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Hybrid Mann–Halpern Iteration Methods for Finding Fixed Points Involving Asymptotically Nonexpansive Mappings and Semigroups

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Abstract

In this paper, we introduce some new iteration methods combining the hybrid method in mathematical programming with Mann’s iterative method and the Halpern method for finding a fixed point of an asymptotically nonexpansive mapping and a common fixed point of an asymptotically nonexpansive semigroup in a Hilbert space. The main results in this paper modify and improve some well-known results in the literature.

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References

  1. Aleyner A., Censor Y. (2005) Best approximation to common fixed points of a semigroup of nonexpansive operators. J. Nonlinear Convex Anal. 6:137–151

    MATH  MathSciNet  Google Scholar 

  2. Buong N. (2011) Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space. Comput. Math. Appl. 61:2546–2554. doi:10.1016/j.camwa.2011.02.047

  3. Goebel K., Kirk W.A. (1972) A fixed point theorem for asymptotically nonexpansive mappings. Proc. Amer. Math. Soc. 35:171–174. doi:10.1090/S0002-9939-1972-0298500-3

    Article  MATH  MathSciNet  Google Scholar 

  4. Halpern B. (1967) Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73:957–961

    Article  MATH  MathSciNet  Google Scholar 

  5. Ishikawa S. (1974) Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44:147–150

    Article  MATH  MathSciNet  Google Scholar 

  6. Kim T.-H., Xu H.-K. (2006) Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 64:1140–1152

    Article  MATH  MathSciNet  Google Scholar 

  7. Lin P.-K., Tan K.-K., Xu H.-K. (1995) Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. Nonlinear Anal. 24:929–946

    Article  MATH  MathSciNet  Google Scholar 

  8. Lions P.-L. (1977) Approximation de points fixes de contractions. C. R. Acad. Sci., Paris, Sér. A 284:1357–1359

    MATH  Google Scholar 

  9. Liu X.F. (2011) Strong convergence theorems for a finite family of relatively nonexpansive mappings. Vietnam J. Math 39:63–69

    MATH  MathSciNet  Google Scholar 

  10. Mann W.R. (1953) Mean value methods in iteration. Proc. Amer. Math. Soc. 4:506–510

    Article  MATH  MathSciNet  Google Scholar 

  11. Martinez-Yanes C., Xu H.-K. (2006) Strong convergence of the CQ method for fixed iteration processes. Nonlinear Anal. 64:2400–2411

    Article  MATH  MathSciNet  Google Scholar 

  12. Nakajo K., Takahashi W. (2003) Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279:372–379

    Article  MATH  MathSciNet  Google Scholar 

  13. Qin X., Su Y., Shang M. (2008) Strong convergence theorems for asymptotically nonexpansive mappings by hybrid methods. Kyungpook Math. J 48:133–142

    Article  MATH  MathSciNet  Google Scholar 

  14. Reich S. (1980) Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75:287–292

    Article  MATH  MathSciNet  Google Scholar 

  15. Saluja G.S., Nashine H.K. (2010) Strong convergence theorems of an implicit iterative algorithm with errors for asymptotically quasi-nonexpansive in the intermediate sense mappings. Vietnam J. Math. 38: 395–402

    MATH  MathSciNet  Google Scholar 

  16. Solodov M.V., Svaiter V.F. (2000) Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program., Ser. A 87:189–202

    MATH  MathSciNet  Google Scholar 

  17. Tan K.-K., Xu H.-K. (1993) Fixed point theorems for Lipschitzian semigroups in Banach spaces. Nonlinear Anal. 20:395–404

    Article  MATH  MathSciNet  Google Scholar 

  18. Wangkeeree R., Preechasilp P. (2012) The general iterative methods for asymptotically nonexpansive semigroups in Banach spaces. Abstr. Appl. Anal. 2012(695183):20

    MathSciNet  Google Scholar 

  19. Wittmann R. (1992) Approximation of fixed points of nonexpansive mappings. Arch. Math. 58:486–491

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The author is extremely grateful to the referees for their useful suggestions and comments by which the contents of this paper are improved.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Thainguyen University, Thainguyen, Vietnam

    Nguyen Thi Thu Thuy

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  1. Nguyen Thi Thu Thuy
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Correspondence to Nguyen Thi Thu Thuy.

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Thuy, N.T.T. Hybrid Mann–Halpern Iteration Methods for Finding Fixed Points Involving Asymptotically Nonexpansive Mappings and Semigroups. Vietnam. J. Math. 42, 219–232 (2014). https://doi.org/10.1007/s10013-014-0071-5

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  • DOI: https://doi.org/10.1007/s10013-014-0071-5

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