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An example on the Mann iteration method for Lipschitz pseudocontractions. (English) Zbl 0972.47062

It was an open problem for the last 20 years whether or not the Mann iterations \(x_{n+1}= (1-c_n)x_n+ c_nTx_n\) for a pseudocontractive self-map \(T\) of a compact convex subset in a Hilbert space converge to a fixed-point of \(T\). In this paper the authors give a negative answer by constructing a counterexample for a Lipschitz pseudo-contractive map on the unit ball in \(\mathbb{R}^2\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

[1] Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041 – 1044. · Zbl 0128.35801
[2] David Borwein and Jonathan Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991), no. 1, 112 – 126. · Zbl 0742.26006 · doi:10.1016/0022-247X(91)90139-Q
[3] Shih Sen Chang, The Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings, Comput. Math. Appl. 37 (1999), no. 9, 17 – 24. · Zbl 0939.47053 · doi:10.1016/S0898-1221(99)00109-1
[4] C. E. Chidume, Global iteration schemes for strongly pseudo-contractive maps, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2641 – 2649. · Zbl 0901.47046
[5] Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251 – 258 (German). · Zbl 0127.08005 · doi:10.1002/mana.19650300312
[6] Troy L. Hicks and John D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (1977), no. 3, 498 – 504. · Zbl 0361.65057 · doi:10.1016/0022-247X(77)90076-2
[7] Shiro Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147 – 150. · Zbl 0286.47036
[8] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004 – 1006. · Zbl 0141.32402 · doi:10.2307/2313345
[9] M. A. Krasnoselski, Two observations about the method of succesive approximations, Uspehi Math. Nauk 10 (1955), no. 1, 123-127. (Russian)
[10] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. · Zbl 0050.11603
[11] Qi Hou Liu, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math. Anal. Appl. 148 (1990), no. 1, 55 – 62. · Zbl 0729.47052 · doi:10.1016/0022-247X(90)90027-D
[12] H. Schaefer, Uber die Methode sukzessiver Approximationen, Jber. Deutsch. Math. Verein. 59 (1957), 131-140. · Zbl 0077.11002
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