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Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities

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Abstract

In this paper, we introduce and study a relaxed extragradient method for finding solutions of a general system of variational inequalities with inverse-strongly monotone mappings in a real Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Second, by using the demi-closedness principle for nonexpansive mappings, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a solution of this system of variational inequalities. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.

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References

  • Browder FE, Petryshyn WV (1967) Construction of fixed points of nonlinear mappings in Hilbert Spaces. J Math Anal Appl 20: 197–228

    Article  MATH  MathSciNet  Google Scholar 

  • Goebel K, Kirk WA (1990) Topics on metric fixed-point theory. Cambridge University Press, Cambridge

    Google Scholar 

  • Korpelevich GM (1976) An extragradient method for finding saddle points and for other problems. Ekon Mate Metody 12: 747–756

    MATH  Google Scholar 

  • Liu F, Nashed MZ (1998) Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Value Analy 6: 313–344

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Osilike MO, Igbokwe DI (2000) Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput Math Appl 40: 559–567

    Article  MATH  MathSciNet  Google Scholar 

  • Suzuki T (2005) Strong convergence of krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without bochner integrals. J Math Anal Appl 305: 227–239

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Verma RU (1999) On a new system of nonlinear variational inequalities and associated iterative algorithms. Math Sci Res, Hot-Line 3(8): 65–68

    MATH  MathSciNet  Google Scholar 

  • Verma RU (2001) Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv Nonlinear Var Inequal 4(1): 117–124

    MATH  MathSciNet  Google Scholar 

  • Xu HK (2004) Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 298: 279–291

    Article  MATH  MathSciNet  Google Scholar 

  • Yao JC (1994) Variational inequalities and generalized monotone operators. Math Opera Res 19: 691–705

    Article  MATH  Google Scholar 

  • Yao JC, Chadli O (2005) Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix JP, Haddjissas N, Schaible S (eds) Handbook of generalized convexity and monotonicity. pp 501–558

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

    Article  MATH  MathSciNet  Google Scholar 

  • Zeng LC, Yao JC (2006) Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J Math 10: 1293–1303

    MATH  MathSciNet  Google Scholar 

  • Zeng LC, Schaible S, Yao JC (2005) Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. J Optim Theory Appl 124: 725–738

    Article  MATH  MathSciNet  Google Scholar 

  • Zeng LC, Wong NC, Yao JC (2006) Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type. Taiwan J Math 10(4): 837–849

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Lu-Chuan Ceng

  2. Institute of Operations Research, Qufu Normal University, Qufu, Shandong, 273165, China

    Chang-yu Wang

  3. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    Jen-Chih Yao

Authors
  1. Lu-Chuan Ceng
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  2. Chang-yu Wang
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  3. Jen-Chih Yao
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Corresponding author

Correspondence to Jen-Chih Yao.

Additional information

L.-C. Ceng’s research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118).

J.-C. Yao’s research was partially supported by a grant from the National Science Council.

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Ceng, LC., Wang, Cy. & Yao, JC. Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math Meth Oper Res 67, 375–390 (2008). https://doi.org/10.1007/s00186-007-0207-4

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  • DOI: https://doi.org/10.1007/s00186-007-0207-4

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