Abstract
The Mann-Ishikawa iterations and the Mann-Ishikawa iterations with errors are equivalent models for several classes of non-Lipschitzian operators.
Authors
B. E. Rhoades
Indiana University, Department of Mathematics, Bloomington, USA
Ştefan M. Şoltuz
Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Mann-Ishikawa iterations; Mann-Ishikawa iterations with errors
Paper coordinates
B.E. Rhoades and Ş.M. Şoltuz, The Mann and Ishikawa iterations and the Mann-Ishikawa with errors are equivalent models dealing with a non-Lipschitzian map, Rev. Anal. Numer. Theor. Approx., 34 (2005) no. 2, 181-193.
About this paper
Journal
Journal of Numerical Analysis and Approximation Theory
Publisher Name
Romanian Academy
Print ISSN
2457-6794
Online ISSN
2501-059X.
google scholar link
[1] Banach S.,Sur les operations dans les ensembles abstraits et leur applications, Fund.Math.3, pp. 133–181, 1922.
[2] 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.
[3] 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.[4] Ishikawa, S.,Fixed Points by a new iteration method, Proc. Amer. Math. Soc.44,pp. 147–150, 1974.
[5] Kannan,R.,Some results on fixed points, Bull. Calcutta Math. Soc.60, pp. 71–76,1968.
[6] Liu, L.S.,Ishikawa and Mann Iterative process with errors for nonlinear strongly ac-cretive mappings in Banach spaces, J. Math. Anal. Appl.194, pp. 114–125, 1995.
[7] Mann, W.R.,Mean value in iteration, Proc. Amer. Math. Soc.4, pp. 506–510, 1953.
[8] Morales, C., Jung, J.S.,Convergence of paths for pseudocontractive mappings inBanach spaces, Proc. Amer. Math. Soc.128:11, pp. 3411–3419, 2000.
[9] Osilike, M.O.,Stability of the Mann and Ishikawa iteration procedures forφ-stronglypseudocontractions and nonlinear equations of theφ-strongly accretive type, J. Math.Anal. Appl.227, pp. 319–334, 1998.
[10] Rhoades, B.E.,Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc.196, pp. 161–176, 1974.
[11] Rhoades, B.E., Soltuz, S.M.,The equivalence of Mann iteration and Ishikawa itera-tion for non-Lipschitzian operators, Internat. J. Math. & Math. Sci.2003, pp. 451–459,2003.
[12] Rhoades, B.E., S ̧oltuz, S ̧.M.,The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Internat. J. Math. & Math. Sci.2003, pp. 2645–2652, 2003.
[13] Rhoades, B.E., S ̧oltuz, S ̧.M.,The equivalence between the convergences of Ishikawaand Mann iterations for asymptotical ly pseudocontractive map, J. Math. Anal.Appl.283, pp. 681–688, 2003.
[14] Rhoades, B.E., S ̧oltuz, S ̧.M.,The equivalence between the convergences of Ishikawaand Mann iterations for asymptotical ly nonexpansive in the intermediate sense andstrongly successively pseudocontractive maps, J. Math. Anal. Appl.289, pp. 266–278,2004.
[15] Rhoades, B.E., S oltuz, S.M.,The Equivalence of Mann and Ishikawa Iterationfor a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly Accretive Maps,Tamkang J. Math.,35, pp. 235–245, 2004.
[16] Rhoades, B.E., S oltuz, S.M.,The equivalence of Mann iteration and Ishikawa iter-ation forψ-uniformly pseudocontractive orψ-uniformly accretive maps, Internat. J.Math. & Math. Sci.2004, pp. 2443–2452, 2004.
[17] S ̧oltuz, S ̧.M.,Mann-Ishikawa iterations and Mann-Ishikawa iterations with errors areequivalent models, Math. Commun.8, pp. 139–150, 2003.
[18] Xu, Y.,Ishikawa and Mann iterative process with errors for nonlinear strongly accretiveoperator equations, J. Math. Anal. Appl.224, pp. 91–101, 1998.
[19] Zhou, H., Jia, Y.,Approximation of fixed points of strongly pseudocontractive mapswithout Lipschitz assumption, Proc. Amer. Math. Soc.125, pp. 1705–1709, 1997.