Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

Shioji, Naoki; Takahashi, Wataru

doi:10.1090/S0002-9939-97-04033-1

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces
HTML articles powered by AMS MathViewer

by Naoki Shioji and Wataru Takahashi PDF

Proc. Amer. Math. Soc. 125 (1997), 3641-3645 Request permission

Abstract:

In this paper, we study the convergence of the sequence defined by \[ x_0\in C , \;\; x_{n+1} = \alpha _{n}x + (1-\alpha _{n})Tx_n , \;\; n=0,1,2, \ldots ,\] where $0 \leq \alpha _n \leq 1$ and $T$ is a nonexpansive mapping from a closed convex subset of a Banach space into itself.

References

S. Banach, Théorie des opérations linéaires, Monografie Mat., PWN, Warszawa, 1932.
Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961. MR 218938, DOI 10.1090/S0002-9904-1967-11864-0
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
Simeon Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287–292. MR 576291, DOI 10.1016/0022-247X(80)90323-6
Simeon Reich, Some problems and results in fixed point theory, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982) Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 179–187. MR 729515, DOI 10.1090/conm/021/729515
Wataru Takahashi and Yoichi Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), no. 2, 546–553. MR 766150, DOI 10.1016/0022-247X(84)90019-2
Rainer Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel) 58 (1992), no. 5, 486–491. MR 1156581, DOI 10.1007/BF01190119