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

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  • Lu-Chuan Ceng
  • Chang-yu Wang
  • Jen-Chih Yao

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. Copyright Springer-Verlag 2008

Suggested Citation

  • Lu-Chuan Ceng & Chang-yu Wang & Jen-Chih Yao, 2008. "Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(3), pages 375-390, June.
    • Handle: RePEc:spr:mathme:v:67:y:2008:i:3:p:375-390
    DOI: 10.1007/s00186-007-0207-4
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    1. L. C. Zeng & S. Schaible & J. C. Yao, 2005. "Iterative Algorithm for Generalized Set-Valued Strongly Nonlinear Mixed Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 725-738, March.
    2. W. Takahashi & M. Toyoda, 2003. "Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 417-428, August.
    3. N. Nadezhkina & W. Takahashi, 2006. "Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 128(1), pages 191-201, January.
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    Cited by:

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    3. Lu-Chuan Ceng & Meijuan Shang, 2019. "Strong Convergence Theorems for Variational Inequalities and Common Fixed-Point Problems Using Relaxed Mann Implicit Iteration Methods," Mathematics, MDPI, vol. 7(5), pages 1-16, May.
    4. Yanlai Song & Luchuan Ceng, 2013. "A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces," Journal of Global Optimization, Springer, vol. 57(4), pages 1327-1348, December.
    5. S. Plubtieng & T. Thammathiwat, 2010. "A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities," Journal of Global Optimization, Springer, vol. 46(3), pages 447-464, March.
    6. Satit Saejung & Kanokwan Wongchan, 2011. "A note on Ceng-Wang-Yao’s result [Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. (2008) 67: 375–390]," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 153-157, April.
    7. Mihai Postolache & Ashish Nandal & Renu Chugh, 2019. "Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space," Mathematics, MDPI, vol. 7(9), pages 1-24, August.
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    9. Rais Ahmad & Mohd Ishtyak & Arvind Kumar Rajpoot & Yuanheng Wang, 2022. "Solving System of Mixed Variational Inclusions Involving Generalized Cayley Operator and Generalized Yosida Approximation Operator with Error Terms in q -Uniformly Smooth Space," Mathematics, MDPI, vol. 10(21), pages 1-17, November.
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    14. Anchalee Sripattanet & Atid Kangtunyakarn, 2019. "Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis," Mathematics, MDPI, vol. 7(10), pages 1-18, October.
    15. Lu-Chuan Ceng & Qamrul Ansari & Siegfried Schaible, 2012. "Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems," Journal of Global Optimization, Springer, vol. 53(1), pages 69-96, May.
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    17. Lu-Chuan Ceng & Qing Yuan, 2019. "Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems," Mathematics, MDPI, vol. 7(2), pages 1-24, February.

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