Skip to main content
Springer Nature Link
Log in

Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Many iterative schemes have already been developed to solve the equilibrium problems, one of which is the most efficient two-step extragradient method. The objective of this research is to propose two new iterative methods with inertial effect to solve equilibrium problems. These iterative methods are based on an extra-gradient method and a Mann-type iterative method. Two strong convergence theorems have been proved in the setting of real Hilbert space, with mild assumptions that the underlying bi-function is Lipschitz-type continuous and pseudo-monotone. The primary advantage of the second method is that it does not require the information of the Lipschitz-type bi-functional constants. We have also studied the applications of our research results to solve particular classes of equilibrium problems. Numerical studies are carried out to show the behaviour of proposed methods and to compare them with the existing ones 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

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Antipin, A.: Equilibrium programming: proximal methods. Comput Math Math Phys 37(11), 1285–1296 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Attouch, F.A.H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Var Anal 9, 3–11 (2001)

    Article  MathSciNet  Google Scholar 

  3. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imagin Sci 2(1), 183–202 (2009a)

    Article  MathSciNet  Google Scholar 

  4. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imagin Sci 2(1), 183–202 (2009b)

    Article  MathSciNet  Google Scholar 

  5. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J Optim Theor Appl 90(1), 31–43 (1996)

    Article  MathSciNet  Google Scholar 

  6. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur J Op Res 227(1), 1–11 (2013)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Browder, F., Petryshyn, W.: Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl 20(2), 197–228 (1967)

    Article  MathSciNet  Google Scholar 

  9. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theor Appl 148(2), 318–335 (2010)

    Article  MathSciNet  Google Scholar 

  10. Dafermos, S.: Traffic equilibrium and variational inequalities. Transp Sci 14(1), 42–54 (1980)

    Article  MathSciNet  Google Scholar 

  11. Fan, K.: A minimax inequality and applications, Inequalities III. Academic Press, New York (1972)

    Google Scholar 

  12. Farhan, M., Omar, Z., Mebarek-Oudina, F., Raza, J., Shah, Z., Choudhari, R.V., Makinde, O.D.: Implementation of the one-step one-hybrid block method on the nonlinear equation of a circular sector oscillator. Comput Math Model 31(1), 116–132 (2020)

    Article  MathSciNet  Google Scholar 

  13. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev 39(4), 669–713 (1997)

    Article  MathSciNet  Google Scholar 

  14. Flåm, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math Program 78(1), 29–41 (1996)

    Article  MathSciNet  Google Scholar 

  15. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium problems nonsmooth optimization and variational inequality models. Springer Science and Business Media, Berlin (2006)

    MATH  Google Scholar 

  16. Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces CMS books in mathematics, 2nd edn. Springer International Publishing, NY (2017)

    Book  Google Scholar 

  17. Hieu, D.V.: Halpern subgradient extragradient method extended to equilibrium problems. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales Serie A Matemáticas 111(3), 823–840 (2016)

    Article  MathSciNet  Google Scholar 

  18. Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math Method Op Res 88(3), 399–415 (2018)

    Article  MathSciNet  Google Scholar 

  19. Hieu, D.V., Cho, Y.J., bin Xiao, Y.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67(11), 2003–2029 (2018)

    Article  MathSciNet  Google Scholar 

  20. Hung, P.G., Muu, L.D.: The tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal: Theory, Method Appl 74(17), 6121–6129 (2011)

    Article  MathSciNet  Google Scholar 

  21. Konnov, I.: Application of the proximal point method to nonmonotone equilibrium problems. J Optim Theory Appl 119(2), 317–333 (2003)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Kreyszig, E.: Introductory functional analysis with applications, 1st edn. Wiley, NJ (1989)

    MATH  Google Scholar 

  24. Maingé, P.-E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Value Anal 16(7–8), 899–912 (2008)

    Article  MathSciNet  Google Scholar 

  25. Mann, W.R.: Mean value methods in iteration. Proc Am Math Soc 4(3), 506–510 (1953)

    Article  MathSciNet  Google Scholar 

  26. Mastroeni, G.: On auxiliary principle for equilibrium problems. Nonconvex optimization and its applications, pp. 289–298. Springer, US (2003)

    Google Scholar 

  27. Mebarek-Oudina, F.: Numerical modeling of the hydrodynamic stability in vertical annulus with heat source of different lengths. Eng Sci Technol, Int J 20(4), 1324–1333 (2017)

    Google Scholar 

  28. Muu, L., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal: Theory, Method Appl 18(12), 1159–1166 (1992)

    Article  MathSciNet  Google Scholar 

  29. Polyak, B.: Some methods of speeding up the convergence of iteration methods. USSR Comput Math Math Phys 4(5), 1–17 (1964)

    Article  Google Scholar 

  30. Stampacchia, G.: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus Hebdomadaires des Seances de L Academie des Sci 258(18), 4413 (1964)

    MathSciNet  MATH  Google Scholar 

  31. Tiel, J.V.: Convex analysis: an introductory text, 1st edn. Wiley, New York (1984)

    MATH  Google Scholar 

  32. Tran, D.Q., Dung, M.L., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  33. ur Rehman, H., Kumam, P., Abubakar, A.B., Cho, Y.J.: The extragradient algorithm with inertial effects extended to equilibrium problems. Comput Appl Math 39(2), 1–26 (2020)

    Article  MathSciNet  Google Scholar 

  34. ur Rehman, H., Kumam, P., Argyros, I.K., Deebani, W., Kumam, W.: Inertial extra-gradient method for solving a family of strongly pseudomonotone equilibrium problems in real Hilbert spaces with application in variational inequality problem. Symmetry 12(4), 503 (2020b)

    Article  Google Scholar 

  35. ur Rehman, H., Kumam, P., Cho, Y.J., Yordsorn, P.: Weak convergence of explicit extragradient algorithms for solving equilibirum problems. J Inequal Appl 2019(1), 1–25 (2019)

    Article  Google Scholar 

  36. ur Rehman, H., Kumam, P., Je Cho, Y., Suleiman, Y.I., Kumam, W.: Modified popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim Method Softw 33, 1–32 (2020)

    MATH  Google Scholar 

  37. Wang, S., Zhang, Y., Ping, P., Cho, Y., Guo, H.: New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces. Filomat 33(6), 1677–1693 (2019)

    Article  MathSciNet  Google Scholar 

  38. Xu, H.-K.: Another control condition in an iterative method for nonexpansive mappings. Bull Aust Math Soc 65(1), 109–113 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon (RMUTP), 1381 Pracharat 1 Road, Wongsawang, Bang Sue, Bangkok, 10800, Thailand

    Kanikar Muangchoo

  2. Fixed Point Research Laboratory, Fixed Point Theory and Applications Research Group, Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand

    Habib ur Rehman & Poom Kumam

  3. Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand

    Poom Kumam

  4. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan

    Poom Kumam

Authors
  1. Kanikar Muangchoo
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Habib ur Rehman
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Poom Kumam
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Poom Kumam.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muangchoo, K., Rehman, H.u. & Kumam, P. Two strongly convergent methods governed by pseudo-monotone bi-function in a real Hilbert space with applications. J. Appl. Math. Comput. 67, 891–917 (2021). https://doi.org/10.1007/s12190-020-01470-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-020-01470-0

Keywords

Mathematics Subject Classification