Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces ✩

The iteration scheme for families of nonexpansive mappings, essentially due to Halpern [Bull. Amer. Math. Soc. 73 (1967) 957–961], is established in a Banach space. The main theorem extends a recent result of O’Hara et al. [Nonlinear Anal. 54 (2003) 1417–1426] to a Banach space setting. For the same iteration scheme, with finitely many mappings, a complementary result to a result of Jung and Kim [Bull. Korean Math. Soc. 34 (1997) 93–102] (also Bauschke [J. Math. Anal. Appl. 202 (1996) 150–159]) is obtained by imposing other condition on the sequence of parameters. Our results also improve results in [C. R. Acad. Sci. Sér A–B Paris 284 (1977) 1357–1359; J. Math. Anal. Appl. 211 (1997) 71–83; Arch. Math. 59 (1992) 486–491] in framework of a Hilbert space.  2004 Elsevier Inc. All rights reserved