Abstract
This paper introduces a composite iteration scheme for approximating a fixed point of nonexpansive mappings in the framework of uniformly smooth Banach spaces and the reflexive Banach spaces which have a weakly continuous duality map, respectively. we establish the strong convergence of the composite iteration scheme. The results improve and extend those of Kim, Xu, Wittmann and some others.
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References
B. Halpern (1967) ArticleTitleFixed points of nonexpansive maps Bull. Amer. Math. Soc. 73 957–961
F. E. Browder (1965) ArticleTitleFixed points theorems for noncompact mappings in Hilbert space in Proc. Natt. Acad. Sci., USA 53 1272–1276 Occurrence Handle10.1073/pnas.53.6.1272
F. E. Browder (1967) ArticleTitleConvergence of approximations to fixed points of nonexpansive mappings in Banach spaces Arch. Rational Mech. Anal. 24 82–90 Occurrence Handle10.1007/BF00251595
S. Reich (1980) ArticleTitleStrong convergence theorems for resolvents of accretive operators in Banach spaces J. Math. Anal. Appl. 75 287–292 Occurrence Handle10.1016/0022-247X(80)90323-6
P. L. Lions (1977) ArticleTitleApproximation de points fixes de contractions C. R. Acad. Sci. Sér A-B Paris 284 1357–1359
R. Wittmann (1992) ArticleTitleApproximation of fixed points nonexpansive mappings Arch. Math. 59 486–491 Occurrence Handle10.1007/BF01190119
S. Reich (1994) ArticleTitleApproximating fixed points of nonexpansive mappings Panamer. Math. J. 4 486–491
T. H. Kim H. K. Xu (2005) ArticleTitleStrong convergence of modified Mann iterations Nonlinear Anal. 61 51–60
H. K. Xu (2006) ArticleTitleStrong convergence of an iterative method for nonexpansive and accretive operators J. Math. Anal. Appl. 314 631–643 Occurrence Handle10.1016/j.jmaa.2005.04.082
T. C. Lim H. K. Xu (1994) ArticleTitleFixed point theorems for asymptotically nonexpansive mappings Nonlinear Anal. 22 1345–1355 Occurrence Handle10.1016/0362-546X(94)90116-3
R. E. Bruck (1973) ArticleTitleNonexpansive projections on subsets of Banach spaces Pacific J. Math. 47 341–355
K. Goebel S. Reich (1984) Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings Marcel Dekker New York
S. Reich (1973) ArticleTitleAsymptotic behavior of contractions in Banach spaces J. Math. Anal. Appl. 44 57–70 Occurrence Handle10.1016/0022-247X(73)90024-3
H. K. Xu (2002) ArticleTitleIterative algorithms for nonlinear operators J. London Math. Soc. 66 240–256 Occurrence Handle10.1112/S0024610702003332
H. K. Xu (2003) ArticleTitleAn iterative approach to quadratic optimization J. Optim. Theory Appl. 116 659–678 Occurrence Handle10.1023/A:1023073621589
Z. Opial (1967) ArticleTitleWeak convergence of the sequence of successive approximations for nonexpansive mappings Bull. Amer. Math. Soc. 73 595–587 Occurrence Handle10.1090/S0002-9904-1967-11761-0
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The project is supported by National Natural Science Foundation of China under Grant No. 60574005.
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Su, Y., Qin, X. Strong Convergence Theorems for Nonexpansive Mapping. Jrl Syst Sci & Complex 20, 85–94 (2007). https://doi.org/10.1007/s11424-007-9007-4
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DOI: https://doi.org/10.1007/s11424-007-9007-4