A Generalized Quasi-Equilibrium Problem

  • Mircea Balaj
  • Donal O’Regan
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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