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

The iteration scheme xn+1=λn+1y+(1−λn+1)Tn+1xn is first considered for infinitely many nonexpansive maps T1,T2,T3,… 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 T1,T2,…,TN, 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 (λn). This condition improves Lions’ condition (C.R. Acad. Sci. Paris Sèr 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 Tis.