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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Q. H. Ansari, A note on generalized vector variational-like inequalities, Optimization 41 (1997), 197-205.
Q. H. Ansari, W. Oettli and D. Schläger, A generalization of vectorial equilibria, Math. Methods Oper. Res. 47 (1997), 147-152.
Q. H. Ansari and F. Flores-Bazán, Generalized vector quasi-equilibrium problems with applications, J. Math. Anal. Appl. 277 (2003), 246-256.
Q. H. Ansari, S. Schaible and J. C. Yao, The systems of vector equilibrium problems and its applications, J. Optim. Theory Appl., 107 (2000), 547-557.
M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions J. Optim. Theory Appl. 92 (1997), 527-542.
E. Blum and W. Oettli, From optimization and variational inequalities problems to equilibrium problems Math. Student 63 (1994), 123-146.
X. P. Ding and J. Y. Park, Generalized vector equilibrium problems in generalized convex spaces, J. Optimiz. Theory Appl. 120 (2004), 327-353.
M. Fakhar and J. Zafarani, Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions, J. Optim. Theory Appl. 126 (2005), no. 1, 109–124.
M. Fang and N.-J. Huang, KKM type theorems with applications to generalized vector equilibrium problems in FC-spaces, Nonlinear Anal. 67 (2007), 809-817.
J. Y. Fu, Generalized vector quasi-equilibrium problems, Math. Methods Oper. Res. 52 (2000), 57-64.
J. Y. Fu and A.-H. Wan, Generalized vector equilibrium problems with set-valued mappings, Math. Methods Oper. Res. 56 (2002), 259-268.
F. Giannessi (ed.), Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000.
B. R. Halpern, G. M. Bergman, A fixed-point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353-358.
S. H. Hou, H. Yu and G. Y. Chen, On vector quasi-equilibrium problems with set-valued maps J. Optim. Theory Appl. 119 (2003), 485-498.
I. V. Konnov and J. C. Yao, Existence of solutions for generalized vector equilibrium problems, J. Math. Anal. Appl. 233 (1999), 328-335.
A. Kristály and C. Varga, Set-valued versions of Ky Fan’s inequality with application to variational inclusion theory, J. Math. Anal. Appl. 282 (2003), 8-20.
M. Lassonde, Fixed points for Kakutani factorizable multifunctions, J. Math. Anal. Appl. 152 (1990), 46-60.
G. M. Lee, D. S. Kim and B. S. Lee, Generalized vector variational inequality, Appl. Math. Lett. 9 (1996), 39-42.
S. J. Li, K. L. Teo and X. Q. Yang, Generalized vector quasi-equilibrium problems, Math. Methods Oper. Res. 61 (2005), 385-397.
G. M. Lee, and S. H. Kum, On implicit vector variational inequalities, J. Optim. Theory Appl. 104 (2000), 409-425.
L. J. Lin, System of generalized vector quasi-equilibrium problems with applications to fixed point theorems for a family of nonexpansive multivalued mappings, J. Global Optim. 34 (2006), 5-32.
L. J. Lin, Variational inclusions problems with applications to Ekeland’s variational principle, fixed point and optimization problems, J. Global Optim, 39 (2007), 509-527.
L. J. Lin and C. S. Chuang, Systems of nonempty intersection theorems with applications, Nonlinear Analysis, 69(2008), 4063-4073 DOI. 10.1016/.2007.10.037.
L. J. Lin and C. I. Tu, The study of variational inclusions problems and variational disclusions problems with applications, Nonlinear Anal., 69 (2008), 1981-1998.
L. J. Lin, S. Y. Wang and C. S. Chuang, Existence theorems of systems of variational inclusion problems with applications, J. Global Optimization, 40 (2008), 751-764.
L. J. Lin, Z. T. Yu and G. Kassay, Existence of equilibria for multivalued mappings and its application to vectorial equilibria, J. Optim. Theory Appl. 114 (2002), 189-208.
S. Park, Fixed points and quasi-equilibrium problems. Nonlinear operator theory, Math. Comput. Modelling 32 (2000), 1297-1304.
N. X. Tan and P. N. Tinh, On the existence of equilibrium points of vector functions, Numerical Functional Analysis Optimiz., 19 (1998), 141-156.
N. C. Yannelis and N. D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econom. 12 (1983), 233-245.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Professor George Isac
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Balaj, M., O’Regan, D. (2010). A Generalized Quasi-Equilibrium Problem. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_15
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0158-3_15
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0157-6
Online ISBN: 978-1-4419-0158-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)