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Existence and convergence theorems for Berinde nonexpansive multivalued mapping on Banach spaces

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Abstract

In this paper, we first prove existence of fixed points for Berinde nonexpansive multivalued mappings on Banach spaces. Moreover, we obtain convergence theorems for common fixed point of Berinde and quasi-nonexpansive mappings. The main results obtained in this paper extened and generalize some of the well-known results in the literature.

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References

  1. Abkar, A., Eslamian, M.: Convergence theorems for a finite family of generalized nonexpansive multivalued mapping in \(CAT(0)\) spaces. Nonlinear Anal. 75, 1895–1903 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Assad, N.A., Kirk, W.A.: Fixed point theorems for set-valued mappings of contractive type. Pac. J. Math. 43, 553562 (1972)

    Article  MathSciNet  Google Scholar 

  3. Abbas, M., Khan, S.H., Khan, A.R., Agarwal, R.P.: Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme. Appl. Math Lett. 24, 97–102 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Agarwal, R., O’Regan, D., Sahu, D.: Fixed Point Theory for Lipschitzian-type Mappings with Applications, vol. 6. Springer, New York (2009)

    MATH  Google Scholar 

  5. Alghamdi M.A., Berinde V., Shahzad N.: Fixed points of multivalued nonself almost contractions. J. Appl. Math. 2013, Article ID 621614 (2013)

  6. Berinde, M., Berinde, V.: On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 326, 772–782 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berinde, V., Pacurar, M.: The role of Pompeiu–Hausdorff metric in fixed point theory. Creative Math. Inform. 22(2), 143–150 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Ciric, L.B.: Fixed Point Theory Contraction Mapping Principle. FME Press, Beograd (2003)

    Google Scholar 

  9. Chen, L., Gao, L., Chen, D.: Fixed point theorems of mean nonexpansive setvalued mappings in Banach spaces. J. Fixed Point Theory Appl. 19, 21292143 (2017)

    Google Scholar 

  10. Cholamjiak, W., Suantai, S.: Approximation of common fixed point of two quasi-nonexpansive multi-valued maps in Banach spaces. Comput. Math. Appl. 61, 941–949 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goebel, K.: On a fixed point theorem for multivalued nonexpansive mappings. Ann. Univ. M. Curie-Sklowdska 29, 70–72 (1975)

    MathSciNet  Google Scholar 

  12. Garcia-Falset, J., Llorens-Fuster, E., Suzuki, T.: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375, 185–195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kirk, W.A., Massa, S.: Remarks on asymptotic and Chebyshev centers. Houston J. Math. 16, 357–364 (1990)

    MathSciNet  MATH  Google Scholar 

  14. Khan, S.H., Abbas, M., Rhoades, B.E.: A new one-step iterative scheme for approximating common fixed points of two multivalued nonexpansive mappings. Rend. Circ. Mat. 59, 149–157 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Lim, T.C.: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bull. Am. Math. Soc. 80, 1123–1126 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nadler, S.: Multi-valued contraction mappings. Pac. J. Math. 20(2), 457–488 (1969)

    MathSciNet  MATH  Google Scholar 

  17. Opial, Z.: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  18. Panyanak, B.: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54, 872877 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Panyanak, B.: Endpoints of multivalued nonexpansive mappings in geodesic spaces. Fixed Point Theory Appl. 147, 111 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Sastry, K.P.R., Babu, G.V.R.: Convergence of Ishikawa iterates for a multivalued mappings with a fixed point. Czechoslovak Math. J. 55, 817826 (2005)

    Article  Google Scholar 

  21. Shahzad, N., ZegeyeH, : On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. 71, 838844 (2009)

    Article  MathSciNet  Google Scholar 

  22. Song, Y., Wang, H.: Erratum to Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54, 872–877 (2007). Comput. Math. Appl. 55, 2999–3002 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Phuengrattana, W., Sopha, S.: Common fixed points for single-valued and multi-valued mappings in complete R-trees. Commun. Korean Math. Soc. 31(3), 507518 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yao, Y., Noor, M.A.: convergence of tree-step iterations for asymptotically nonexpansive mappings. Apl. Math. Comput. 187, 883–892 (2007)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Dr. Bancha Panyanak for a useful discussion and Chiang Mai University for the financial support.

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  1. Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

    Nuttawut Bunlue & Suthep Suantai

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  1. Nuttawut Bunlue
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  2. Suthep Suantai
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Correspondence to Suthep Suantai.

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Bunlue, N., Suantai, S. Existence and convergence theorems for Berinde nonexpansive multivalued mapping on Banach spaces. Afr. Mat. 30, 483–494 (2019). https://doi.org/10.1007/s13370-019-00661-8

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  • DOI: https://doi.org/10.1007/s13370-019-00661-8

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