Skip to main content
SpringerLink
Log in

Common solution to an equilibrium problem and a fixed point problem for an asymptotically quasi-\(\phi \)-nonexpansive mapping in intermediate sense

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we introduce and study an iterative method for solving an equilibrium problem and a fixed point problem for an asymptotically quasi-\(\phi \)-nonexpansive mapping in intermediate sense in Banach space. We prove that the sequences generated by iterative algorithm converge strongly to a common solution of these problems in a uniformly smooth and strictly convex Banach space.

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. Alber, Y.I.: Metric and generalized projection operators in Banach spaces. In: Properties and applications, Lect. Notes Pure Appl. Math., pp. 15–50 (1996)

  2. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Global Optim. 46, 635–646 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ceng, L.C., Guu S.M., Yao, J.C.: Finding common solution of variational inequality, a general system of variational inequalities and fixed point problem via a hybrid extragradient method, Fixed Point Theory Appl. Article ID 626159, p. 22 (2011)

  5. Ceng, L.C., Wang, C.Y., Yao, J.C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, 375–390 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daniele, P., Giannessi, F., Mougeri, A. (Eds): Equilibrium problems and variational models. Nonconvex Optimization and its Application Vol. 68. Kluwer Academic Publications, Norwell (2003)

  7. Dong, Q.L., Lu, Y.Y.: A new hybrid algorithm for anonexpansive mapping. Fixed Point Theory Appl. 37, 22 (2015)

    Google Scholar 

  8. Hao, Y.: Some results on a modified Mann iterative scheme in reflexive Banach space. Fixed Point Theory Appl. 2013, 227 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equation. Acta Mathenatica 115, 271–310 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithim in a Banach space. SIAM J. Optim. 13, 938–945 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kazmi, K.R., Rizvi, S.H., Ali, R.: A hybrid iterative method without extrapolating step for solving mixed equilibrium problem. Creative Math. Inf. 24(2), 163–170 (2015)

    MathSciNet  Google Scholar 

  12. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  13. Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolating step for solving variational inequality problems. J. Global Optim. 61, 193–202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Moudafi, A.: Second order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4(1), 1–7 (2003)

  15. Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mapping. SIAM J. Optim. 16(40), 1230–1241 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Qin, X.L., Wang, L.: On asymptotically quasi-\(\phi \)-nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal, Article ID 636217 (2012)

  17. Rouhani, B.D., Kazmi, K.R., Rizvi, S.H.: A hybrid-extragradient-convex approximation method for a system of unrelated mixed equilibrium problems. Trans. Math. Pogram. Appl. 1(8), 82–95 (2013)

    Google Scholar 

  18. Takahashi, W., Zembayashi, K.: Strong and weak convergence theorem for equilibrium problems and relatively nonexpansive mmapings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are very grateful to the anonymous referee for his valuable comments toward the improvement of the paper. The second author is thankful to the University Grant Commission, Government of India, New Delhi for BSR (Basic Scientific Research) fellowship to support this research work.

Author information

Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    K. R. Kazmi & Rehan Ali

Authors
  1. K. R. Kazmi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rehan Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. R. Kazmi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kazmi, K.R., Ali, R. Common solution to an equilibrium problem and a fixed point problem for an asymptotically quasi-\(\phi \)-nonexpansive mapping in intermediate sense. RACSAM 111, 877–889 (2017). https://doi.org/10.1007/s13398-016-0331-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-016-0331-1

Keywords

Mathematics Subject Classification

Search

Navigation