Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Hong-Kun Xu 13 February 2006 2006 2 13 371 379 723-5157-Xu-2005 10.wxyz/C2005V72P3p371 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5157-Xu/ We establish a Banach space version of a theorem of Suzuki \cite {Su}. More precisely we prove that if $X$ is a uniformly convex Banach space with a weakly continuous duality map (for example, $l^p$ for $1 < p < \infty $), if $C$ is a closed convex subset of $X$, and if $\mathcal {F}=\bigl \{T(t):t\ge 0\bigr \}$ is a contraction semigroup on $C$ such that $\Fix (\mathcal {F})\not =\emptyset $, then under certain appropriate assumptions made on the sequences $\{\alpha _n\}$ and $\{t_n\}$ of the parameters, we show that the sequence $\{x_n\}$ implicitly defined by $$x_n=\alpha _n u+(1-\alpha _n)T(t_n)x_n$$ for all $n\ge 1$ converges strongly to a member of $\Fix (\mathcal {F})$. 47H20, 47H09 Supported in part by the National Research Foundation of South Africa. F.E. Browder Proc. Nat. Acad. Sci. U.S.A. MR178324 F.E. Browder; Fixed point theorems for noncompact mappings in Hilbert space, \textit{Proc. Nat. Acad. Sci. U.S.A.} \textbf{53} (1965), pp.~1272--1276. F.E. Browder Math. Z. MR215141 F.E. Browder; Convergence theorems for sequences of nonlinear operators in Banach spaces, \textit{Math. Z.} \textbf{100} (1967), pp.~201--225. R.E. Bruck Israel J. Math. MR531254 R.E. Bruck; A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, \textit{Israel J. Math.} \textbf{32} (1979), pp.~107--116. K. Goebel and S. Reich Marcel Dekker MR744194 K. Goebel and S. Reich; \textit{Uniform convexity, hyperbolic geometry, and nonexpansive mappings} (Marcel Dekker, New York, 1984). T.C. Lim and H.K. Xu Nonlinear Anal. MR1280202 T.C. Lim and H.K. Xu; Fixed point theorems for asymptotically nonexpansive mappings, \textit{Nonlinear Anal.} \textbf{22} (1994), pp.~1345--1355. S. Reich J. Math. Anal. Appl. MR576291 S. Reich; Strong convergence theorems for resolvents of accretive operators in Banach spaces, \textit{J. Math. Anal. Appl.} \textbf{75} (1980), pp.~287--292. N. Shioji and W. Takahashi Nonlinear Anal. MR1631657 N. Shioji and W. Takahashi; Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces, \textit{Nonlinear Anal.} \textbf{34} (1998), pp.~87--99. T. Suzuki Proc. Amer. Math. Soc. MR1963759 T. Suzuki; On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, \textit{Proc. Amer. Math. Soc.} \textbf{131} (2002), pp.~2133--2136. H.K. Xu Numer. Funct. Anal. Optim. MR1606953 H.K. Xu; Approximations to fixed points of contraction semigroups in Hilbert spaces, \textit{Numer. Funct. Anal. Optim.} \textbf{19} (1998), pp.~157--163. H.K. Xu J. London Math. Soc. MR1911872 H.K. Xu; Iterative algorithms for nonlinear operators, \textit{J. London Math. Soc.} \textbf{66} (2002), pp.~240--256.