We study descent-like approximation methods and proximal methods of the retraction type for solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. We prove strong and weak convergences for weakly contractive and nonexpansive maps, respectively. We also establish the stability of these methods with respect to perturbations of the operators and the constraint sets.

Let $\phi$ be a semiflow of holomorphic maps of a bounded domain $D$ in a complex Banach space. The general question arises under which conditions the existence of a periodic orbit of $\phi$ implies that $\phi$ itself is periodic. An answer is provided, in the first part of this paper, in the case in which $D$ is the open unit ball of a $J^{\ast }$-algebra and $\phi$ acts isometrically. More precise results are provided when the $J^{\ast }$-algebra is a Cartan factor of type one or a spin factor. The second part of this paper deals essentially with the discrete semiflow $\phi$ generated by the iterates of a holomorphic map. It investigates how the existence of fixed points determines the asymptotic behaviour of the semiflow. Some of these results are extended to continuous semiflows.