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The modified viscosity implicit rules for variational inequality problems and fixed point problems of nonexpansive mappings in Hilbert spaces

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


The purpose of this paper is to introduce a modified viscosity implicit rules for finding a common element of the set of solutions of variational inequality problems for two inverse-strongly monotone operators and the set of fixed points of one nonexpansive mapping in Hilbert spaces. Under some suitable assumptions imposed on the parameters, we obtain some strong convergence theorems. We also apply our main results to solve fixed point problems for strict pseudocontractive mappings and equilibrium problems in Hilbert spaces. A numerical example is given for supporting our main results.

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  1. Ceng, L.C., Wang, C., 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  Google Scholar 

  2. Yao, Y., Noor, M.A., Noor, K.I., Liou, Y.-C., Yaqoobet, H.: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 110, 1211–1224 (2010)

    Article  MathSciNet  Google Scholar 

  3. Noor, M.A.: Some development in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Yao, Y., Noor, M.A.: On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 325, 776–787 (2007)

    Article  MathSciNet  Google Scholar 

  5. Yao, Y., Maruster, S.: Strong convergence of an iterative algorithm for variational inequalities in Banach spaces. Math. Comput. Model. 54, 325–329 (2011)

    Article  MathSciNet  Google Scholar 

  6. Noor, M.A.: Some algorithms for general monotone mixed variational inequalities. Math. Comput. Model. 29, 1–9 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Noor, M.A.: On iterative methods for solving a system of mixed variational inequalities. Appl. Anal. 87, 99–108 (2008)

    Article  MathSciNet  Google Scholar 

  8. Zhu, D.L., Marcotte, P.: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 774–726 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Iiduka, H., Takahashi, W., Toyoda, M.: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14, 49–61 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Zegeye, H., Shahzad, N., Yao, Y.: Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization 64, 453–471 (2015)

    Article  MathSciNet  Google Scholar 

  11. Yao, Y., Agarwal, R.P., Postolache, M., Liou, Y.-C.: Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theory Appl. 183, 14 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Yao, Y., Shahzad, N.: Strong convergence of a proximal point algorithm with general errors. Optim. Lett. 6, 621–628 (2012)

    Article  MathSciNet  Google Scholar 

  13. Yao, Y., Postolache, M., Yao, C.: An iterative algorithm for solving the generalized variational inequalities and xed points problems. Mathematics 7, 61 (2019).

    Article  Google Scholar 

  14. Deuflhard, P.: Recent progress in extrapolation methods for ordinary differential equations. SIAM Rev. 27(4), 505–535 (1985)

    Article  MathSciNet  Google Scholar 

  15. Bader, G., Deuflhard, P.: A semi-implicit mid-point rule for stiff systems of ordinary differential equations. Numer. Math. 41, 373–398 (1983)

    Article  MathSciNet  Google Scholar 

  16. Somalia, S.: Implicit midpoint rule to the nonlinear degenerate boundary value problems. Int. J. Comput. Math. 79(3), 327–332 (2002)

    Article  MathSciNet  Google Scholar 

  17. Xu, H.K., Aoghamdi, M.A., Shahzad, N.: The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2015, 41 (2015)

    Article  MathSciNet  Google Scholar 

  18. Ke, Y., Ma, C.: The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2015, 190 (2015)

    Article  MathSciNet  Google Scholar 

  19. Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  Google Scholar 

  20. Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119(1), 185–201 (2003)

    Article  MathSciNet  Google Scholar 

  21. Goebel, K., Kirk, W.A.: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  22. Zhou, H.: Convergence theorems of fixed points for \(k\)-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69(2), 456–462 (2008)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Chancelier, J.P.: Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 353, 141–153 (2009)

    Article  MathSciNet  Google Scholar 

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Correspondence to Gang Cai.

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This work was supported by the NSF of China (Grant nos. 11401063, 11771063), the Natural Science Foundation of Chongqing (Grant nos. cstc2017jcyjAX0006, KJZDM201800501), Science and Technology Project of Chongqing Education Committee (Grant no. KJ1703041), the University Young Core Teacher Foundation of Chongqing (Grant no. 020603011714), Talent Project of Chongqing Normal University (Grant no. 02030307-00024).

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Cai, G., Shehu, Y. & Iyiola, O.S. The modified viscosity implicit rules for variational inequality problems and fixed point problems of nonexpansive mappings in Hilbert spaces. RACSAM 113, 3545–3562 (2019).

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