On the convergence of some iteration processes in uniformly convex Banach spaces

Gwinner, J.

doi:10.1090/S0002-9939-1978-0477899-4

On the convergence of some iteration processes in uniformly convex Banach spaces
HTML articles powered by AMS MathViewer

by J. Gwinner PDF

Proc. Amer. Math. Soc. 71 (1978), 29-35 Request permission

Abstract:

For the approximation of fixed points of a nonexpansive operator T in a uniformly convex Banach space E the convergence of the Mann-Toeplitz iteration ${x_{n + 1}} = {\alpha _n}T({x_n}) + (1 - {\alpha _n}){x_n}$ is studied. Strong convergence is established for a special class of operators T. Via regularization this result can be used for general nonexpansive operators, if E possesses a weakly sequentially continuous duality mapping. Furthermore strongly convergent combined regularization-iteration methods are presented.

References

Haïm Brezis and Moïse Sibony, Méthodes d’approximation et d’itération pour les opérateurs monotones, Arch. Rational MEch. Anal. 28 (1967/1968), 59–82 (French). MR 0220110, DOI 10.1007/BF00281564
A. B. Bakušinskiĭ and B. T. Poljak, The solution of variational inequalities, Dokl. Akad. Nauk SSSR 219 (1974), 1038–1041 (Russian). MR 0377619
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Ronald E. Bruck Jr., A strongly convergent iterative solution of $0\in U(x)$ for a maximal monotone operator $U$ in Hilbert space, J. Math. Anal. Appl. 48 (1974), 114–126. MR 361941, DOI 10.1016/0022-247X(74)90219-4
J. C. Dunn, Convexity, monotonicity, and gradient processes in Hilbert space, J. Math. Anal. Appl. 53 (1976), no. 1, 145–158. MR 388176, DOI 10.1016/0022-247X(76)90152-9
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
Shiro Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), no. 1, 65–71. MR 412909, DOI 10.1090/S0002-9939-1976-0412909-X
Konrad Knopp, Theorie und Anwendung der Unendlichen Reihen, Springer-Verlag, Berlin-Heidelberg, 1947 (German). 4th ed. MR 0028430
Pierre-Louis Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 21, A1357–A1359 (French, with English summary). MR 470770
Simeon Reich, Approximating zeros of accretive operators, Proc. Amer. Math. Soc. 51 (1975), no. 2, 381–384. MR 470762, DOI 10.1090/S0002-9939-1975-0470762-1