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Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1011.47039

The authors suggest a new class of three step iterative schemes for solving nonlinear equations \(Tx=x\) with asymptotically nonexpansive mappings in Banach spaces. The Ishikawa-type and Mann-type iteration schemes are included as particular cases. The convergence of the new schemes is proved.

MSC:

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

References:

[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] S. S. Chang, Weak Convergence to the Fixed Point for Nonexpansive and Asymptotically Nonexpansive Mappings in Banach Spaces, preprint.; S. S. Chang, Weak Convergence to the Fixed Point for Nonexpansive and Asymptotically Nonexpansive Mappings in Banach Spaces, preprint.
[3] Chidume, C. E.; Moor, Chika, Fixed point iteration for pseudocontractive maps, Proc. Amer. Math. Soc., 127, 1163-1170 (1999) · Zbl 0913.47052
[4] Glowinski, R.; Le Tallec, P., Augemented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics (1989), SIAM: SIAM Philadelphia · Zbl 0698.73001
[5] Goeble, K.; Kirk, W. A., A fixed point theorem fo 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 operaors, 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] Liu, Q. H., Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal., 26, 1835-1842 (1996) · Zbl 0861.47047
[10] Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc., 4, 506-510 (1953) · Zbl 0050.11603
[11] Noor, M. Aslam, New aproximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 217-229 (2000) · Zbl 0964.49007
[12] Noor, M. Aslam, Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math. Anal. Appl. (2001) · Zbl 0986.49006
[13] Noor, M. Aslam, Some predictor-corrector algorithms for multivalued variational inequalites, J. Optim. Theory Appl., 108 (2001) · Zbl 0996.47055
[14] Rhoades, B. E., Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183, 118-120 (1994) · Zbl 0807.47045
[15] Rhoades, B. E., Comments on two fixed point iteration methods, J. Math. Anal. Appl., 56, 741-750 (1976) · Zbl 0353.47029
[16] Rhoades, B. E., A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226, 257-290 (1977) · Zbl 0365.54023
[17] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158, 407-413 (1991) · Zbl 0734.47036
[18] Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43, 153-159 (1991) · Zbl 0709.47051
[19] 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
[20] 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 applicaitons, 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
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