×

Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces. (English) Zbl 1215.65104

Authors’ abstract: We modify Halpern and Mann’s iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space. Consequently, a strong convergence theorem for a nonspreading mapping is deduced. Using a concept of duality theorems, we also obtain analogue results for certain generalized nonexpansive and generalized nonexpansive type mappings. Finally, we discuss two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
46B20 Geometry and structure of normed linear spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alber, Y.I., Metric and generalized projection operators in Banach spaces: properties and applications, (), 15-50 · Zbl 0883.47083
[2] Genel, A.; Lindenstrauss, J., An example concerning fixed points, Israel. J. math., 22, 81-86, (1975) · Zbl 0314.47031
[3] Halpren, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[4] Honda, T.; Ibaraki, T.; Takahashi, W., Duality theorems and convergence theorems for nonlinear mappings in Banach spaces and applications, Int. J. math. stat., 6, 46-64, (2010)
[5] Ibaraki, T.; Takahashi, W., A new projection and convergence theorems for projections in Banach spaces, J. approx. theory, 149, 1-14, (2007) · Zbl 1152.46012
[6] Ibaraki, T.; Takahashi, W., Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces, Adv. math. econom., 10, 51-64, (2007) · Zbl 1131.49016
[7] Ibaraki, T.; Takahashi, W., Weak convergence theorem for new nonexpansive mappings in Banach spaces and its applications, Taiwanese J. math., 11, 929-944, (2007) · Zbl 1219.47115
[8] Ibaraki, T.; Takahashi, W., Fixed point theorems for nonlinear mappings of nonexpansive type in Banach spaces, J. nonlinear convex anal., 10, 21-32, (2009) · Zbl 1168.47046
[9] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2002) · Zbl 1101.90083
[10] Kohsaka, F.; Takahashi, W., Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. appl. anal., 2004, 239-249, (2004) · Zbl 1064.47068
[11] Kohsaka, F.; Takahashi, W., Generalized nonexpansive retractions and a proximal-type algorithm in Banach spaces, J. nonlinear convex anal., 8, 197-209, (2007) · Zbl 1132.47051
[12] Kohsaka, F.; Takahashi, W., Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. math. (basel), 91, 66-177, (2008) · Zbl 1149.47045
[13] Maingé, P.E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-valued anal., 16, 899-912, (2008) · Zbl 1156.90426
[14] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[15] Matsushita, S.; Takahashi, W., Weak and strong convergence theorems for relatively nonexpansive mappings in a Banach space, Fixed point theory appl., 2004, 37-47, (2004) · Zbl 1088.47054
[16] S. Matsushita, W. Takahashi, An iterative algorithm for relatively nonexpansive mappings by hybrid method and applications, in: Proceedings of the Third International Conference on Nonlinear Analysis and Convex Analysis, (2004), pp. 305-313. · Zbl 1086.47055
[17] Matsushita, S.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. approx. theory, 134, 257-266, (2005) · Zbl 1071.47063
[18] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[19] Nilsrakoo, W.; Saejung, S., Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings, Fixed point theory appl, (2008), Art. ID 312454, 19 pp. · Zbl 1203.47061
[20] Nilsrakoo, W.; Saejung, S., On the fixed-point set of a family of relatively nonexpansive and generalized nonexpansive mappings, Fixed point theory appl., (2010), Art. ID 414232, 14 pp. · Zbl 1203.47027
[21] Plubtieng, S.; Sriprad, W., Strong and weak convergence of modified Mann iteration for new resolvents of maximal monotone operators in Banach spaces, Abstr. appl. anal., (2009), Art. ID 795432, 20 pp. · Zbl 1186.47074
[22] Rockafellar, R.T., On the maximality of sums of nonlinear monotone operators, Tran. amer. math. soc., 149, 75-88, (1970) · Zbl 0222.47017
[23] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama
[24] Xu, H.K., Inequalities in Banach spaces with applications, Nonlinear anal., 16, 1127-1138, (1991) · Zbl 0757.46033
[25] Xu, H.K., Another control condition in an iterative method for nonexpansive mappings, Bull. austral. math. soc., 65, 109-113, (2002) · Zbl 1030.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.