Abstract
In this paper, we prove a strong convergence theorem of modified Ishikawa iterations for relatively asymptotically nonexpansive mappings in Banach space. Our results extend and improve the recent results by Nakajo, Takahashi, Kim, Xu, Matsushita and some others.
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Alber Ya I, Metric and generalized projection operators in Banach spaces: Properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (ed.) A G Kartsatos, (New York: Marcel Dekker) (1996) 15–50
Alber Ya I and Reich S, An iterative method for solving a class of nonlinear operatore-quations in Banach spaces, Panamer. Math. J. 4 (1994) 39–54
Alber Ya and Guerre-Delabriere S, On the projection methods for fixed point problems, Analysis 21 (2001) 17–39
Butnariu D, Reich S and Zaslavski A J, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7 (2001) 151–174
Butnariu D, Reich S and Zaslavski A J, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim. 24 (2003) 489–508
Censor Y and Reich S, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996) 323–339
Chidume C E and Mutangadura S A, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Am. Math. Soc. 129 (2001) 2359–2363
Cioranescu I, Geometry of Banach spaces, duality mappings and nonlinear problems, (Dordrecht: Kluwer) (1990)
Genel A and Lindenstrass J, An example concerning fixed points, Israel J. Math. 22 (1975) 81–86
Goebel K and Kirk W A, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc. 35 (1972) 171–174
Ishikawa S, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974) 147–150
Kamimura S and Takahashi W, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002) 938–945
Kim T H and Xu H K, Strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152
Mann W R, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953) 506–510
Nakajo K and Takahashi W, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379
Reich S, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276
Reich S, A weak convergence theorem for the alternating method with Bregman distance, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (ed.) A G Kartsatos (New York: Marcel Dekker) (1996) 313–318
Takahashi W, Nonlinear Functional Analysis (Yokohama-Publishers) (2000)
Yanes C Martinez and Xu H K, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400–2411
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Su, Y., Qin, X. Strong convergence of modified Ishikawa iterations for nonlinear mappings. Proc Math Sci 117, 97–107 (2007). https://doi.org/10.1007/s12044-007-0008-y
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DOI: https://doi.org/10.1007/s12044-007-0008-y