Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces

The iteration scheme x(n+1) = lambda(n+1) y + (1 - lambda(n+1))T(n+1)x(n) is first considered for infinitely many nonexpansive maps T-1, T-2, T-3,... in a Hilbert space. A result of Shimizu and Takahashi (J: Math. Anal. Appl. 211 (1997) 71) is generalized, and it is shown that the sequence of iterates converges to Py, where P is some projection. For this same iteration scheme, with finitely many maps T-1, T-2,..., T-N, a complementary result to a result of Bauschke (J. Math. Anal. Appl. 202 (1996) 150) is proved by introducing a new condition on the sequence of parameters (lambda(n)). This condition improves Lions' condition (C.R. Acad. Sci. Paris Ser A-B 284 (1977) 1357). The iterates converge to Py, where P is the projection onto the intersection of the fixed point sets of the T(i)s. (C) 2003 Elsevier Ltd. All rights reserved.