Journal of Global Optimization

, Volume 39, Issue 3, pp 393–407 | Cite as

On quasivariational inclusion problems of type I and related problems

  • Lai-Jiu LinEmail author
  • Nguyen Xuan Tan
Original Paper


The 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 a general vector ideal (proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of vector quasi-equilibrium problems. Further, we prove theorems on the existence for solutions of the sum of these inclusions. As corollaries, we shall show several results on the existence of solutions to another problems in the vector optimization problems concerning multivalued mappings.


Upper quasivariational inclusions Lower quasivariational 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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Blum E., Oettli W. (1993). From Optimization and Variational Inequalities to Equilibrium Problems. The Mathematical Student 64: 1–23 Google Scholar
  2. Browder F.E. (1968). The fixed point theory of multivalued mappings in topological vector spaces. Math. Ann. 177: 283–301 CrossRefGoogle Scholar
  3. Chan D., Pang J.S. (1982). The generalized quasi-variational inequality problem. Math. Oper. Res. 7: 211–222 CrossRefGoogle Scholar
  4. Fan K. (1972). A Minimax Inequality and Application. In: Shisha, O (eds) Inequalities III, pp 33. Academic Press, New-York Google Scholar
  5. Gurraggio A., Tan N.X. (2002). On general vector quasi–optimization problems. Math. Meth. Operation Res. 55: 347–358 CrossRefGoogle Scholar
  6. Kakutani S. (1941). A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8: 457–459 CrossRefGoogle Scholar
  7. Lin L.J., Yu Z.T., Kassay G. (2002). Existence of equilibria for monotone multivalued mappings and its applications to vectorial equilibria. J. Optimization Theory Appl. 114: 189–208 CrossRefGoogle Scholar
  8. Luc D.T.: Theory of vector optimization. Lectures Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, vol 319 (1989)Google Scholar
  9. Minh N.B., Tan N.X. (2000). Some sufficient conditions for the existence of equilibrium points concerning multivalued mappings. vietnam J. Math. 28: 295–310 Google Scholar
  10. Parida J., Sen A. (1987). A variational-like inequality for multifunctions with applications. J. Math. Anal. Appl. 124: 73–81 CrossRefGoogle Scholar
  11. Park S. (2000). Fixed points and quasi-equilibrium problems. Nonlinear operator theory. Math. Comput. Model. 32: 1297–1304 CrossRefGoogle Scholar
  12. Tan N.X. (2004). On the existence of solutions of quasi-variational inclusion problems. J. Optimization Theory Appl. 123: 619–638 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

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

Personalised recommendations