Quasi-equilibrium Inclusion Problems of the Blum–Oettli-Type and Related Problems

  • Nguyen Xuan TanEmail author
  • Lai-Jiu Lin
Part of the Springer Optimization and Its Applications book series (SOIA, volume 39)


The quasi-equilibrium inclusion problems of Blum–Oettli type are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of general vector ideal (resp. proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of quasivariational inclusion problems.


upper and lower quasivariational inclusions inclusions α-quasi-optimization problems vector optimization problem, quasi-equilibrium problems upper and lower C-quasiconvex multivalued mappings upper and lower C-continuous multivalued mappings 


  1. 1.
    Blum, E., Oettli, W.: From optimization and student. 64, 1–23 (1993)Google Scholar
  2. 2.
    Chan, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Fan, K.: A Minimax Inequality and Application. In: O. Shisha (Ed.) Inequalities III, Academic, New York, NY (pp. 33), (1972)Google Scholar
  4. 4.
    Gurraggio, A., Tan, N. X.: On general vector quasi-optimization problems. Math. Meth. Oper. Res. 55, 347–358 (2002)Google Scholar
  5. 5.
    Lin, L.J., Yu, Z. T., Kassay, G.: Existence of equilibria for monotone multivalued mappings and its applications to vectorial equilibria. J. Optim. Theory Appl. 114, 189–208 (2002)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Luc, D.T.: Theory of vector optimization. Lect. Notes Eco. Math. Syst. 319, Springer-Verlag, (1989)Google Scholar
  7. 7.
    Luc, D.T., Tan, N.X.: Existence conditions in variational inclusions with constraints. Optimization 53 (5–6), 505–515 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Minh, N.B., Tan, N.X.: Some sufficient conditions for the existence of equilibrium points concerning multivalued mappings. Vietnam. J. Math. 28, 295–310, (2000)zbMATHGoogle Scholar
  9. 9.
    Minh, N.B., Tan, N.X.: On the existence of solutions of quasivariational inclusion problems of Stampacchia type. Adv. Nonlinear Var. Inequal. 8, 1–16 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Parida, J., Sen, A.: A Variational-like inequality for multifunctions with applications. J. Math. Anal. Appl. 124, 73–81 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Park, S.: Fixed points and quasi-equilibrium problems. Nonlinear Oper. Theory. Math. Com. Model. 32, 1297–1304 (2000)zbMATHGoogle Scholar
  12. 12.
    Tan, N.X.: On the existence of solutions of quasi-variational inclusion problems. J. Optim. Theory Appl. 123, 619–638 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan

Personalised recommendations