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)
Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. (English) Zbl 1281.47050
Summary: Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in [Y. Kurokawa and W. Takahashi, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 6, 1562–1568 (2010; Zbl 1229.47117)]. We explore an auxiliary mapping in our theorems and proofs and this also yields a strong convergence theorem of Halpern type for our class of mappings and hence resolves in the affirmative an open problem posed by Kurokawa and Takahashi [loc. cit.] in their final remark for the case where the mapping T is averaged.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H25Nonlinear ergodic theorems
47H09Mappings defined by “shrinking” properties
References:
[1]Kohsaka, F.; Takahashi, W.: Fixed point theorems for a class of nonlinear mappings relate to maximal monotone operators in Banach spaces, Arch. math. (Basel) 91, 166-177 (2008) · Zbl 1149.47045 · doi:10.1007/s00013-008-2545-8
[2]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
[3]Igarashi, T.; Takahashi, W.; Tanaka, K.: Weak convergence theorems for nonspreading mappings and equilibrium problems, Nonlinear analysis and optimization, 75-85 (2009)
[4]Iemoto, S.; Takahashi, W.: Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear anal. 71, 2080-2089 (2009)
[5]Kurokawa, Y.; Takahashi, W.: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear anal. 73, 1562-1568 (2010) · Zbl 1229.47117 · doi:10.1016/j.na.2010.04.060
[6]Baillon, J.: Un théorème de type ergodique pour LES contractions nonlinéaires dans un espace de Hilbert, C. R. Acad. sci., Paris ser. A-B 280, No. Aii, A1511-A1514 (1975) · Zbl 0307.47006
[7]Halpern, B.: Fixed points of nonexpanding mappings, Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[8]Takahashi, W.; Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings, J. optim. Theory appl. 118, 417-428 (2003) · Zbl 1055.47052 · doi:10.1023/A:1025407607560
[9]Aoyama, K.; Kimura, Y.; Takahashi, W.; Toyoda, M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear anal. 67, 2350-2360 (2007) · Zbl 1130.47045 · doi:10.1016/j.na.2006.08.032
[10]Xu, H. K.: Iterative algorithms for nonlinear operators, J. lond. Math. soc. 66, No. 2, 240-256 (2002)
[11]Browder, F. E.; Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert spaces, J. math. Anal. appl. 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[12]Hicks, T. L.; Kubicek, J. R.: On the Mann iterative process in Hilbert spaces, J. math. Anal. appl. 59, 498-504 (1977) · Zbl 0361.65057 · doi:10.1016/0022-247X(77)90076-2
[13]Naimpally, S. A.; Singh, K. L.: Extensions of some fixed point theorems of rhoades, J. math. Anal. appl. 96, 437-446 (1983) · Zbl 0524.47033 · doi:10.1016/0022-247X(83)90052-5
[14]Song, Y.; Chai, X.: Halpern iteration for firmly type nonexpansive mappings, Nonlinear anal. 71, 4500-4506 (2009) · Zbl 1169.49010 · doi:10.1016/j.na.2009.03.018
[15]Saejung, S.: Halpern’s iteration in Banach spaces, Nonlinear anal. 73, 3431-3439 (2010)