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Generalized viscosity extragradient algorithm for pseudomonotone equilibrium and fixed point problems for finite family of demicontractive operators

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Abstract

In this paper, we propose a generalized viscosity extragradient algorithm in such a way that it does not require prior knowledge of the Lipschitz constants and prove that the generated sequences converges strongly to a common solution of a pseudomonotone equilibrium problem and fixed point problem for a finite family of demicontractive operators in a real Hilbert space. Further, we apply our results to solve various nonlinear analysis problems. Our results generalize and improve results in many different directions. We present numerical results to demonstrate the numerical behaviour of proposed method and to compare it with other methods.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Blum, E., Oettli, W.: Variational principles for equilibrium problems. In: Parametric Optimization and Related Topics, III, Güstrow, 1991. Approx. Optim. 3, 79-88 (1993)

  3. Fan, K.: A minimax inequality and applications, inequalities III (O. Shisha, ed.). Academic Press, New York (1972)

  4. Pathak, H.K.: An introduction to nonlinear analysis and fixed point theory, Springer Nature Singapore Pte Ltd. (2018)

  5. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium problems: Nonsmooth optimization and variational inequality models. Kluwer (2004)

  6. Konnov, I.V.: Equilibrium models for multi-commodity auction market problems. Adv. Model. Optim. 15, 511–524 (2013)

    MATH  Google Scholar 

  7. Nandal, A., Chugh, R., Postolache, M.: Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 11, 655 (2019)

    Article  MATH  Google Scholar 

  8. Postolache, M., Nandal, A., Chugh, R.: Strong convergence of a new generalized viscosity implicit rule and some applications in hilbert space. Mathematics 7, 773 (2019)

    Article  Google Scholar 

  9. Nandal, A., Chugh, R.: On zeros of accretive operators with application to the convex feasibility problem. UPB Sci. Bull. A: Appl. Math. Phys. 81, 95-106(2019)

  10. Hussain, N., Nandal, A., Kumar, V., Chugh, R.: Multistep generalized viscosity iterative algorithm for solving convex feasibility problems in banach spaces. J. Nonlinear Convex Anal. 21, 587–603 (2020)

    MathSciNet  MATH  Google Scholar 

  11. Nandal, A., Chugh, R., Kumari, S.: Convergence analysis of algorithms for variational inequalities involving strictly pseudocontractive operators. Poincare J. Anal. Appl. 2019, 123–136 (2019)

    MATH  Google Scholar 

  12. Gupta, N., Postolache, M., Nandal, A., Chugh, R.: A cyclic iterative algorithm for multiple-sets split common fixed point problem of demicontractive mappings without prior knowledge of operator norm. Mathematics 9, 372 (2021)

    Article  Google Scholar 

  13. Rehman, H.U., Pakkaranang, N., Kumam, P., Cho, Y.J.: Modified subgradient extragradient method for a family of pseudomonotone equilibrium problems in real a Hilbert space. J. Nonlinear Convex Anal. 9, 2011–2025 (2020)

    MathSciNet  MATH  Google Scholar 

  14. 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, 503 (2020)

    Article  Google Scholar 

  15. Khammahawong, K., Kumam, P., Chaipunya, P., Yao, J.C., Wen, C.F., Jirakitpuwapat, W.: An extragradient algorithm for strongly pseudomonotone equilibrium problems on Hadamard manifolds. Thai J. Math. 18, 350–371 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Rehman, H.U., Kumam, P., Je Cho, Y., Suleiman, Y.I., Kumam, W.: Modified popov’s explicit iterative algorithms for solving pseudomonotone equilibrium problems. Optim. Methods Softw. 36, 82–113 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sitthithakerngkiet, K., Deepho, J., Tanaka, T., Kumam, P.: A viscosity extragradient method for variational inequality and split generalized equilibrium problems with a sequence of nonexpansive mappings. J. Nonlinear Sci. Appl. 5, 78–89 (2016)

    Google Scholar 

  18. Kitkuan, D., Kumam, P., Berinde, V., Padcharoen, A.: Adaptive algorithm for solving the SCFPP of demicontractive operators without a priori knowledge of operator norms. Ann. Univ. Ovidius Math. Ser 27, 153–175 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Padcharoen, A., Kumam, P., Cho, Y.J.: Split common fixed point problems for demicontractive operators. Numer. Algorithms 82, 297–320 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4, 154–158 (1970)

    MathSciNet  MATH  Google Scholar 

  21. Moudafi, A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geom. 15, 91–100 (1999)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Nonconvex Optimization and Its Applications, 289-298. Springer, New York (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62, 271–283 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, S., Zhao, M., Kumam, P., Cho, Y.J.: A viscosity extragradient method for an equilibrium problem and fixed point problem in Hilbert space. J. Fixed Point Theory Appl. 20, 1–14 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rehman, H.U., Kumam, P., Özdemir, M., Karahan, I.: Two generalized non-monotone explicit strongly convergent extragradient methods for solving pseudomonotone equilibrium problems and applications. Math. Comput. Simul., 1-24 (2021)

  30. Rehman, H.U., Kumam, P., Dong, Q.L., Cho, Y.J.: A modified self-adaptive extragradient method for pseudomonotone equilibrium problem in a real Hilbert space with applications. Math. Methods Appl. Sci. 44, 3527–3547 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Rehman, H.U., Kumam, P., Gibali, A., Kumam, W.: Convergence analysis of a general inertial projection-type method for solving pseudomonotone equilibrium problems with applications. J. Inequal. Appl. 2021, 1–27 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yordsorn, P., Kumam, P., Hassan, I.A.: A weak convergence self-adaptive method for solving pseudomonotone equilibrium problems in a real Hilbert space. Mathematics 8, 1165 (2020)

    Article  Google Scholar 

  33. Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert space. Springer, New York (2011)

    Book  MATH  Google Scholar 

  34. Dinh, B.V.: Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. 302, 106–117 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Isiogugu, F.O., Izuchukwu, C., Okeke, C.C.: New iteration scheme for approximating a common fixed point of a finite family of mappings. J. Math. 2020, 1–14 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Rehman, H.U., Kumam, P., Sitthithakerngkiet, K.: Viscosity-type method for solving pseudomonotone equilibrium problems in a real Hilbert space with applications. AIMS Math. 6, 1538–1560 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  41. Hieu, D.V., Quy, P.K., Van, V.L.: Explicit iterative algorithms for solving equilibrium problems. Calcolo 56, 1–21 (2019)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author thank the referee for his/her useful comments, remarks and important suggestions on this article. The first author delightedly acknowledges the University Grants Commission (UGC), New Delhi for providing financial support.

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Authors and Affiliations

  1. Department of Mathematics, Maharshi Dayanand University, Rohtak, 124001, India

    Charu Batra, Renu Chugh & Rajeev Kumar

  2. Department of Mathematics, Pt. NRS Government College, Rohtak, 124001, India

    Nishu Gupta

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  1. Charu Batra
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  2. Nishu Gupta
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  3. Renu Chugh
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  4. Rajeev Kumar
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Correspondence to Nishu Gupta.

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Batra, C., Gupta, N., Chugh, R. et al. Generalized viscosity extragradient algorithm for pseudomonotone equilibrium and fixed point problems for finite family of demicontractive operators. J. Appl. Math. Comput. 68, 4195–4222 (2022). https://doi.org/10.1007/s12190-022-01699-x

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