondence and duality

Abstract: The notion of a firmly nonexpansive mapping is central in fixed point theory because of attractive convergence properties for iterates and the correspondence with maximal monotone operators due to Minty. In this paper, we systematically analyze the relationship between properties of firmly nonexpansive mappings and associated maximal monotone operators. Dual and self-dual properties are also identified. The results are illustrated through several examples.