ON THE CONVERGENCE THEOREMS OFAN IMPLICIT ITERATION PROCESS FORASYMPTOTICALLY QUASII-NONEXPANSIVE MAPPINGS

Year 2013, Volume: 42 Issue: 6, 617 - 626, 01.06.2013

İsa Yildirim

Abstract

In this paper, we establish an implicit iterative process for convergenceto a common fixed point of two asymptotically quasi I-nonexpansivemappings in Banach spaces, and prove weak and strong convergenceof this process to a common fixed point of such mappings. Our results improve and extend corresponding results of [7, 10, 14, 18] to twoasymptotically quasi I-nonexpansive mappings.

Keywords

Asymptotically quasi I-nonexpansive mappings , Common fixed point; , Uniformly convex Banach space , 2000 AMS Classification: 47H09

ON THE CONVERGENCE THEOREMS OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY QUASI I-NONEXPANSIVE MAPPINGS

Abstract

References

• Browder, F. E., Nonexpansive nonlinear operators in a Banach space, Proceedings of the National Academy of Sciences of the United States of America, 54, 1041–1044, 1965. Chidume, C. E., Geometric Properties of Banach Spaces and Nonlinear Iterations, (Lecture Notes in Mathematics, Springer, London, UK, 2009, 1965).
• Diaz, J. B. and Metcalf, F. T., On the structure of the set of subsequential limit points of successive approximations, Bull. Amer. Math. Soc., 73, 516–519, 1967.
• Ghosh, M. K. and Debnath, L., Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl., 207, 96–103, 1997.
• Goebel, K. and Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35, 171–174, 1972.
• Goebel, K. and Kirk, W. A., Topics in Metric Fixed Point Theory, (Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990).
• Gu, F. and Lu, J., A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces, Fixed Point Theo. Appl., Article ID 82738, 2006. Lami Dozo E., Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc., 1973, 38, 286-292.
• Liu, Q., Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259, 1–7, 2001.
• Mukhamedov, F. and Saburov, M., Weak and strong convergence of an implicit iteration process for an asymptotically quasi I-nonexpansive mapping in Banach spaces, Fixed Point Theo. Appl. doi:10.1155/2010/719631, 2010.
• Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591–597, 1967.
• Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158, 407–413, 1991.
• Shahzad, N., Generalized I-nonexpansive maps and best approximations in Banach spaces, Demonstratio Mathematica, bf 37, 597–600, 2004.
• Sun, Z. -H., Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286, 351–358, 2003.
• Tan, K. -K. and Xu, H. -K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178, 301–308, 1993.
• Temir, S., On the convergence theorems of implicit iteration process for a finite family of I-asymptotically nonexpansive mappings, J. Comp. Appl. Math., 225, 398–405, 2009. Wittmann, R., Approximation of fixed points of nonexpansive mappings, Archiv der Mathematik, 58, 486–491, 1992.
• Xu, H. -K., Ori R. G., An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22, 767–773, 2001.