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A System of Generalized Mixed Equilibrium Problems, Maximal Monotone Operators, and Fixed Point Problems with Application to Optimization Problems

Author

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  • Pongsakorn Sunthrayuth
  • Poom Kumam

Abstract

We introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, zero set of the sum of a maximal monotone operators and inverse‐strongly monotone mappings, and the set of common fixed points of an infinite family of nonexpansive mappings with infinite real number. Furthermore, we prove under some mild conditions that the proposed iterative algorithm converges strongly to a common element of the above four sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results presented in the paper improve and extend the recent ones announced by many others.

Suggested Citation

  • 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).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:316276
    DOI: 10.1155/2012/316276
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    References listed on IDEAS

    as
    1. Atid Kangtunyakarn, 2011. "A New Iterative Algorithm for the Set of Fixed-Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-24, April.
    2. 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).
    3. Atid Kangtunyakarn, 2011. "A New Iterative Algorithm for the Set of Fixed‐Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    4. Thanyarat Jitpeera & Poom Kumam, 2011. "A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-25, April.
    5. Wanpen Chantarangsi & Chaichana Jaiboon & Poom Kumam, 2010. "A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-39, July.
    6. Wanpen Chantarangsi & Chaichana Jaiboon & Poom Kumam, 2010. "A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
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    Citations

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    Cited by:

    1. 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).
    2. 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).
    3. 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).
    4. 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).

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