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Krasnoselski-Mann Iterations in Normed Spaces

Published online by Cambridge University Press:  20 November 2018

Jonathan Borwein ,
Simeon Reich  and
Itai Shafrir
Jonathan Borwein
Affiliation:
Department of Mathematics Statistics and Computing Science Dalhousie University Halifax, Nova Scotia B3H3J5
Simeon Reich
Affiliation:
Department of Mathematics University of Southern California Los Angeles, California U.S.A. 90089 and Department of Mathematics Technion—Israel Institute of Technology 32000 Haifa Israel
Itai Shafrir
Affiliation:
Department of Mathematics Technion—Israel Institute of Technology 32000 Haifa Israel
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Abstract

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We provide general results on the behaviour of the Krasnoselski-Mann iteration process for nonexpansive mappings in a variety of normed settings.

Keywords

47H0947H1047H17.
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 1 , 01 March 1992 , pp. 21 - 28
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Dotson, W. G., On the Mann iterative process, Trans. Amer. Math. Soc. 149(1970),655–73.Google Scholar
2. Edelstein, M. and R. C. O'Brien Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc. 17(1978),547–554.Google Scholar
3. Fujihara, T., Asymptotic behavior of nonexpansive mappings in Banach spaces, Tokyo J. Math. 7(1984),119–128.Google Scholar
4. Goebel, K., and Kirk, W. A., Iteration processes for nonexpansive mappings, Contemporary Mathematics 21(1983),115–123.Google Scholar
5. Groetsch, C.W., A note on segmenting Mann iterates, J. Math. Anal, and Appl. 40(1972),369–372.Google Scholar
6. Hicks, T. L. and Kubicek, J. D., On the Mann iteration process in Hilbert space, J. Math. Anal, and Appl. 59(1977),498–504.Google Scholar
7. Hillam, B. P., A generalization ofKrasnoselski s theorem on the real line, Mathematics Magazine 48( 1975), 167–168.Google Scholar
8. Ishikawa, S., Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59(1976),65–71.Google Scholar
9. Kirk, W. A., KrasnoselskiVs iteration process in hyperbolic space, Numer. Funct. Anal. Optimiz. 4(1982),371–381.Google Scholar
10. Kirk, W. A., Fixed point theory for nonexpansive mappings, I and II, (I) appears in Lecture Notes in Mathematics 886 (Springer-Verlag, 1981), pp. 484-505; (II) appears in: Fixed Points and Nonexpansive Mappings (R. Sine, éd.), Contemporary Math. 18, Amer. Math. Soc, Providence RI, (1983), pp. 121–140.Google Scholar
11. Kohlberg, E., and Neyman, A., Asymptotic behaviour of nonexpansive mappings in normed linear spaces, Israel J. Math. 38(1981),269–275.Google Scholar
12. Krasnoselski, M. A., Two observations about the method of successive approximations, Usp. Math. Nauk, 10(1955),123–127.Google Scholar
13. Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc. 4(1953),506–510.Google Scholar
14. Reich, S., On the asymptotic behaviour of nonlinear semigroups and the range of accretive operators II, J. Math. Anal. Appl. 87(1982),134–146.Google Scholar
15. Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67(1979),274–276.Google Scholar
16. Reich, S., The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88(1983),44–45.Google Scholar
17. Reich, S. and Shafrir, I., On the method of successive approximationsfor nonexpansive mappings, Nonlinear and Convex Analysis, Marcel Dekker, New York, (1987), 193–201.Google Scholar
18. Reich, S. and Shafrir, I., Nonexpansive iterations in hyperbolic spaces, Nonlinear Analysis 15(1990),537–558.Google Scholar
19. Shafrir, I., The approximate fixed point property in Banach and hyperbolic spaces, Israel J. Math, 71( 1990), 211–223.Google Scholar