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Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

Author

Listed:
  • S. Al-Homidan

    (King Fahd University of Petroleum and Minerals)

  • Q. H. Ansari

    (King Fahd University of Petroleum and Minerals
    Aligarh Muslim University)

  • S. Schaible

    (University of California)

Abstract

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques.

Suggested Citation

  • S. Al-Homidan & Q. H. Ansari & S. Schaible, 2007. "Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 515-531, September.
    • Handle: RePEc:spr:joptap:v:134:y:2007:i:3:d:10.1007_s10957-007-9236-7
    DOI: 10.1007/s10957-007-9236-7
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    References listed on IDEAS

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    1. Q. H. Ansari & J> C> Yao, 2000. "On Nondifferentiable and Nonconvex Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 475-488, September.
    2. X.Q. Yang & J.C. Yao, 2002. "Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 407-417, November.
    3. G. M. Lee & S. H. Kum, 2000. "On Implicit Vector Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 104(2), pages 409-425, February.
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    Cited by:

    1. Jia-Wei Chen & Zhongping Wan & Yeol Cho, 2013. "Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(1), pages 33-64, February.
    2. J. W. Chen & Y. J. Cho & S. A. Khan & Z. Wan & C. F. Wen, 2015. "The Levitin-Polyak well-posedness by perturbations for systems of general variational inclusion and disclusion problems," Indian Journal of Pure and Applied Mathematics, Springer, vol. 46(6), pages 901-920, December.

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