From a Dieudonne theorem concerning the Cauchy problem to an open problem in the theory of weakly Picard operators

Let (X, d) be a complete metric space and let f : X -> X be a self operator. In this paper we study the following two problems: Problem 1. Let f be such that its fixed points set is a singleton, i.e., F-f = {x*}. Under which conditions the next implication does hold: f is asymptotically regular double right arrow f is a Picard operator? Problem 2. Let f be such that, F-f not equal phi. Under which conditions the following implication does hold: f is asymptotically regular double right arrow f is a weakly Picard operator? The case of operators defined on a linear L*-space is also studied.