Strong convergence theorems for multi-step Noor iterations with errors in Banach spaces. (English) Zbl 1095.47042

Plubtieng, Somyot; Wangkeeree, Rabian

J. Math. Anal. Appl. 321, No. 1, 10-23 (2006).

Summary: We establish two strong convergence theorems for a multi-step Noor iterative scheme with errors for mappings which are asymptotically nonexpansive in the intermediate sense (or asymptotically quasi-nonexpansive, respectively) in Banach spaces. Our results extend and improve recent ones announced by B.–L. Xu and M. A. Noor [J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039)], Y. J. Cho, H.–Y. Zhou and G.–T. Guo, Comput. Math. Appl. 47, No. 4–5, 707–717 (2004; Zbl 1081.47063)], and many others.

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10	Fixed-point theorems

Keywords:

asymptotically nonexpansive in the intermediate sense; asymptotically quasi-nonexpansive mappings; completely continuous; uniformly convex; uniformly \(L\)-Lipschitzian

Citations:

Zbl 1011.47039; Zbl 1081.47063

References:

[1]	Bruck, R.E.; Kuczumow, T.; Reich, S., Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. math., 65, 169-179, (1993) · Zbl 0849.47030
[2]	Cho, Y.J.; Zhou, H.; Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. math. appl., 47, 707-717, (2004) · Zbl 1081.47063
[3]	Glowinski, R.; Le Tallec, P., Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, (1989), SIAM Philadelphia · Zbl 0698.73001
[4]	Goebel, K.; Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. amer. math. soc., 35, 171-174, (1972) · Zbl 0256.47045
[5]	Haubruge, S.; Nguyen, V.H.; Strodiot, J.J., Convergence analysis and applications of the glowinski – le tallec splitting method for finding a zero of the sum of two maximal monotone operators, J. optim. theory appl., 97, 645-673, (1998) · Zbl 0908.90209
[6]	Ishikawa, S., Fixed point by a new iterations, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036
[7]	Jeong, J.U.; Aslam Noor, M.; Rafiq, A., Noor iterations for nonlinear Lipschitzian strongly accretive mappings, J. Korea soc. math. educ. ser. B pure appl. math., 11, 339-350, (2004) · Zbl 1142.49301
[8]	Kim, T.H.; Choi, J.W., Asymptotic behavior of almost-orbits of non-Lipschitzian mappings in Banach spaces, Math. japon., 38, 191-197, (1993) · Zbl 0774.47026
[9]	Kim, G.E.; Kim, T.H., Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comput. math. appl., 42, 1565-1570, (2001) · Zbl 1001.47048
[10]	Liu, Q., Iterations sequence for asymptotically quasi-nonexpansive mapping with an error member, J. math. anal. appl., 259, 18-24, (2001)
[11]	Mann, W.R., Mean value methods in iterations, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[12]	Kirk, W.A., Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. math., 17, 339-346, (1974) · Zbl 0286.47034
[13]	Noor, M.A., New approximation schemes for general variational inequalities, J. math. anal. appl., 251, 217-229, (2000) · Zbl 0964.49007
[14]	Noor, M.A., Three-step iterative algorithms for multivalued quasi variational inclusions, J. math. anal. appl., 255, (2001) · Zbl 0986.49006
[15]	Noor, M.A.; Rassias, T.M.; Huang, Z., Three-step iterations for nonlinear accretive operator equations, J. math. anal. appl., 274, 59-68, (2002) · Zbl 1028.65063
[16]	Rhoades, B.E., Fixed point iterations for certain nonlinear mappings, J. math. anal. appl., 183, 118-120, (1994) · Zbl 0807.47045
[17]	Schu, J., Iterative construction of fixed points of strictly quasicontractive mapping, Appl. anal., 40, 67-72, (1991) · Zbl 0697.47061
[18]	S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., in press · Zbl 1086.47057
[19]	Xu, B.L.; Noor, M.A., Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. math. anal. appl., 267, 444-453, (2002) · Zbl 1011.47039