ABSTRACT. Let T_{i}(i = 1, 2, ..., N) be nonexpansive mappings on a Hilbert sapce H, and let Θ : H → R ∪ {∞} be a function which has a uniformly strongly positive and uniformly bounded second ( Fréchet ) derivative over the convex hull of T_{i}(H) for some i. We first prove thet Θ has a unique minimum over the intersection of the fixed point sets of all the T_{i}’s at some point u^{∗}. Then a cyclic hybrid steepest descent algorithm is proposed and we prove that it converges to u^{∗}. This generalizes some recent results of Wittmann (1992), Combettes (1995), Bauschke (1996), and Yamada, Ogura, Yamashita, and Sakaniwa (1997).

In particular, the minimization of Θ over the intersection ∩_{1}^{N}C_{i} of closed convex sets C_{i} can be handled by taking T_{i} to be the metric projection P_{C_{i}} onto C_{i}. We also propose a modification of our algorithm to handle the inconsistent case (i.e., when ∩_{1}^{N}C_{i} is empty) as well.