Skip to main content
SpringerLink
Log in

Fixed-Point Methods for a Certain Class of Operators

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We introduce in this paper a new class of nonlinear operators, which contains, among others, the class of operators with semimonotone additive inverse and also the class of nonexpansive mappings. We study this class and discuss some of its properties. Then we present iterative procedures for computing fixed points of operators in this class, which allow for inexact solutions of the subproblems and relative error criteria. We prove weak convergence of the generated sequences in the context of Hilbert spaces. Strong convergence is also discussed.

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. Gárciga Otero, R., Iusem, A.N.: Regularity results for semimonotone operators. Commun. Appl. Math. 30, 1–13 (2011)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Reich, S.: Weak convergence theorems for nonexpansive mappings in banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal.: Theory Methods Appl. 56, 715–738 (2004)

    Article  MATH  Google Scholar 

  5. Genel, A., Lindenstrauss, J.: An example concerning fixed points. Isr. J. Math. 22, 81–86 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Nevanlinna, O., Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in banach spaces. Isr. J. Math. 32, 44–58 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Iusem, A.N., Pennanen, T., Svaiter, B.F.: Inexact variants of the proximal point algorithm without monotonicity. SIAM J. Optim. 13, 1080–1097 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

R. Gárciga Otero was partially supported by PRONEX-Optimization. A. Iusem was partially supported by CNPq Grant No. 301280/86.

Author information

Authors and Affiliations

  1. Instituto de Economia da Universidade Federal de Rio de Janeiro, Avenida Pasteur 250, Urca, Rio de Janeiro, RJ, 22290-240, Brazil

    Rolando Gárciga Otero

  2. Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

    Alfredo Iusem

Authors
  1. Rolando Gárciga Otero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alfredo Iusem
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alfredo Iusem.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gárciga Otero, R., Iusem, A. Fixed-Point Methods for a Certain Class of Operators. J Optim Theory Appl 159, 656–672 (2013). https://doi.org/10.1007/s10957-012-0146-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0146-y

Keywords

Search

Navigation