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A Generalized Quasi-Equilibrium Problem

In: Nonlinear Analysis and Variational Problems

Author

Listed:
  • Mircea Balaj

    (University of Oradea)

  • Donal O’Regan

    (National University of Ireland)

Abstract

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 (∀)y∈R( $$\widetilde{x}$$ ), (α)z∈Q( $$\widetilde{x}$$ ), 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.

Suggested Citation

  • Mircea Balaj & Donal O’Regan, 2010. "A Generalized Quasi-Equilibrium Problem," Springer Optimization and Its Applications, in: Panos M. Pardalos & Themistocles M. Rassias & Akhtar A. Khan (ed.), Nonlinear Analysis and Variational Problems, chapter 0, pages 201-211, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-0158-3_15
    DOI: 10.1007/978-1-4419-0158-3_15
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    Cited by:

    1. Alfredo Iusem & Mostafa Nasri, 2011. "Korpelevich’s method for variational inequality problems in Banach spaces," Journal of Global Optimization, Springer, vol. 50(1), pages 59-76, May.

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