Skip to main content
SpringerLink
Log in

Fixed points of nonexpansive and quasi-nonexpansive mappings

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In the paper Krasnoselskii–Mann method for non-self mappings in the journal of Fixed Point Theory and Applications, Colao and Marino proved strong convergence of Krasnoselskii–Mann algorithm defined by \(x_{n+1}=\alpha _nx_n+(1-\alpha _n)Tx_n\) for a non-expansive non-self mapping in a Hilbert space and they proposed three open questions. In this paper we have proved theorems that are answers to all the open questions raised in that paper by relaxing the space, involved map and inward condition to be uniformly convex Banach space, quasi-nonexpansive and weakly inward condition respectively. An application of non-linear parabolic partial differential equation is discussed.We also provide numerical example to verify our main result.

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. Bauschke, H.H. 1996. The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 202 (1): 150–159.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauschke, H.H., and P.L. Combettes. 2001. A weak-to-strong convergence principle for Fejr-monotone methods in Hilbert spaces. Mathematical Operations Research 26 (2): 248–264.

    Article  MATH  Google Scholar 

  3. Chidume, C. 2009. Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965., Lecture Notes in Mathematics, Berlin: Springer.

  4. Dukic, Dusan, Ljiljana Paunovic, and Stojan Radenovic. 2011. Convergence of iterates with errors of uniformly quasi-Lipschitzian mappings in cone metric spaces. Kragujevac Journal of Mathematics 35 (3): 399–410.

    MathSciNet  MATH  Google Scholar 

  5. Edelstein, M., and R.C. Obrien. 1978. Nonexpansive mappings, asymptotic regularity and successive approximations. Journal of the London Mathematical Society 2 (3): 547–554.

    Article  MathSciNet  MATH  Google Scholar 

  6. Groetsch, C.W. 1972. A note on segmenting Mann iterates. Journal of Mathematical Analysis and Applications 40 (2): 369–372.

    Article  MathSciNet  MATH  Google Scholar 

  7. Gunduz, B., and S. Akbulut. 2015. On weak and strong convergence theorems for a finite family of nonself I-asymptotically nonexpansive mappings. Mathematica Moravica 19: 49–64.

    Article  MathSciNet  MATH  Google Scholar 

  8. Gunduz, B., and S. Akbulut. 2017. Common fixed points of a finite family of I-asymptotically nonexpansive mappings by S iteration process in Banach spaces. Thai Journal of Mathematics 15 (3): 673–687.

    MathSciNet  Google Scholar 

  9. He, S., and C. Yang. 2014. Boundary point algorithms for minimum norm fixed points of nonexpansive mappings. Fixed Point Theory and Application 2014: 56.

    Article  MathSciNet  MATH  Google Scholar 

  10. He, S., W. Zhu. 2013. A modified Mann iteration by boundary point method for finding minimum-norm fixed point of nonexpansive mappings. Abstract and Applied Analysis 2013: Article ID 768595.

  11. Hicks, T.L., and J.D. Kubicek. 1977. On the Mann iteration process in a Hilbert space. Journal of Mathematical Analysis and Applications 59 (3): 498–504.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hillam, B.P. 1975. A generalization of Krasnoselskis theorem on the real line. Mathematics Magazine 48 (3): 167–168.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ishikawa, S. 1976. Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 59 (1): 65–71.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kirk, W., and B. Sims. 2001. Handbook of Metric Fixed Point Theory. Berlin: Springer.

    Book  MATH  Google Scholar 

  15. Mann, W.R. 1953. Mean value methods in iteration. Proceedings of the American Mathematical Society 4 (3): 506–510.

    Article  MathSciNet  MATH  Google Scholar 

  16. Marino, G., and G. Trombetta. 1992. On approximating fixed points for nonexpansive mappings. Indian Journal of Mathematics 34: 91–98.

    MathSciNet  MATH  Google Scholar 

  17. Marino, G., and H.-K. Xu. 2007. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. Journal of Mathematical Analysis and Applications 329 (1): 336–346.

    Article  MathSciNet  MATH  Google Scholar 

  18. Meifang Guo, Xia Li and Yongfu Su. 2016. On an open question of V. Colao and G. Marino presented in the paper “Krasnoselskii-Mann method for non-self mappings”. Springer Plus 5:1328. https://doi.org/10.1186/s40064-016-2977-8.

  19. Pazy, A. 1983. Semigroups of Linear Operators and Applications to Partial Differential Equations. New York: Springer.

    Book  MATH  Google Scholar 

  20. Reich, S. 1979. Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 67 (2): 274–276.

    Article  MathSciNet  MATH  Google Scholar 

  21. Schu, J. 1991. Iterative construction of fixed points of asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 158 (2): 407–413.

    Article  MathSciNet  MATH  Google Scholar 

  22. Suzuki, T. 2005. Strong convergence of Krasnoselskii and Manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 305 (1): 227–239.

    Article  MathSciNet  MATH  Google Scholar 

  23. Takahashi, W., and G.-E. Kim. 1998. Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces. Nonlinear Analysis 32 (3): 447–454.

    Article  MathSciNet  MATH  Google Scholar 

  24. Vittorio, Colao, and G. Marino. 2015. Krasnoselskii–Mann method for non-self mappings. Fixed Point Theory and Applications 2015: 39. https://doi.org/10.1186/s13663-015-0287-4.

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu, H.-K. 1997. Approximating curves of nonexpansive nonself-mappings in Banach spaces. Comptes Rendus de l Academie des Sciences - Series I - Mathematics 325 (2): 151–156.

    MathSciNet  MATH  Google Scholar 

  26. Xu, H.-K., and X.-M. Yin. 1995. Strong convergence theorems for nonexpansive nonself-mappings. Nonlinear Analysis 24 (2): 223–228.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Tamil Nadu, 624 302, India

    M. Sankara Narayanan

  2. Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamil Nadu, 620 024, India

    M. Marudai

Authors
  1. M. Sankara Narayanan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Marudai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Sankara Narayanan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Narayanan, M.S., Marudai, M. Fixed points of nonexpansive and quasi-nonexpansive mappings. J Anal 27, 75–87 (2019). https://doi.org/10.1007/s41478-018-0104-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-018-0104-7

Keywords

Mathematics Subject Classification

Search

Navigation