The Mann-Ishikawa iterations and the Mann-Ishikawa iterations with errors are equivalent models dealing with a non-Lipschitzian map

Authors

  • B. E. Rhoades Indiana University, Department of Mathematics, Bloomington, United States
  • Ştefan M. Şoltuz Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy

Keywords:

Mann-Ishikawa iterations, Mann-Ishikawa iterations with errors

Abstract

The Mann-Ishikawa iterations and the Mann-Ishikawa iterations with errors are equivalent models for several classes of non-Lipschitzian operators.

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References

Banach S., Sur les opérations dans les ensembles abstraits et leur applications, Fund. Math. 3, pp.133-181, 1922,https://doi.org/10.4064/fm-3-1-133-181

Chidume, C.E., Muntangadura, S.A., An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129, pp. 2359-2363, 2001, https://doi.org/10.1090/s0002-9939-01-06009-9

Chidume, C.E., Moore, C., Steepest descent method for equilibrium points of nonlinear systems with accretive operators, J. Math. Anal. Appl. 245, pp. 142-160, 2000, https://doi.org/10.1006/jmaa.2000.6744

Ishikawa, S., Fixed Points by a new iteration method, Proc. Amer. Math. Soc. 44, pp. 147-150, 1974, https://doi.org/10.1090/s0002-9939-1974-0336469-5

Kannan,R., Some results on fixed points, Bull. Calcutta Math. Soc. 60, pp. 71-76, 1968.

Liu, L.S., Ishikawa and Mann Iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194, pp. 114-125, 1995, https://doi.org/10.1006/jmaa.1995.1289

Mann, W.R., Mean value in iteration, Proc. Amer. Math. Soc. 4, pp. 506-510, 1953, https://doi.org/10.1090/s0002-9939-1953-0054846-3

Morales, C., Jung, J.S., Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128:11, pp. 3411-3419, 2000, https://doi.org/10.1090/s0002-9939-00-05573-8

Osilike, M.O., Stability of the Mann and Ishikawa iteration procedures for φ-strongly pseudocontractions and nonlinear equations of the φ-strongly accretive type, J. Math. Anal. Appl. 227, pp. 319-334, 1998, https://doi.org/10.1006/jmaa.1998.6075

Rhoades, B.E., Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196, pp. 161-176, 1974, https://doi.org/10.1090/s0002-9947-1974-0348565-1

Rhoades, B.E., Şoltuz, Ş.M., The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Internat. J. Math. & Math. Sci. 2003, pp. 451-459, 2003, https://doi.org/10.1155/s0161171203110198

Rhoades, B.E., Şoltuz, Ş.M., The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Internat. J. Math. & Math. Sci. 2003, pp. 2645-2652, 2003, https://doi.org/10.1155/s0161171203211418

Rhoades, B.E., Şoltuz, Ş.M., The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically pseudocontractive map, J. Math. Anal. Appl.283, pp. 681-688, 2003, https://doi.org/10.1016/s0022-247x(03)00338-x

Rhoades, B.E., Şoltuz, Ş.M., The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289, pp. 266-278, 2004, https://doi.org/10.1016/j.jmaa.2003.09.057

Rhoades, B.E., Şoltuz, Ş.M., The Equivalence of Mann and Ishikawa Iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly Accretive Maps, Tamkang J. Math., 35, pp. 235-245, 2004, https://doi.org/10.5556/j.tkjm.35.2004.204

Rhoades, B.E., Şoltuz, Ş.M., The equivalence of Mann iteration and Ishikawa iteration for ψ- uniformly pseudocontractive or ψ-uniformly accretive maps, Internat. J. Math. & Math. Sci. 2004, pp. 2443-2452, 2004, https://doi.org/10.1155/s0161171204312020

Şoltuz, Ş.M., Mann-Ishikawa iterations and Mann-Ishikawa iterations with errors are equivalent models, Math. Commun. 8, pp. 139-150, 2003.

Xu, Y., Ishikawa and Mann iterative process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224, pp. 91-101, 1998,https://doi.org/10.1006/jmaa.1998.5987

Zhou, H., Jia, Y., Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125, pp. 1705-1709, 1997.

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Published

2005-08-01

How to Cite

Rhoades, B. E., & Şoltuz, Ştefan M. (2005). The Mann-Ishikawa iterations and the Mann-Ishikawa iterations with errors are equivalent models dealing with a non-Lipschitzian map. Rev. Anal. Numér. Théor. Approx., 34(2), 181–193. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2005-vol34-no2-art7

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