Research Article | Open Access
Random Three-Step Iteration Scheme and Common Random Fixed Point of Three Operators
We construct random iterative processes with errors for three asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. The results presented in this paper extend and improve the recent ones announced by I. Beg and M. Abbas (2006), and many others.
- A. Špaček, “Zufällige Gleichungen,” Czechoslovak Mathematical Journal, vol. 5(80), pp. 462–466, 1955.
- A. T. Bharucha-Reid, “Fixed point theorems in probabilistic analysis,” Bulletin of the American Mathematical Society, vol. 82, no. 5, pp. 641–657, 1976.
- O. Hanš, “Random fixed point theorems,” in Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Liblice, Prague, 1956), pp. 105–125, Czechoslovak Academy of Sciences, Prague, Czech Republic, 1957.
- O. Hanš, “Random operator equations,” in Proceedings 4th Berkeley Symposium Math. Statist. and Prob., Vol. II, pp. 185–202, University California Press, Berkeley, Calif, USA, 1961.
- B. S. Choudhury, “Convergence of a random iteration scheme to a random fixed point,” Journal of Applied Mathematics and Stochastic Analysis, vol. 8, no. 2, pp. 139–142, 1995.
- I. Beg, “Approximation of random fixed points in normed spaces,” Nonlinear Analysis, vol. 51, no. 8, pp. 1363–1372, 2002.
- B. S. Choudhury, “Random Mann iteration scheme,” Applied Mathematics Letters, vol. 16, no. 1, pp. 93–96, 2003.
- H. Duan and G. Li, “Random Mann iteration scheme and random fixed point theorems,” Applied Mathematics Letters, vol. 18, no. 1, pp. 109–115, 2005.
- G. Li and H. Duan, “On random fixed point theorems of random monotone operators,” Applied Mathematics Letters, vol. 18, no. 9, pp. 1019–1026, 2005.
- S. Itoh, “A random fixed point theorem for a multivalued contraction mapping,” Pacific Journal of Mathematics, vol. 68, no. 1, pp. 85–90, 1977.
- I. Beg and M. Abbas, “Iterative procedures for solutions of random operator equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 181–201, 2006.
- S. Plubtieng, R. Wangkeeree, and R. Punpaeng, “On the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 1018–1029, 2006.
- J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.
- H.-K. Xu, “Random fixed point theorems for nonlinear uniformly Lipschitzian mappings,” Nonlinear Analysis, vol. 26, no. 7, pp. 1301–1311, 1996.
- P. L. Ramírez, “Some random fixed point theorems for nonlinear mappings,” Nonlinear Analysis, vol. 50, no. 2, pp. 265–274, 2002.
Copyright © 2007 Somyot Plubtieng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.