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.
View at: Google Scholar | Zentralblatt MATH | MathSciNetA. T. Bharucha-Reid, “Fixed point theorems in probabilistic analysis,” Bulletin of the American Mathematical Society, vol. 82, no. 5, pp. 641–657, 1976.
View at: Google Scholar | Zentralblatt MATH | MathSciNetO. 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.
View at: Google Scholar | Zentralblatt MATH | MathSciNetO. 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.
View at: Google Scholar | Zentralblatt MATH | MathSciNetB. 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.
View at: Google Scholar | Zentralblatt MATH | MathSciNetI. Beg, “Approximation of random fixed points in normed spaces,” Nonlinear Analysis, vol. 51, no. 8, pp. 1363–1372, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. S. Choudhury, “Random Mann iteration scheme,” Applied Mathematics Letters, vol. 16, no. 1, pp. 93–96, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetH. Duan and G. Li, “Random Mann iteration scheme and random fixed point theorems,” Applied Mathematics Letters, vol. 18, no. 1, pp. 109–115, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. Li and H. Duan, “On random fixed point theorems of random monotone operators,” Applied Mathematics Letters, vol. 18, no. 9, pp. 1019–1026, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Itoh, “A random fixed point theorem for a multivalued contraction mapping,” Pacific Journal of Mathematics, vol. 68, no. 1, pp. 85–90, 1977.
View at: Google Scholar | Zentralblatt MATH | MathSciNetI. 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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. 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.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. 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.
View at: Google Scholar | Zentralblatt MATH | MathSciNetH.-K. Xu, “Random fixed point theorems for nonlinear uniformly Lipschitzian mappings,” Nonlinear Analysis, vol. 26, no. 7, pp. 1301–1311, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetP. L. Ramírez, “Some random fixed point theorems for nonlinear mappings,” Nonlinear Analysis, vol. 50, no. 2, pp. 265–274, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet