IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i10p916-d272962.html
   My bibliography  Save this article

Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis

Author

Listed:
  • Anchalee Sripattanet

    (Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand)

  • Atid Kangtunyakarn

    (Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand)

Abstract

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.

Suggested Citation

  • 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.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:916-:d:272962
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/10/916/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/7/10/916/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. 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.
    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.
    4. Zhao-Rong Kong & Lu-Chuan Ceng & Ching-Feng Wen, 2012. "Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-32, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. L. Zeng & J. Yao, 2009. "A hybrid extragradient method for general variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(1), pages 141-158, March.
    4. 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.
    5. Yonghong Yao & Yeong-Cheng Liou & Ngai-Ching Wong, 2013. "Superimposed optimization methods for the mixed equilibrium problem and variational inclusion," Journal of Global Optimization, Springer, vol. 57(3), pages 935-950, November.
    6. 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.
    7. 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.
    8. Bunyawee Chaloemyotphong & Atid Kangtunyakarn, 2019. "Modified Halpern Iterative Method for Solving Hierarchical Problem and Split Combination of Variational Inclusion Problem in Hilbert Space," Mathematics, MDPI, vol. 7(11), pages 1-26, November.
    9. Jinzuo Chen & Mihai Postolache & Li-Jun Zhu, 2019. "Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption," Mathematics, MDPI, vol. 7(9), pages 1-13, August.
    10. 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.
    11. Z. Y. Huang & M. A. Noor & E. Al-Said, 2010. "On an Open Question of Takahashi for Nonexpansive Mappings and Inverse Strongly Monotone Mappings," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 194-204, October.
    12. Atid Kangtunyakarn, 2013. "A new iterative scheme for fixed point problems of infinite family of κ i -pseudo contractive mappings, equilibrium problem, variational inequality problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1543-1562, August.
    13. Bin-Chao Deng & Tong Chen, 2013. "Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings," Journal of Mathematics, Hindawi, vol. 2013, pages 1-7, July.
    14. Y. Censor & A. Gibali & S. Reich, 2011. "The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 318-335, February.
    15. Lateef Olakunle Jolaoso & Maggie Aphane, 2020. "A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems," Mathematics, MDPI, vol. 8(11), pages 1-29, November.
    16. Lu-Chuan Ceng & Adrian Petruşel & Jen-Chih Yao, 2019. "On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities," Mathematics, MDPI, vol. 7(10), pages 1-14, October.
    17. X. Ding & Y. Liou & J. Yao, 2012. "Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces," Journal of Global Optimization, Springer, vol. 53(2), pages 331-346, June.
    18. Yuanheng Wang & Cancan Li & Lirong Lu, 2020. "A New Algorithm for the Common Solutions of a Generalized Variational Inequality System and a Nonlinear Operator Equation in Banach Spaces," Mathematics, MDPI, vol. 8(11), pages 1-21, November.
    19. S. Schaible & J. C. Yao & L. C. Zeng, 2006. "Iterative Method for Set-Valued Mixed Quasi-variational Inequalities in a Banach Space," Journal of Optimization Theory and Applications, Springer, vol. 129(3), pages 425-436, June.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:916-:d:272962. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.