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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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    1. Gang Cai & Shangquan Bu, 2013. "Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces," Journal of Global Optimization, Springer, vol. 55(2), pages 437-457, February.
    2. 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.
    3. Lu-Chuan Ceng & Xiaoye Yang, 2019. "Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems," Mathematics, MDPI, vol. 7(3), pages 1-20, February.
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    6. 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.
    7. Lu-Chuan Ceng & Qing Yuan, 2019. "Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities, and Fixed Point Problems," Mathematics, MDPI, vol. 7(2), pages 1-19, February.
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    10. 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.
    11. 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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    13. Lu-Chuan Ceng & Mihai Postolache & Xiaolong Qin & Yonghong Yao, 2018. "System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces," Mathematics, MDPI, vol. 7(1), pages 1-14, December.
    14. Lu-Chuan Ceng & Meijuan Shang, 2019. "Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems," Mathematics, MDPI, vol. 7(10), pages 1-18, October.
    15. Gang Cai & Shangquan Bu, 2013. "Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces," Journal of Global Optimization, Springer, vol. 56(4), pages 1529-1542, August.
    16. 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.
    17. 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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