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Common solution to generalized mixed equilibrium problem and fixed point problems in Hilbert space

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

Abstract

In this paper, we deal with an extra-gradient iterative method for finding a common solution to a generalized mixed equilibrium problem and fixed point problems for a nonexpansive mapping and for a finite family of k-strict pseudo-contraction mappings in Hilbert space. We prove a strong convergence theorem for the extra-gradient iterative method under some mild conditions. Further, we give a numerical example to illustrate the main result.

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References

  1. Baiocchi, C., Capelo, A.: Variational and Quasi-Variational Inequalities. Wiley, New York (1984)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24, 82–90 (1967)

    Article  Google Scholar 

  4. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)

    Article  MathSciNet  Google Scholar 

  5. Daniele, P., Giannessi, F., Mougeri, A. (eds.): Equilibrium Problems and Variational Models. Nonconvex Optimization and Its Application, vol. 68. Kluwer Academic Publishers, Norwell (2003)

    Google Scholar 

  6. Djafari-Rouhani, B., 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 

  7. Djafari-Rouhani, B., Farid, M., Kazmi, K.R.: Common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. J. Korean Math. Soc. 53(1), 89–114 (2016)

    Article  MathSciNet  Google Scholar 

  8. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physcis. Springer, Berlin (1976)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity. SIAM, Philadelphia (1998)

    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. Marino, G., Xu, H.K.: Weak and strong convergence theorems for \(k\)-strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–349 (2007)

    Article  MathSciNet  Google Scholar 

  14. Moudafi, A.: Second order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4(1), 18 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Moudafi, A., Thera, M.: Proximal and Dynamical Approaches to Equilibrium Problems. Lecture Notes in Economics and Mathematical systems, vol. 477, pp. 187–201. Springer, New York (1999)

    MATH  Google Scholar 

  16. Nadezhkina, N., Takahashi, W.: Weak convergence theorem by a extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Oden, J.T., Pires, E.B.: Contact problems in elastostatics with nonlocal friction laws, TICOM, Report 81–12. University of Texas, Austin (1981)

  20. Osilike, M.O., Igbokwe, D.I.: Weak and strong convergence theorems for fixed points of pseudo-contractions and solutions of monotone type operator equations. Comput. Math. Appl. 40, 559–567 (2000)

    Article  MathSciNet  Google Scholar 

  21. Scherzer, O.: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 194, 911–933 (1991)

    Article  MathSciNet  Google Scholar 

  22. Xu, H.K.: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67, 2258–2271 (2007)

    Article  MathSciNet  Google Scholar 

  23. Xu, W., Wang, Y.: Strong convergence of the iterative methods for hierarchial fixed point problems of an infinite family of strictly nonself pseudocontractions. Abstr. Appl. Anal. 2012, 457024 (2012). https://doi.org/10.1155/2012/457024

    Article  MATH  Google Scholar 

  24. Zegeye, H., Boikanyo, A.: Approximating solutions of variational inequalities and fixed points of nonexpansive semigroups. Adv. Nonlinear Var. Inequal. 20, 26–40 (2017)

  25. Zhou, H.Y.: Convergence theorems of fixed points for \(k\)-strict pseudo-contractions in Hilbert space. Nonlinear Anal. 69, 456–462 (2008)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Authors are very grateful to the anonymous referees for their critical comments which led to substantial improvements in the original version of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

    K. R. Kazmi

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

    K. R. Kazmi & Saleem Yousuf

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  1. K. R. Kazmi
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  2. Saleem Yousuf
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Correspondence to K. R. Kazmi.

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Kazmi, K.R., Yousuf, S. Common solution to generalized mixed equilibrium problem and fixed point problems in Hilbert space. RACSAM 113, 3699–3715 (2019). https://doi.org/10.1007/s13398-019-00725-1

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