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Approximation of zeros of accretive operators in a Banach space. (English) Zbl 1487.47096

Summary: We consider the problem of finding a zero of an accretive operator in a Banach space and prove strong convergence results for resolvents of the accretive operator.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
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[1] K. Aoyama, Y. Kimura and F. Kohsaka, Strong convergence theorems for strongly relatively nonexpansive sequences and applications, Journal of Nonlinear Analysis and Optimization3 (2012), 67-77. · Zbl 1413.47100
[2] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis67 (2007), 2350-2360. · Zbl 1130.47045 · doi:10.1016/j.na.2006.08.032
[3] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, On a strongly nonexpansive sequence in Hilbert spaces, Journal of Nonlinear and Convex Analysis8 (2007), 471-489. · Zbl 1142.47030
[4] Aoyama, K.; Kimura, Y.; Takahashi, W.; Toyoda, M., Strongly nonexpansive sequences and their applications in Banach spaces, 1-18 (2008) · Zbl 1200.47071
[5] H. H. Bauschke, E. Matoušková and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Analysis56 (2004), 715-738. · Zbl 1059.47060 · doi:10.1016/j.na.2003.10.010
[6] O. A. Boikanyo and G. Moroşanu, A proximal point algorithm converging strongly for general errors, Optimization Letters4 (2010), 635-641. · Zbl 1202.90271 · doi:10.1007/s11590-010-0176-z
[7] H. Brézis and P.-L. Lions, Produits infinis de résolvantes, Israel Journal of Mathematics29 (1978), 329-345. · Zbl 0387.47038 · doi:10.1007/BF02761171
[8] R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston Journal of Mathematics3 (1977), 459-470. · Zbl 0383.47035
[9] García Falset, J., Strong convergence theorems for resolvents of accretive operators, 87-94 (2006), Yokohama · Zbl 1115.47039
[10] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and AppliedMathematics, Vol. 83, Marcel Dekker, Inc., New York, (1984). · Zbl 0537.46001
[11] O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization29 (1991), 403-419. · Zbl 0737.90047 · doi:10.1137/0329022
[12] B. Halpern, Fixed points of nonexpanding maps, Bulletin of the American Mathematical Society73 (1967), 957-961. · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[13] S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, Journal of Approximation Theory106 (2000), 226-240. · Zbl 0992.47022 · doi:10.1006/jath.2000.3493
[14] S. Kamimura and W. Takahashi, Iterative schemes for approximating solutions of accretive operators in Banach spaces, Scientiae Mathematicae3 (2000), 107-115 (electronic). · Zbl 0994.47049
[15] S. Kamimura and W. Takahashi, Weak and strong convergence of solutions to accretive operator inclusions and applications, Set-Valued Analysis8 (2000), 361-374. · Zbl 0981.47036 · doi:10.1023/A:1026592623460
[16] Kopecká, E.; Reich, S., Nonexpansive retracts in Banach spaces, 161-174 (2007), Warsaw · Zbl 1125.46019 · doi:10.4064/bc77-0-12
[17] N. Lehdili and A. Moudafi, Combining the proximal algorithm and Tikhonov regularization, Optimization,37 (1996), 239-252. · Zbl 0863.49018 · doi:10.1080/02331939608844217
[18] P.-E. Maingé, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis16 (2008), 899-912. · Zbl 1156.90426 · doi:10.1007/s11228-008-0102-z
[19] B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Revue Fran¸caise d’Informatique et de Recherche Opérationnelle4 (1970), 154-158. · Zbl 0215.21103 · doi:10.1051/m2an/197004R301541
[20] J.-J. Moreau, Proximité et dualité dans un espace hilbertien, Bulletin de la Société Mathématique de France93 (1965), 273-299. · Zbl 0136.12101 · doi:10.24033/bsmf.1625
[21] K. Nakajo Strong convergence to zeros of accretive operators in Banach spaces, Journal of Nonlinear and Convex Analysis7 (2006), 71-81. · Zbl 1126.49025
[22] O. Nevanlinna and S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel Journal of Mathematics32 (1979), 44-58. · Zbl 0427.47049 · doi:10.1007/BF02761184
[23] S. Reich, Asymptotic behavior of contractions in Banach spaces, Journal of Mathematical Analysis and Applications44 (1973), 57-70. · Zbl 0275.47034 · doi:10.1016/0022-247X(73)90024-3
[24] Reich, S., Constructive techniques for accretive and monotone operators, 335-345 (1979), New York-London · Zbl 0444.47042
[25] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications67 (1979), 274-276. · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6
[26] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, Journal of Mathematical Analysis and Applications75 (1980), 287-292. · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[27] S. Reich, On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, Journal of Mathematical Analysis and Applications79 (1981), 113-126. · Zbl 0457.47053 · doi:10.1016/0022-247X(81)90013-5
[28] R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific Journal of Mathematics17 (1966), 497-510. · Zbl 0145.15901 · doi:10.2140/pjm.1966.17.497
[29] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization14 (1976), 877-898. · Zbl 0358.90053 · doi:10.1137/0314056
[30] S. Saejung, Halpern’s iteration in Banach spaces, Nonlinear Analysis73 (2010), 3431-3439. · Zbl 1234.47054 · doi:10.1016/j.na.2010.07.031
[31] D. R. Sahu and J. C. Yao, The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces, Journal of Global Optimization51 (2011), 641-655. · Zbl 1247.47048 · doi:10.1007/s10898-011-9647-8
[32] N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proceedings of the American Mathematical Society125 (1997), 3641-3645. · Zbl 0888.47034 · doi:10.1090/S0002-9939-97-04033-1
[33] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, (2000). · Zbl 0997.47002
[34] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, (2009). · Zbl 1183.46001
[35] W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Analysis70 (2009), 719-734. · Zbl 1170.47048 · doi:10.1016/j.na.2008.01.005
[36] W. Takahashi and Y. Ueda, On Reich’s strong convergence theorems for resolvents of accretive operators, Journal of Mathematical Analysis and Application104 (1984), 546-553. · Zbl 0599.47084 · doi:10.1016/0022-247X(84)90019-2
[37] C. Tian and Y. Song, Strong convergence of a regularization method for Rockafellar’s proximal point algorithm, Journal of Global Optimization55 (2013), 831-837. · Zbl 1288.90127 · doi:10.1007/s10898-011-9827-6
[38] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Archiv der Mathematik58 (1992), 486-491. · Zbl 0797.47036 · doi:10.1007/BF01190119
[39] H.-K. Xu, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society66 (2002), 240-256. · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[40] H.-K. Xu, A regularization method for the proximal point algorithm, Journal of Global Optimization36 (2006), 115-125. · Zbl 1131.90062 · doi:10.1007/s10898-006-9002-7
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