Taiwanese Journal of Mathematics

WEAK AND STRONG CONVERGENCE THEOREMS OF A MANN-TYPE ITERATIVE ALGORITHM FOR k-STRICT PSEUDO-CONTRACTIONS

Yeol Je Cho, Xiaolong Qin, Jung Im Kang, and Meijuan Shang

Full-text: Open access

Abstract

In this paper, we consider a Mann-type iterative algorithm for nonself strict pseudo-contractions. Weak and strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the results announced by many others.

Article information

Source
Taiwanese J. Math., Volume 14, Number 4 (2010), 1439-1455.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405959

Digital Object Identifier
doi:10.11650/twjm/1500405959

Mathematical Reviews number (MathSciNet)
MR2663923

Zentralblatt MATH identifier
1217.47113

Subjects
Primary: 47H05: Monotone operators and generalizations 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Keywords
nonexpansive mapping strict pseudo-contraction Hilbert space fixed point

Citation

Cho, Yeol Je; Qin, Xiaolong; Kang, Jung Im; Shang, Meijuan. WEAK AND STRONG CONVERGENCE THEOREMS OF A MANN-TYPE ITERATIVE ALGORITHM FOR k-STRICT PSEUDO-CONTRACTIONS. Taiwanese J. Math. 14 (2010), no. 4, 1439--1455. doi:10.11650/twjm/1500405959. https://projecteuclid.org/euclid.twjm/1500405959


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References

  • G. L. Acedo and H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal., 67 (2007), 2258-2271.
  • F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.
  • Y. J. Cho, S. M. Kang and X. Qin, Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Comput. Math. Appl., 56 (2008), 2058-2064.
  • Y. J. Cho, S. M. Kang and X. Qin, Some results on $k$-strictly pseudo-contractive mappings in Hilbert spaces, Nonlinear Anal., (2008), doi:10.1016/j.na.2008.02.094.
  • Y. J. Cho, S. M. Kang and X. Qin, Convergence theorems of fixed points for a finite family of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 856145.
  • S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung and S. M. Kang, Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl., 224 (1998), 149-165.
  • L. C. Ceng, S. Al-Homidan, Q. H. Ansari and J. C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math., (2008), doi:10.1016/j.cam.2008.03.032.
  • T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., 61 (2005), 51-60.
  • T. H. Kim and H. K. Xu, Convergence of the modified Manns iteration method for asymptotically strict pseudo-contractions, Nonlinear Anal., 68 (2008), 2828-2836.
  • G. Marino and H. K. Xu, Weak and strong convergence theorems for $k$-strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-349.
  • G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52.
  • W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
  • J. W. Peng and J. C. Yao, Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings, to appear in Taiwanese J. Math..
  • X. Qin and Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329 (2007), 415-424.
  • B. E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc., 196 (1974), 162-176.
  • S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276.
  • T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239.
  • H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.
  • Y. Yao, R. Chen and J. C. Yao, Strong convergence and certain control conditions for modified Mann iteration, Nonlinear Anal., 68 (2008), 1687-1693.
  • Y. Yao, R. Chen and H. Zhou, Iterative algorithms to fixed point of nonexpansive mapping, Acta Math. Sinica $($Chin. Ser.$)$, 50 (2007), 139-144.
  • H. Y. Zhou, Convergence theorems of fixed points for $k$-strict pseudo-contractions in Hilbert space, Nonliner Anal., 69 (2008), 456-462.
  • H. Y. Zhou, Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl., 343 (2008), 546-556.