Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. (English) Zbl 1086.47057

The reviewer introduced and analyzed three-step iterative schemes for solving nonlinear operator equations in Hilbert spaces in [M. A.Noor, J.Math.Anal.Appl.251, 217–229 (2000; Zbl 0964.49007)]. These three-step iterative schemese are also known as Noor iterations. These iterations have been extended and modified for several classes of nonexpansive mappings. It is well-known that Noor iterations include the Mann and Ishikawa iterations as special cases. B. Xu and M. A.Noor [J.Math.Anal.Appl.267, 444–453 (2002; Zbl 1011.47039)] studied Noor iterations for a class of nonexpansive for asymptotically mappings.
In the paper under review, modified Noor-iterations are suggested and analyzed. In particular, the author investigates weak and strong convergence criteria of his modified Noor iterations under some suitable mild conditions. The results proved in this paper are interesting and present an improvement of the previously known results.


47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H05 Monotone operators and generalizations
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[1] Bose, S. C., Weak convergence to the fixed point of an asymptotically nonexpansive map, Proc. Amer. Math. Soc., 68, 305-308 (1978) · Zbl 0377.47037
[2] Chidume, C. E.; Moor, C., Fixed point iteration for pseudocontractive maps, Proc. Amer. Math. Soc., 127, 1163-1170 (1999) · Zbl 0913.47052
[3] Cho, Y. J.; Zhou, H. Y.; 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
[4] Glowinski, R.; Le Tallec, P., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (1989), SIAM: SIAM Philadelphia · Zbl 0698.73001
[5] Goebel, K.; Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35, 171-174 (1972) · Zbl 0256.47045
[6] 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
[7] Hicks, T.; Kubicek, J., On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl., 59, 498-504 (1977) · Zbl 0361.65057
[8] Ishikawa, S., Fixed point by a new iteration, Proc. Amer. Math. Soc., 44, 147-150 (1974) · Zbl 0286.47036
[9] Kruppel, M., On an inequality for nonexpansive mappings in uniformly convex Banach spaces, Rostock Math. Kolloq., 51, 25-32 (1997) · Zbl 0891.47037
[10] Liu, Q. H., Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member in a uniformly convex Banach space, J. Math. Anal. Appl., 266, 468-471 (2002) · Zbl 1057.47057
[11] Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603
[12] Noor, M. A., New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 217-229 (2000) · Zbl 0964.49007
[13] Opial, Z., Weak convergence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, 591-597 (1967) · Zbl 0179.19902
[14] Rhoades, B. E., Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183, 118-120 (1994) · Zbl 0807.47045
[15] Rhoades, R. E.; Soltuz, S. M., The equivalence between Mann-Ishikawa iterations and multistep iterations, Nonlinear Anal., 58, 59-68 (2004) · Zbl 1064.47070
[16] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158, 407-413 (1991) · Zbl 0734.47036
[17] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43, 153-159 (1991) · Zbl 0709.47051
[18] Tan, K. K.; Xu, H. K., Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl., 178, 301-308 (1993) · Zbl 0895.47048
[19] Tan, K. K.; Xu, H. K., Fixed point iteration processes for asymptotically nonexpansive mapping, Proc. Amer. Math. Soc., 122, 733-739 (1994) · Zbl 0820.47071
[21] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 1127-1138 (1991) · Zbl 0757.46033
[22] Xu, H. K., Existence and convergence for fixed points of asymptotically nonexpansive type, Nonlinear Anal., 16, 1139-1146 (1991) · Zbl 0747.47041
[23] Xu, B. L.; Aslam Noor, M., Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267, 444-453 (2002) · Zbl 1011.47039
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