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Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces

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  • Y.P. Fang

    (Sichuan University)

  • N.J. Huang

    (Sichuan University)

Abstract

In this paper, we introduce two classes of variational-like inequalities with generalized monotone mappings in Banach spaces. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed η–α monotone mappings in reflexive Banach spaces. We present also the solvability of variational-like inequalities with relaxed η–α semimonotone mappings in arbitrary Banach spaces by means of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve the corresponding results of Refs. 1–6.

Suggested Citation

  • Y.P. Fang & N.J. Huang, 2003. "Variational-Like Inequalities with Generalized Monotone Mappings in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 118(2), pages 327-338, August.
    • Handle: RePEc:spr:joptap:v:118:y:2003:i:2:d:10.1023_a:1025499305742
    DOI: 10.1023/A:1025499305742
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    References listed on IDEAS

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    1. K. L. Lin & D. P. Yang & J. C. Yao, 1997. "Generalized Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 117-125, January.
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    Cited by:

    1. Syed Shakaib Irfan & Mohammad Firdosh Khan, 2016. "Variational-Like Inequalities for Weakly Relaxed Pseudomonotone Set-Valued Mappings in Banach Space," International Journal of Analysis, Hindawi, vol. 2016, pages 1-6, September.
    2. S. K. Mishra & Vivek Laha, 2013. "On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 156(2), pages 278-293, February.
    3. D. L. Zhu & L. L. Zhu & Q. Xu, 2008. "Generalized Invex Monotonicity and Its Role in Solving Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 453-464, May.
    4. Nicuşor Costea & Daniel Alexandru Ion & Cezar Lupu, 2012. "Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 79-99, October.
    5. Gayatri Pany & Ram N. Mohapatra & Sabyasachi Pani, 2018. "Solution of a class of equilibrium problems and variational inequalities in FC spaces," Annals of Operations Research, Springer, vol. 269(1), pages 565-582, October.
    6. Jiawei Chen & Shengjie Li & Zhongping Wan & Jen-Chih Yao, 2015. "Vector Variational-Like Inequalities with Constraints: Separation and Alternative," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 460-479, August.
    7. Suhel Ahmad Khan, 2013. "Vector Variational-Like Inequalities with Generalized Semimonotone Mappings," International Journal of Analysis, Hindawi, vol. 2013, pages 1-7, January.
    8. T. Jabarootian & J. Zafarani, 2008. "Generalized Vector Variational-Like Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 15-30, January.
    9. Farajzadeh, A.P. & Plubtieng, S. & Ungchittrakool, K. & Kumtaeng, D., 2015. "Generalized mixed equilibrium problems with generalized α -η -monotone bifunction in topological vector spaces," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 313-319.
    10. Ayed E. Hashoosh & Mohsen Alimohammady & M. K. Kalleji, 2016. "Existence Results for Some Equilibrium Problems Involving -Monotone Bifunction," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2016, pages 1-5, February.
    11. Somyot Plubtieng & Tipphawan Thammathiwat, 2014. "Existence of Solutions of New Generalized Mixed Vector Variational-Like Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 589-604, August.

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