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On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps

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  • P. H. Sach

    (Institute of Mathematics)

Abstract

This paper considers the following generalized vector quasiequilibrium problem: find a point (z 0,x 0) of a set E×K such that x 0∈A(z 0,x 0) and $$\forall \eta \in A(z_{0},x_{0}),\quad \exists v\in B(z_{0},x_{0},\eta),\quad (F(v,x_{0},\eta),C(v,x_{0},\eta))\in \alpha ,$$ where α is a subset of 2 Y ×2 Y , A:E×K→2 K , B:E×K×K→2 E , C:E×K×K→2 Y , F:E×K×K→2 Y are set-valued maps and Y is a topological vector space. Existence theorems are established under suitable assumptions, one of which is the requirement of the openness of the lower sections of some set-valued maps which can be satisfied with maps B,C, F being discontinuous. It is shown that, in some special cases, this requirement can be verified easily by using the semicontinuity property of these maps. Another assumption in the obtained existence theorems is assured by appropriate notions of diagonal quasiconvexity.

Suggested Citation

  • P. H. Sach, 2008. "On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 337-350, November.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:2:d:10.1007_s10957-008-9424-0
    DOI: 10.1007/s10957-008-9424-0
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    References listed on IDEAS

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    1. Jun-Yi Fu & An-Hua Wan, 2002. "Generalized vector equilibrium problems with set-valued mappings," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(2), pages 259-268, November.
    2. P. H. Sach & L. A. Tuan, 2007. "Existence Results for Set-Valued Vector Quasiequilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(2), pages 229-240, May.
    3. L.J. Lin & Q.H. Ansari & J.Y. Wu, 2003. "Geometric Properties and Coincidence Theorems with Applications to Generalized Vector Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 121-137, April.
    4. S. H. Hou & H. Yu & G. Y. Chen, 2003. "On Vector Quasi-Equilibrium Problems with Set-Valued Maps," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 485-498, December.
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    Cited by:

    1. Le Tuan & Gue Lee & Pham Sach, 2010. "Upper semicontinuity result for the solution mapping of a mixed parametric generalized vector quasiequilibrium problem with moving cones," Journal of Global Optimization, Springer, vol. 47(4), pages 639-660, August.

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