×

Strong convergence of a modified Halpern’s iteration for nonexpansive mappings. (English) Zbl 1203.47049

Summary: The purpose of this paper is to prove that a modified Halpern’s iterative sequence \({x_{n}}\) converges strongly to a fixed point of nonexpansive mappings in Banach spaces which have a uniformly Gâteaux differentiable norm. Our result is an extension of the corresponding results of several authors.
Editorial remark: A counterexample to the main theorem has been found by S.Wang [Fixed Point Theory Appl.2010, Article ID 805326 (2010; Zbl 1203.47080)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1203.47080
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] doi:10.1007/BF00251595 · Zbl 0148.13601 · doi:10.1007/BF00251595
[2] doi:10.1016/j.jmaa.2005.05.023 · Zbl 1095.47034 · doi:10.1016/j.jmaa.2005.05.023
[3] doi:10.1090/S0002-9904-1967-11864-0 · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[4] doi:10.1155/2008/824607 · Zbl 1203.47050 · doi:10.1155/2008/824607
[5] doi:10.1155/2008/167535 · Zbl 1203.47053 · doi:10.1155/2008/167535
[6] doi:10.1016/j.na.2004.11.011 · Zbl 1091.47055 · doi:10.1016/j.na.2004.11.011
[9] doi:10.1090/S0002-9939-00-05573-8 · Zbl 0970.47039 · doi:10.1090/S0002-9939-00-05573-8
[10] doi:10.1016/0022-247X(80)90323-6 · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[12] doi:10.1090/S0002-9939-97-04033-1 · Zbl 0888.47034 · doi:10.1090/S0002-9939-97-04033-1
[13] doi:10.1155/FPTA.2005.103 · Zbl 1123.47308 · doi:10.1155/FPTA.2005.103
[14] doi:10.1090/S0002-9939-06-08435-8 · Zbl 1117.47041 · doi:10.1090/S0002-9939-06-08435-8
[15] doi:10.1016/0022-247X(84)90019-2 · Zbl 0599.47084 · doi:10.1016/0022-247X(84)90019-2
[16] doi:10.1007/BF01190119 · Zbl 0797.47036 · doi:10.1007/BF01190119
[17] doi:10.1112/S0024610702003332 · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[18] doi:10.1016/j.jmaa.2007.03.078 · doi:10.1016/j.jmaa.2007.03.078
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.