Skip to main content
SpringerLink
Log in

Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

A Mann-type iterative scheme which converges strongly to a fixed point of a multi-valued nonexpansive non-self mapping T is constructed in a real Hilbert space H. We also constructed a Mann-type sequence which converges to a fixed point of a multi-valued quasi-nonexpansive non-self mapping under appropriate conditions. In addition, an Ishikawa-type iterative scheme which approximates the fixed points of multi-valued Lipschitz pseudocontractive non-self mappings is constructed in Banach spaces. The results obtained in this paper improve and extend the known results in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbas, M., Cho, Y.J.: Fixed point results for multi-valued non-expansive mappings on an unbounded set. Analele Stiintifice Ale Universitatii Ovidius Constanta 18(2), 5–14 (2010)

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

  3. Beg I., Abbas M.: Fixed-point theorem for weakly inward multi-valued maps on a convex metric space. Demonstr. Math. 39(1), 149–160 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Benavides T.D., Ramrez P.L.: Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions. J. Math. Anal. Appl. 291(1), 100–108 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Browder E.F.: Convergence theorems for sequences of nonlinear operators in banach spaces. Math. Zeitscher 100, 201–225 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chidume, C.E., Chidume, C.O., Djitte, N., Minjibir, M.S.: Convergence theorems for fixed points of multivalued strictly pseudocontractive mappings in Hilbert spaces. In: Abstract and Applied Analysis, vol. 2013. Hindawi Publishing Corporation, Cairo (2013)

  7. Colao V., Marino G.: Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory Appl. 2015(1), 1–7 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Djitte, N., Sene, M.: Convergence theorems for fixed points of multivalued mappings in Hilbert spaces. Int. J. Anal. 2014 (2014)

  9. Garca-Falset J., Llorens-Fuster E., Suzuki T.: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375(1), 185–195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Isiogugu F.O., Osilike M.O.: Convergence theorems for new classes of multivalued hemicontractive-type mappings. Fixed Point Theory Appl. 2014(1), 1–12 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Khan S.H., Yildirim I.: Fixed points of multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2012(1), 1–9 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Khan S.H., Yildirim I., Rhoades B.E.: A one-step iterative process for two multivalued nonexpansive mappings in Banach spaces. Comput. Math. Appl. 61(10), 3172–3178 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Marino G.: Fixed points for multivalued mappings defined on unbounded sets in Banach spaces. J. Math. Anal. Appl. 157(2), 555–567 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Marino G., Trombetta G.: On approximating fixed points for nonexpansive mappings. Indian J. Math. 34, 91–98 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Nadler S.B. Jr: Multi-valued contraction mappings. Pacific J. Math. 30(2), 475–488 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  16. Panyanak B.: Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54(6), 872–877 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sastry K.P.R., Babu G.V.R.: Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point. Czechoslovak Math. J. 55(4), 817–826 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sebisebe, T.W, Mengistu, G.S., Habtu, Z.: Strong Convergence Theorems for a Common Fixed Point of a Finite Family of Lipschitz Hemicontractive-type Multivalued Mappings (submitted).

  19. Shahzad N., Zegeye H.: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. Theory Methods Appl. 71(3), 838–844 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Song Y., Cho Y.J.: Some notes on Ishikawa iteration for multi-valued mappings. Bull. Kor. Math. Soc. 48(3), 575–584 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Song Y., Chen R.: Viscosity approximation methods for nonexpansive nonself-mappings. J. Math. Anal. Appl. 321(1), 316–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Song Y.S., Cho Y.J.: Averaged iterates for non-expansive nonself mappings in Banach spaces. J. Comput. Anal. Appl. 11, 451–460 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Song Y., Wang H.: Erratum to GuMann and Ishikawa iterative processes for multivalued mappings in Banach spaces. Comput. Math. Appl. 54(2007), 872–877 (2007)

    MathSciNet  Google Scholar 

  24. Takahashi W., Kim G.E.: Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces. Nonlinear Anal. Theory Methods Appl. 32(3), 447–454 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu H.K, Yin X.M: Strong convergence theorems for nonexpansive non-self mappings. Nonlinear Anal. Theory Methods Appl. 24(2), 223–228 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu H.K.: Approximating curves of nonexpansive nonself-mappings in Banach spaces. C. R. Acad. Sci. Paris Sr. I Math. 325(2), 151–156 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Botswana, Pvt. Bag 00704, Gaborone, Botswana

    Abebe R. Tufa & H. Zegeye

Authors
  1. Abebe R. Tufa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Zegeye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Zegeye.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tufa, A.R., Zegeye, H. Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings. Mediterr. J. Math. 13, 4369–4384 (2016). https://doi.org/10.1007/s00009-016-0750-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-016-0750-4

Mathematics Subject Classification

Keywords

Search

Navigation