Skip to main content
SpringerLink
Log in

A New Hybrid Relaxed Extragradient Algorithm for Solving Equilibrium Problems, Variational Inequalities and Fixed Point Problems

  • Conference paper
  • First Online:
Transactions on Engineering Technologies

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 275))

  • 888 Accesses

Abstract

The purpose of this paper is to introduce a new hybrid relaxed extragradient iterative method for finding a common element of the solution of set a equilibrium, variational inequality and the set of fixed point of a \(\xi \)-strict pseudocontraction mapping in Hilbert spaces. We obtain a strong convergence theorem of the purposed iterative algorithm under some suitable conditions. The results presented in this paper generalize, improve and extend some well-known results in the literature.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ceng L-C, Al-Homidan S, Ansari QH, Yao J-C (2009) An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J Comput Appl Math 223(2):967–974

    Article  MathSciNet  MATH  Google Scholar 

  2. Fan K (1972) A minimax inequality and applications. In: Shisha O (ed) Inequality III. Academic Press, New York, pp 103–113

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Flam SD, Antipin AS (1997) Equilibrium progamming using proximal-link algorithms. Math Program 78:29–41

    Article  MathSciNet  MATH  Google Scholar 

  5. Moudafi A, Thera M (1999) Proximal and dynamical approaches to equilibrium problems. In: Lecture note in economics and mathematical systems, vol 477. Springer, New York, pp 187–201.

    Google Scholar 

  6. Takahashi S, Takahashi W (2007) Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 331:506–515

    Article  MathSciNet  MATH  Google Scholar 

  7. Scherzer O (1995) Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J Mathe Anal Appl 194(3):911–933

    Article  MathSciNet  MATH  Google Scholar 

  8. Mann WR (1953) Mean value methods in iteration. Proc Amer Math Soc 4:506–510

    Article  MathSciNet  MATH  Google Scholar 

  9. Reich S (1979) Weak convergence theorems for nonexpansive mappings. J Math Anal Appl 67:274–276

    Article  MathSciNet  MATH  Google Scholar 

  10. Genel A, Lindenstrass J (1975) An example concerning fixed points. Isael J Math 22:81–86

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Takahashi W, Toyoda M (2003) Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 118:417–428

    Article  MathSciNet  MATH  Google Scholar 

  13. Ceng LC, Ansari QH, Yao JC (2011) Relaxed extragradient iterative methods for variational inequalities. Appl Math Comput 218:1112–1123

    Article  MathSciNet  MATH  Google Scholar 

  14. Vuong PT, Strodiot JJ, Nguyen VH (2012) Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J Optim Theory Appl 1–23.

    Google Scholar 

  15. Phiangsungnoen S, Kumam P (2013) A hybrid extragradient method for solving Ky Fan inequalities, variational inequalities and fixed point problems. In: Proceedings of the international multiConference of engineers and computer scientists, vol II, IMECS 2013, March 13–15, 2013. Hong Kong, pp 1042–1047.

    Google Scholar 

  16. Rockafellar RT (1970) On the maximality of sums of nonlinear monotone operators. Trans Amer Math Sco 149:75–88

    Article  MathSciNet  MATH  Google Scholar 

  17. Rockafellar RT (1976) Monotone operators and proximal point algorithm. SIAM J Control Optim 14:877–898

    Article  MathSciNet  MATH  Google Scholar 

  18. Opial Z (1967) Weak convergence of successive approximations for nonexpansive mappings. Bull Amer Math Soc 73:591–597

    Article  MathSciNet  MATH  Google Scholar 

  19. Goebel K, Kirk WA (1990) Topics in metric fixed point theory. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  20. Takahashi W (2000) Nonlinear functional analysis. Yokohama Publishers, Yokohama

    MATH  Google Scholar 

  21. Mastroeni G (2003) On auxiliary principle for equilibrium problems. In: Daniele P, Gianessi F, Maugeri A (eds) Equilibrium problems and variational models. Kluwer Academic, Dordrecht, pp 289–298

    Chapter  Google Scholar 

  22. Tran DQ, Muu LD, Nguyen VH (2008) Extragradient algorithms extended to equilibrium problems. Optimization 57:749–776

    Article  MathSciNet  MATH  Google Scholar 

  23. Anh PN (2011) A hybrid extragradient method extended to fixed point problems and equilibrium problem. Optimization. doi:url10.1080/02331934.2011.607497

    Google Scholar 

  24. Jaiboon C, Kumam P (2010) Strong convergence theorems for solving equilibrium problems and fixed point problems of \(\xi \)-strict pseudo-contraction mappings by two hybrid projection methods. J Comput Appl Math 230:722–732

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. 55000613).

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, 126 Pracha-Uthit Road, Bangmod, Thung khru, Bangkok, 10140, Thailand

    Supak Phiangsungnoen & Poom Kumam

Authors
  1. Supak Phiangsungnoen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Poom Kumam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Poom Kumam .

Editor information

Editors and Affiliations

  1. Department of Multimedia Engineering College of Engineering, Mokpo National University, Chonnam, Korea, Republic of (South Korea)

    Gi-Chul Yang

  2. International Association of Engineers, Hong Kong, Hong Kong SAR

    Sio-Iong Ao

  3. Faculty of Information Sciences and Engineering, University of Canberra, Canberra, Aust Capital Terr, Australia

    Xu Huang

  4. Computer Sci. in the Graduate Division, Tijuana Institute of Technology, Tijuana, Baja California, Mexico

    Oscar Castillo

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this paper

Cite this paper

Phiangsungnoen, S., Kumam, P. (2014). A New Hybrid Relaxed Extragradient Algorithm for Solving Equilibrium Problems, Variational Inequalities and Fixed Point Problems. In: Yang, GC., Ao, SI., Huang, X., Castillo, O. (eds) Transactions on Engineering Technologies. Lecture Notes in Electrical Engineering, vol 275. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7684-5_26

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-7684-5_26

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-7683-8

  • Online ISBN: 978-94-007-7684-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation