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Convergence theorems for a new iteration scheme for mixed-type asymptotically nonexpansive map**s

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Abstract

In this paper, we introduce a new iteration scheme of mixed type for two asymptotically nonexpansive self-map**s and two asymptotically nonexpansive nonself-map**s, and prove some weak and strong convergence theorems of the proposed iteration scheme in uniformly convex Banach spaces. Our results improve and extend the corresponding results given by some authors.

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References

  1. Beauzamy, B., Maurey, B.: Points minimaux et ensembles optimaux dans les espaces de Banach. J. Funct. Anal. 24, 107–139 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bruck, R.E., Kuczumow, T., Reich, S.: Convergence of iterates of asymptotically nonexpansive map**s in Banach spaces with the uniform Opial property. Colloquium Math. 65, 169–179 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chidume, C.E., Ofoedu, E.U., Zegeye, H.: Strong and weak convergence theorems for asymptotically nonexpansive map**s. J. Math. Anal. Appl. 280, 364–374 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Davis, W. J., Enflo, P.: Contractive Projections on lp Spaces, Analysis at Urbana. Vol. I (Urbana, IL, 19861987), 151161, London Math. Soc. Lecture Note Ser., vol 137. Cambridge Univ. Press, Cambridge (1989)

  5. Falset, J.G., Kaczor, W., Kuczumow, T., Reich, S.: Weak convergence theorems for asymptotically nonexpansive map**s and semigroups. Nonlinear Anal. 43, 377–401 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive map**s. Proc. Am. Math. Soc. 35, 171–174 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goebel, K., Kirk, W. A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

  8. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Map**s. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  9. Guo, W., Cho, Y.J., Guo, W.: Convergence theorems for mixed type asymptotically nonexpansive map**s. Fixed Point Theory Appl. 2012, 224 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, W., Guo, W.: Weak convergence theorems for asymptotically nonexpansive nonself-map**s. Appl. Math. Lett. 24, 2181–2185 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Jung, J.S., Kim, S.S.: Strong convergence theorems for nonexpansive nonself-map**s in Banach spaces. Nonlinear Anal. 33, 321–329 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kopecká, E., Reich, S.: Nonexpansive retracts in Banach spaces. Banach Center Publ. 77, 161–174 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, Z., Feng, C., Ume, J.S., Kang, S.M.: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive map**s. Taiwan. J. Math. 11, 27–42 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Matsushita, S.Y., Kuroiwa, D.: Strong convergence of averaging iteration of nonexpansive nonself-map**s. J. Math. Anal. Appl. 294, 206–214 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Opial, Z.: Weak convergence of successive approximations for nonexpansive mappins. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MATH  Google Scholar 

  17. Osilike, M.O., Udomene, A.: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive map**s. Math. Comput. Model. 32, 1181–1191 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Osilike, M.O., Udomene, A.: Demiclosedness principle and convergence theorems for strictly pseudocontractive map**s of Browder-Petryshyn type. J. Math. Anal. Appl. 256, 431–445 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Reich, S.: Fixed points in locally convex spaces. Math. Z. 125, 17–31 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  20. Reich, S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44, 57–70 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  21. Reich, S.: Fixed point iterations of nonexpansive map**s. Pac. J. Math. 60(2), 195–198 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reich, S.: Minimal displacement of points under weakly inward pseudo-lipschitzian map**s I. Atti. Accad. Naz. Linzei Rend. Cl. Sci. Fis. Mat. Natur. 59, 40–44 (1975)

    MathSciNet  MATH  Google Scholar 

  23. Reich, S.: Minimal displacement of points under weakly inward pseudo-lipschitzian map**s II. Atti. Accad. Naz. Linzei Rend. Cl. Sci. Fis. Mat. Natur. 60, 95–96 (1976)

    MathSciNet  MATH  Google Scholar 

  24. Reich, S.: On fixed point theorems obtained from existence theorems for differential equations. J. Math. Anal. Appl. 54, 26–36 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  25. Reich, S.: Weak convergence theorems for nonexpansive map**s in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rhoades, B.E.: Fixed point iterations for certain nonlinear map**s. J. Math. Anal. Appl. 183, 118–120 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive map**s. Bull. Aust. Math. Soc. 43, 153–159 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Schu, J.: Iterative construction of a fixed points of asymptotically nonexpansive map**s. J. Math. Anal. Appl. 158, 407–413 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shahzad, N.: Approximating fixed points of non-self nonexpansive map**s in Banach spaces. Nonlinear Anal. 61, 1031–1039 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Suantai, S.: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive map**s. J. Math. Anal. Appl. 311, 506–517 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sun, Z.H.: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive map**s. J. Math. Anal. Appl. 286, 351–358 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Takahashi, W.: Nonlinear Functional Analysis. Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama, Japan (2000)

    MATH  Google Scholar 

  33. Takahashi, W., Kim, G.E.: Strong convergence of approximants to fixed points of nonexpansive nonself-map**s. Nonlinear Anal. 32, 447–454 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive map** by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thianwan, S.: Common fixed points of new iterations for two asymptotically nonexpansive nonself-map**s in a Banach space. J. Comput. Appl. Math. 224, 688–695 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, L.: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive map**s. J. Math. Anal. Appl. 323, 550–557 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, H.K., Yin, X.M.: Strong convergence theorems for nonexpansive nonself-map**s. Nonlinear Anal. 242, 23–228 (1995)

    MATH  Google Scholar 

  38. Zhou, H. Y., Cho, Y. J., Kang, S. M.: A New Iterative Aloritem for Approximating Common Fixed Points for Asymptotically Nonexpansive Map**s. Fixed Point Theory Appl. 10 (2007) (Article ID 64874)

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Acknowledgements

The author is extremely grateful to Professor Simeon Reich and the anonymous referee for their valuable comments and useful suggestions which improve the presentation of this paper. The author is grateful to University of Phayao for supporting research project RD60062.

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  1. School of Science, University of Phayao, Phayao, 56000, Thailand

    Tanakit Thianwan

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  1. Tanakit Thianwan
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Thianwan, T. Convergence theorems for a new iteration scheme for mixed-type asymptotically nonexpansive map**s. J. Fixed Point Theory Appl. 20, 145 (2018). https://doi.org/10.1007/s11784-018-0625-3

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