Abstract
We point out that results on the asymptotic behaviour of an alternative iterative method are corollaries of corresponding results on the well-known Halpern iteration
References
- V. Colao, L. Leuştean, G. Lopez-Acedo, V. Martin-Marquez: Alternative iterative methods for nonexpansive mappings, rates of convergence and applications, J. Convex Analysis 18 (2011) 465–487.https://doi.org/10.68381/jca18030Google Scholar
- P. L. Combettes, S. A. Hirstoaga: Approximating curves for nonexpansive and monotone operators, J. Convex Analysis 13 (2006) 633–646.https://doi.org/10.68381/jca13049Google Scholar
- B. Halpern: Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 957–961.https://doi.org/10.1090/S0002-9904-1967-11864-0Google Scholar
- U. Kohlenbach: On quantitative versions of theorems due to F. E. Browder and R. Wittmann, Adv. Math. 226 (2011) 2764–2795.https://doi.org/10.1016/j.aim.2010.10.002Google Scholar
- U. Kohlenbach, L. Leuştean: Effective metastability of Halpern iterates in CAT(0) spaces, Adv. Math. 231 (2012) 2526–2556; Addendum in: Adv. Math. 250 (2014) 650–651.https://doi.org/10.1016/j.aim.2012.06.028https://doi.org/10.1016/j.aim.2013.07.023Google Scholar
- D. Körnlein: Quantitative results for Halpern iterations of nonexpansive mappings, J. Math. Anal. Appl. 428 (2015) 1161–1172.https://doi.org/10.1016/j.jmaa.2015.03.020Google Scholar
- L. Leuştean, A. Nicolae: Effective results on compositions of nonexpansive mappings, J. Math. Anal. Appl. 410 (2014) 902–907.https://doi.org/10.1016/j.jmaa.2013.09.006Google Scholar
- L. Leuştean, A. Nicolae: Effective results on nonlinear ergodic averages in CAT(κ) spaces, Ergodic Theory Dyn. Syst., to appear; arXiv:1305.5916 [math.FA] (2015).https://doi.org/10.1017/etds.2015.31Google Scholar
- S. Reich: Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292.https://doi.org/10.1016/0022-247X(80)90323-6Google Scholar
- N. Shioji, W. Takahashi: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641–3645.https://doi.org/10.1090/S0002-9939-97-04033-1Google Scholar
- R. Wittmann: Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491.https://doi.org/10.1007/BF01190119Google Scholar
- H. K. Xu: Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66 (2002) 240–256.https://doi.org/10.1112/S0024610702003332Google Scholar
- H. K. Xu: An alternative regularization method for nonexpansive mappings with applications, in: Nonlinear Analysis and Optimization I: Nonlinear Analysis (Haifa, 2008), A. Leizarowitz et al. (ed.), Cont. Math. 513, Amer. Math. Soc., Providence (2010) 239–263.https://doi.org/10.1090/conm/513/10087Google Scholar
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Published by Heldermann Verlag, 2017. Rights now held by Banach Press.
