A Generalized Quasi-Equilibrium Problem

Part of the Springer Optimization and Its Applications book series (SOIA, volume 35)


In this paper, using the Kakutani–Fan–Glicksberg fixed point theorem, we obtain an existence theorem for a generalized vector quasi-equilibrium problem of the following type: for a suitable choice of the sets X, Z and V and of the mappings T:X ⊸ X, R:X ⊸ X, Q:X ⊸ Z, F:X× X×Z ⊸ V, C:X ⊸ V, find \(\widetilde{x}\)X such that \(\widetilde{x}\)T(\(\widetilde{x}\)) and (∀)yR(\(\widetilde{x}\)), (α)zQ(\(\widetilde{x}\)), ρ(F((\(\widetilde{x}\),y,z), C(\(\widetilde{x}\))), where ρ is a given binary relation on 2V and α is any of the quantifiers ∈, ∃. Finally, several particular cases are discussed and some applications are given.


Topological Vector Space Vector Variational Inequality Vector Equilibrium Problem Nonempty Convex Hausdorff Topological Vector Space 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OradeaOradeaRomania
  2. 2.Department of MathematicsNational University of IrelandGalwayIreland

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