Journal of Optimization Theory and Applications

, Volume 123, Issue 3, pp 619–638 | Cite as

On the Existence of Solutions of Quasivariational Inclusion Problems

  • N. X. Tan


Quasivariational inclusion problems 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, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.

Upper quasivariational inclusions lower quasivariational inclusions α quasioptimization problems vector optimization problem quasiequilibrium problems upper and lower C-quasiconvex multivalued mappings upper and lower C-continuous multivalued mappings 


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  1. 1.
    Park, S., Fixed Points and Quasiequilibrium Problems: Nonlinear Operator Theory, Mathematical and Computer Modelling, Vol. 32, pp. 1297–1304, 2000.Google Scholar
  2. 2.
    Corley, H. W., An Existence Result for Maximization with Respect to Cones, Journal of Optimzation Theory and Applications, Vol. 31, pp. 277–281, 1980.Google Scholar
  3. 3.
    Parida, J., and Sen, A., A Variational-Like Inequality for Multifunctions with Applications, Journal of Mathematical Analysis and Applications, Vol. 124, pp. 73–81, 1987.Google Scholar
  4. 4.
    Gurraggio, A., and Tan, N. X., On General Vector Quasi-Optimization Problems, Mathematical Methods of Operation Research, Vol. 55, pp. 347–358, 2002.Google Scholar
  5. 5.
    Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 64, pp. 1–23, 1993.Google Scholar
  6. 6.
    Lin, L. J., Yu, Z. T., and Kassay, G., Existence of Equilibria for Monotone Multivalued Mappings and Its Application to Vectorial Equilibria, Journal of Optimization Theory and Applications, Vol. 114, pp. 189–208, 2002.Google Scholar
  7. 7.
    Tan, N. X., and Tinh, P. N., On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Optimization, Vol. 19, pp. 141–156, 1998.Google Scholar
  8. 8.
    Minh, N. B., and Tan, N. X., Some Sufficient Conditions for the Existence of Equilibrium Points Concerning Multivalued Mappings, Vietnam Journal of Mathematics, Vol. 28, pp. 295–310, 2000.Google Scholar
  9. 9.
    Fan, K., A Minimax Inequality and Applications, Inequalities III, Edited by O. Shisha, Academic Press, New York, NY, p. 33, 1972.Google Scholar
  10. 10.
    Luc, D. T., Theory of Vector Optimization, Lectures Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 319, 1989.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • N. X. Tan
    • 1
  1. 1.Institute of MathematicsVietnam

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