Recently there has been an extensive research concerning p-nonexpansive mappings of B, see [5, 7, 14, 161. In fact the more general nonexpansive mappings of B”, the Cartesian product of n Hilbert balls, with respect to its hyperbolic metric had also been studied, see [lo, 11, 171. In this paper we shall present a new approach to the study of p-nonexpansive mappings, and more generally coaccretive operators (see the definition below). This approach involves the structure of (B, p) as a Hilbert manifold and can be used to simplify and unify a large part of the known basic results of the theory which can be found in the book [7]. However, we prefer to concentrate on the new results that our methods lead to, relying on the known results when needed. These new results include a study of a new class of (generally multivalued) operators, namely coaccretive operators, which are closely related to accretive operators in Banach spaces, and of the asymptotic behavior, both at the origin and at infinity, of certain families of firmly nonexpansive mappings (of both kinds). In particular we answer a question raised by Goebel and Reich [7, p. 135; 6; 13, problem 161. Our methods lead also to results of ergodic nature for p-nonexpansive mappings which will appear elsewhere.