IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2012y2012i1n804538.html

Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

Author

Listed:
  • Kamonrat Nammanee
  • Suthep Suantai
  • Prasit Cholamjiak

Abstract

We introduce hybrid‐iterative schemes for solving a system of the zero‐finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

Suggested Citation

  • Kamonrat Nammanee & Suthep Suantai & Prasit Cholamjiak, 2012. "Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:804538
    DOI: 10.1155/2012/804538
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2012/804538
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2012/804538?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Siwaporn Saewan & Poom Kumam, 2010. "A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-31, November.
    2. Kriengsak Wattanawitoon & Poom Kumam, 2012. "A New Iterative Scheme for Generalized Mixed Equilibrium, Variational Inequality Problems, and a Zero Point of Maximal Monotone Operators," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Kriengsak Wattanawitoon & Poom Kumam, 2012. "A New Iterative Scheme for Generalized Mixed Equilibrium, Variational Inequality Problems, and a Zero Point of Maximal Monotone Operators," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-27, January.
    4. Pongsakorn Sunthrayuth & Poom Kumam, 2012. "A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-39, December.
    5. Prasit Cholamjiak & Suthep Suantai, 2010. "Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi-Nonexpansive Mappings in Banach Spaces," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-17, September.
    6. Pongsakorn Sunthrayuth & Poom Kumam, 2012. "A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    7. Prasit Cholamjiak & Suthep Suantai, 2010. "Convergence Analysis for a System of Equilibrium Problems and a Countable Family of Relatively Quasi‐Nonexpansive Mappings in Banach Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    8. Uamporn Witthayarat & Yeol Je Cho & Poom Kumam, 2012. "Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-17, April.
    9. Yekini Shehu, 2011. "A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-Nonexpansive Mappings and System of Equilibrium Problems," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-23, June.
    10. Siwaporn Saewan & Poom Kumam, 2010. "A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi‐Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    11. Kriengsak Wattanawitoon & Poom Kumam, 2011. "Hybrid Proximal-Point Methods for Zeros of Maximal Monotone Operators, Variational Inequalities and Mixed Equilibrium Problems," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-31, February.
    12. Uamporn Witthayarat & Yeol Je Cho & Poom Kumam, 2012. "Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    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. Zhangsong Yao & Sun Young Cho & Shin Min Kang & Li-Jun Zhu, 2015. "Approximating Iterations for Nonexpansive and Maximal Monotone Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2015(1).
    2. Yaqin Wang, 2012. "A New Hybrid Method for Equilibrium Problems, Variational Inequality Problems, Fixed Point Problems, and Zero of Maximal Monotone Operators," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Kriengsak Wattanawitoon & Poom Kumam, 2012. "A New Iterative Scheme for Generalized Mixed Equilibrium, Variational Inequality Problems, and a Zero Point of Maximal Monotone Operators," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    4. Chaichana Jaiboon & Poom Kumam, 2012. "Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed‐Point Problems of a Countable Family of Nonexpansive Mappings," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    5. Utith Inprasit & Weerayuth Nilsrakoo, 2011. "A Halpern‐Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    6. Pongsakorn Sunthrayuth & Poom Kumam, 2012. "A New General System of Generalized Nonlinear Mixed Composite‐Type Equilibria and Fixed Point Problems with an Application to Minimization Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    7. Poom Kumam & Phayap Katchang, 2012. "A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    8. Prasit Cholamjiak & Suthep Suantai, 2013. "Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups," Journal of Global Optimization, Springer, vol. 57(4), pages 1277-1297, December.
    9. Yekini Shehu, 2010. "A New Iterative Scheme for Countable Families of Weak Relatively Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    10. S. Saewan & P. Kumam, 2013. "A generalized f-projection method for countable families of weak relatively nonexpansive mappings and the system of generalized Ky Fan inequalities," Journal of Global Optimization, Springer, vol. 56(2), pages 623-645, June.
    11. Siwaporn Saewan & Poom Kumam, 2011. "Strong Convergence Theorems for Countable Families of Uniformly Quasi‐ϕ‐Asymptotically Nonexpansive Mappings and a System of Generalized Mixed Equilibrium Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).

    More about this item

    Statistics

    Access and download statistics

    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:wly:jnljam:v:2012:y:2012:i:1:n:804538. 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: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    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.