Journal of Optimization Theory and Applications

, Volume 179, Issue 3, pp 761–777 | Cite as

Intersection Theorems with Applications in Optimization

  • Ravi P. Agarwal
  • Mircea Balaj
  • Donal O’Regan


In this paper, we establish two intersection theorems which are useful in considering some optimization problems (complementarity problems, variational inequalities, minimax inequalities, saddle point problems).


Weak KKM map Variational inequality Minimax inequality 

Mathematics Subject Classification

47H10 49J53 


  1. 1.
    Fan, K.: A generalization of Tychonoff s fixed point theorem. Math. Ann. 142, 305–310 (1961)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Park, S.: Some coincidence theorems on acyclic multifunctions and applications to KKM theory. In: Fixed Point Theory and Applications, pp. 248–277. World Scientific Publications, River Edge, NJ (1992)Google Scholar
  3. 3.
    Chang, T.H., Yen, C.L.: KKM property and fixed point theorems. J. Math. Anal. Appl. 203, 224–235 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Balaj, M.: Weakly \(G\)-KKM mappings, \(G\)-KKM property, and minimax inequalities. J. Math. Anal. Appl. 294, 237–245 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide. Springer, Berlin (2006)zbMATHGoogle Scholar
  6. 6.
    Agarwal, R.P., Balaj, M., O’Regan, D.: Common fixed point theorems in topological vector spaces via intersection theorems. J. Optim. Theory Appl. 173, 443–458 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Isac, G.: Topological Methods in Complementarity Theory. Nonconvex Optimization and its Applications, vol. 41. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  8. 8.
    Isac, G., Kalashnikov, V.V.: Exceptional family of elements, Leray–Schauder alternative, pseudomonotone operators and complementarity. J. Optim. Theory Appl. 109, 69–83 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Peng, J., Yang, X.: On multivalued complementarity problems in Banach spaces. J. Math. Anal. Appl. 307, 245–254 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Khan, S.A.: Generalized vector implicit quasi complementarity problems. J. Glob. Optim. 49, 695–705 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fu, J., Wang, S.: Generalized strong vector quasi-equilibrium problem with domination structure. J. Glob. Optim. 55, 839–847 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Agarwal, R.P., Balaj, M., O’Regan, D.: An intersection theorem for set-valued mappings. Appl. Math. 58, 269–278 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Browder, F.: The fixed point theory of multi-valued mappings in topological vector spaces. Math. Ann. 177, 283–301 (1968)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Klee, V.: External structure of convex sets. Math. Z. 69, 90–104 (1958)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, F.C.: On a form of KKM principle and Sup Inf Sup inequalities of von Neumann and of Ky Fan type. J. Math. Anal. Appl. 155, 420–436 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fan, K.: A minimax inequality and applications. Inequalities III, 103–113 (1972)MathSciNetGoogle Scholar
  17. 17.
    Sion, M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
    • 2
  • Mircea Balaj
    • 3
  • Donal O’Regan
    • 4
  1. 1.Texas A&M University-KingsvilleKingsvilleUSA
  2. 2.Florida Institute of TechnologyMelbourneUSA
  3. 3.University of OradeaOradeaRomania
  4. 4.National University of IrelandGalwayIreland

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