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Two algorithms for solving single-valued variational inequalities and fixed point problems

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Abstract

In this paper, we suggest two new iterative methods for finding a common element of the solution set of a variational inequality problem and the set of fixed points of a contraction mapping in Hilbert space. We also present weak and strong convergence theorems for these new methods, provided that the fixed point mapping is a θ-strict pseudocontraction and the mapping associated with the variational inequality problem is monotone. The results presented in this paper improve and unify important recent results announced by many authors.

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References

  1. Bnouhachem A.: A self-adaptive method for solving general mixed variational inequalities. J. Math. Anal. Appl. 309, 136–150 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Cegielski, A. Gibali, S. Reich and R. Zalas, An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space. Numer. Funct. Anal. Optim. 34 (2013), 1067–1096.

  3. L.-C. Ceng, N. Hadjisavvas and N.-C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Global Optim. 46 (2010), 635–646.

  4. Ceng L.-C., Huang S.: Modified extragradient methods for strict pseudocontractions and monotone mappings. Taiwanese J. Math. 13, 1197–1211 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Ceng L.-C., Yao J.-C.: An extragradient-like approximation method for variational inequality problems and fixed point problems. Appl. Math. Comput. 190, 205–215 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Y. Censor, A. Gibali, and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26 (2011), 827–845.

  7. Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148 (2011), 318–335.

  8. Y. Censor, A. Gibali, and S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61 (2012), 1119–1132.

  9. J. Y. Bello Cruz and A. N. Iusem, Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46 (2010), 247–263.

  10. F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. 1, 2, Springer-Verlag, New York, 2003.

  11. Fang J. C., Chen L. S.: A subgradient extragradient algorithm for solving multi-valued variational inequality. Appl. Math. Comput. 229, 123–130 (2014)

    Article  MathSciNet  Google Scholar 

  12. C. J. Fang, S. L. Chen and J. M. Zheng, A projection-type method for multivalued variational inequality. Abstr. Appl. Anal. 2013 (2013), Article ID 836720, 6 pages.

  13. C. J. Fang and Y. R. He, A double projection algorithm for multi-valued variational inequalities and a unified framework of the method. Appl. Math. Comput. 217 (2011), 9543–9511.

  14. Fang C. J., He Y. R.: An extragradient method for generalized variational inequality. Pac. J. Optim. 9, 47–59 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Fukushima M.: The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem. Math. Program. 72, 1–15 (1996)

    MathSciNet  MATH  Google Scholar 

  16. He Y. R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185, 166–173 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Iiduka and W. Takahashi, Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications. Adv. Nonlinear Var. Inequal. 9 (2006), 1–10.

  18. H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19 (2008), 1881–1893.

  19. Iiduka H., Yamada I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58, 251–261 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Jaiboon and P. Kumam, Strong convergence theorems for solving equilibrium problems and fixed point problems of ξ-strict pseudo-contraction mappings by two hybrid projection methods. J. Comput. Appl. Math. 234 (2010), 722–732.

  21. Karamardian S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 12 (2011), 1319–1344.

  23. G. M. Korpelevič, An extragradient method for finding saddle points and for other problems. Èkonom. i Mat. Metody 12 (1976), 747–756 (in Russian).

  24. Maingé P.-E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Maingé P.-E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. European J. Oper. Res. 205, 501–506 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Marino G., Xu H.-K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Petruşel and J.-C. Yao, An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequalitity problems. Cent. Eur. J. Math. 7 (2009), 335–347.

  31. Rockafellar R. T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems. SIAM J. Control Optim. 37 (1999), 765–776.

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

  34. Tseng P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. P. T. Vuong, J. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155 (2012), 605–627.

  36. Y. J. Wang, N. H. Xiu and J. Z. Zhang, Modified extragradient method for variational inequalities and verification of solution existence. J. Optim. Theory Appl. 119 (2003), 167–183.

  37. F. H. Wang and H.-K. Xu, Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng’s extragradient method. Taiwanese J. Math. 16, 1125–1136 (2012)

  38. X. Y. Wang, S. J. Li and X. P. Kou, An extension of subgradient method for variational inequality problems in Hilbert space. Abstr. Appl. Anal. 2013 (2013), Article ID 531912, 7 pages.

  39. Xia F.-Q., Huang N.-J.: A projection-proximal point algorithm for solving generalized variational inequalities. J. Optim. Theory Appl. 150, 98–117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yao Y., Yao J.-C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186, 1551–1558 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. II. Spectral theory. In: Contributions to Nonlinear Functional Analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Academic Press, New York, 1971, 343–424.

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Authors and Affiliations

  1. College of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Changjie Fang, Ying Wang & Shaokang Yang

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  1. Changjie Fang
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  2. Ying Wang
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  3. Shaokang Yang
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Correspondence to Changjie Fang.

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Fang, C., Wang, Y. & Yang, S. Two algorithms for solving single-valued variational inequalities and fixed point problems. J. Fixed Point Theory Appl. 18, 27–43 (2016). https://doi.org/10.1007/s11784-015-0258-8

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