zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces. (English) Zbl 1236.47074
Summary: We note that the main convergence theorem in [C.-J. Zhang, J.-L. Li and B.-Q. Liu, Comput. Math. Appl. 61, No. 2, 262–276 (2011; Zbl 1211.65063)] is incorrect and we prove a correction. We also modify Halpern’s iteration for finding a fixed point of a strongly relatively nonexpansive mapping in a Banach space. Consequently, two strong convergence theorems for a relatively nonexpansive mapping and for a mapping of firmly nonexpansive type are deduced. Using the concept of duality theorems, we obtain analogous results for strongly generalized nonexpansive mappings and for mappings of firmly generalized nonexpansive type. In addition, we study two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
References:
[1]Mann, W. R.: Mean value methods in iteration, Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603 · doi:10.2307/2032162
[2]Genel, A.; Lindenstrauss, J.: An example concerning fixed points, Israel J. Math. 22, 81-86 (1975) · Zbl 0314.47031 · doi:10.1007/BF02757276
[3]Halpren, B.: Fixed points of nonexpansive maps, Bull. amer. Math. soc. 73, 957-961 (1967)
[4]Bauschke, H. H.; Combettes, P. L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. oper. Res. 26, 248-264 (2001) · Zbl 1082.65058 · doi:10.1287/moor.26.2.248.10558
[5]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 · doi:10.1155/S1687182004310089
[6]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
[7]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 · doi:10.1016/j.jat.2005.02.007
[8]Alber, Y. I.: Metric and generalized projection operators in Banach spaces: properties and applications, Lecture notes in pure and appl. Math. 178, 15-50 (1996) · Zbl 0883.47083
[9]Reich, S.: A weak convergence theorem for the alternating method with Bregman distances, Lecture notes in pure and appl. Math. 178, 313-318 (1996) · Zbl 0943.47040
[10]Aoyama, K.; Kohsaka, F.; Takahashi, W.: Strongly nonexpansive sequences in Banach spaces and applications, J. fixed point theory appl. 5, 201-225 (2009)
[11]Kimura, Y.; Takahashi, W.: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space, J. math. Anal. appl. 357, No. 2, 356-363 (2009) · Zbl 1166.47058 · doi:10.1016/j.jmaa.2009.03.052
[12]Kohsaka, F.; Takahashi, W.: Approximating common fixed points of countable families of strongly nonexpansive mappings, Nonlinear stud. 14, 219-234 (2007) · Zbl 1147.47050
[13]Lewicki, G.; Marino, G.: On some algorithms in Banach spaces finding fixed points of nonlinear mappings, Nonlinear anal. 71, 3964-3972 (2009) · Zbl 1171.41009 · doi:10.1016/j.na.2009.02.066
[14]Li, X.; Huang, N.; O’regan, D.: Strong convergence theorems for relatively nonexpansive mappings in Banach spaces with applications, Comput. math. Appl. 60, 1322-1331 (2010) · Zbl 1201.65091 · doi:10.1016/j.camwa.2010.06.013
[15]Nilsrakoo, W.; Saejung, S.: Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings, Fixed point theory appl. (2008)
[16]Nilsrakoo, W.; Saejung, S.: Strong convergence theorems by halpern–Mann iterations for relatively nonexpansive mappings in Banach spaces, Appl. math. Comput. 217, 6577-6586 (2011) · Zbl 1215.65104 · doi:10.1016/j.amc.2011.01.040
[17]Qin, X.; Cho, Y. J.; Kang, S. M.; Zhou, H.: Convergence of a modified halpern-type iteration algorithm for quasi-ϕ-nonexpansive mappings, Appl. math. Lett. 22, 1051-1055 (2009) · Zbl 1179.65061 · doi:10.1016/j.aml.2009.01.015
[18]Qin, X.; Su, Y.: Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear anal. 67, 1958-1965 (2007) · Zbl 1124.47046 · doi:10.1016/j.na.2006.08.021
[19]Reich, S.; Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces, Nonlinear anal. 73, 122-135 (2010) · Zbl 1226.47089 · doi:10.1016/j.na.2010.03.005
[20]Su, Y.; Wang, D.; Shang, M.: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed point theory appl. (2008) · Zbl 1203.47078 · doi:10.1155/2008/284613
[21]Zhang, C.; Li, J.; Liu, B.: Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Comput. math. Appl. 61, 262-276 (2011) · Zbl 1211.65063 · doi:10.1016/j.camwa.2010.11.002
[22]Censor, Y.; Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37, 323-339 (1996) · Zbl 0883.47063 · doi:10.1080/02331939608844225
[23]Bruck, R. E.; Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3, 459-470 (1977) · Zbl 0383.47035
[24]Reich, S.: A limit theorem for projections, Linear multilinear algebra 13, 281-290 (1983) · Zbl 0523.47040 · doi:10.1080/03081088308817526
[25]Aoyama, K.; Takahashi, W.: Strong convergence theorems for a family of relatively nonexpansive mappings in Banach spaces, Fixed point theory 8, 143-160 (2007) · Zbl 1143.47044
[26]Lee, M. B.; Park, S. H.: Convergence of sequential parafirmly nonexpansive mappings in reflexive Banach spaces, J. optim. Theory appl. 123, 549-571 (2004) · Zbl 1141.90499 · doi:10.1007/s10957-004-5723-2
[27]Kohsaka, F.; Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19, 824-835 (2008) · Zbl 1168.47047 · doi:10.1137/070688717
[28]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, 166-177 (2008) · Zbl 1149.47045 · doi:10.1007/s00013-008-2545-8
[29]Reich, S.; Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces, Fixed-point algorithms for inverse problems in science and engineering 49 (2011)
[30]Reich, S.; Sabach, S.: A projection method for solving nonlinear problems in reflexive Banach spaces, J. fixed point theory appl. 9, 101-116 (2011) · Zbl 1206.47043 · doi:10.1007/s11784-010-0037-5
[31]Bauschke, H. H.; Borwein, J. M.; Combettes, P. L.: Bregman monotone optimization algorithms, SIAM J. Control optim. 42, 596-636 (2003) · Zbl 1049.90053 · doi:10.1137/S0363012902407120
[32]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)
[33]Nilsrakoo, W.; Saejung, S.: On the fixed-point set of a family of relatively nonexpansive and generalized nonexpansive mappings, Fixed point theory appl. (2010) · Zbl 1203.47027 · doi:10.1155/2010/414232
[34]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 · doi:http://www.ybook.co.jp/online/jncae/vol10/p21.html
[35]Takahashi, W.: Nonlinear functional analysis, (2000) · Zbl 0997.47002
[36]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 · doi:10.1137/S105262340139611X
[37]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 · doi:10.1155/S1085337504309036
[38]Xu, H. K.: Another control condition in an iterative method for nonexpansive mappings, Bull. aust. Math. soc. 65, 109-113 (2002) · Zbl 1030.47036 · doi:10.1017/S0004972700020116
[39]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 · doi:10.1007/s11228-008-0102-z
[40]Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. Anal. appl. 178, 301-308 (1993) · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[41]Saejung, S.: Halpern’s iteration in Banach spaces, Nonlinear anal. 73, 3431-3439 (2010)
[42]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 · doi:10.1016/j.jat.2007.04.003
[43]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
[44]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
[45]Rockafellar, R. T.: On the maximality of sums of nonlinear monotone operators, Trans. amer. Math. soc. 149, 75-88 (1970) · Zbl 0222.47017 · doi:10.2307/1995660
[46]Ibaraki, T.; Takahashi, W.: Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces, Adv. math. Econ. 10, 51-64 (2007) · Zbl 1131.49016