Applications of Mathematics

, Volume 58, Issue 3, pp 269–278

An intersection theorem for set-valued mappings



Given a nonempty convex set X in a locally convex Hausdorff topological vector space, a nonempty set Y and two set-valued mappings T: XX, S: YX we prove that under suitable conditions one can find an xX which is simultaneously a fixed point for T and a common point for the family of values of S. Applying our intersection theorem, we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.


intersection theorem fixed point saddle point equilibrium problem complementarity problem 

MSC 2010

47H10 49J53 


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  1. [1]
    R.P. Agarwal, M. Balaj, D. O’Regan: Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces. Appl. Anal. 88 (2009), 1691–1699.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    C.D. Aliprantis, K.C. Border: Infinite Dimensional Analysis. A Hitchhiker’s Guide. 3rd ed. Springer, Berlin, 2006.MATHGoogle Scholar
  3. [3]
    Q.H. Ansari, A.P. Farajzadeh, S. Schaible: Existence of solutions of vector variational inequalities and vector complementarity problems. J. Glob. Optim. 45 (2009), 297–307.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Q.H. Ansari, J.C. Yao: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett. 12 (1999), 53–56.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    M. Balaj: An intersection theorem with applications in minimax theory and equilibrium problem. J. Math. Anal. Appl. 336 (2007), 363–371.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    M. Balaj: A fixed point-equilibrium theorem with applications. Bull. Belg. Math. Soc. —Simon Stevin 17 (2010), 919–928.MathSciNetMATHGoogle Scholar
  7. [7]
    M. Balaj, D. O’Regan: Inclusion and intersection theorems with applications in equilibrium theory in G-convex spaces. J. Korean Math. Soc. 47 (2010), 1017–1029.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    K. Fan: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142 (1961), 305–310.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    A.P. Farajzadeh, M.A. Noor, S. Zainab: Mixed quasi complementarity problems in topological vector spaces. J. Glob. Optim. 45 (2009), 229–235.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    C. J. Himmelberg: Fixed points of compact multifunctions. J. Math. Anal. Appl. 38 (1972), 205–207.MathSciNetCrossRefGoogle Scholar
  11. [11]
    V. Jeyakumar, W. Oettli, M. Natividad: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179 (1993), 537–546.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    S.A. Khan: Generalized vector implicit quasi complementarity problems. J. Glob. Optim. 49 (2011), 695–705.MATHCrossRefGoogle Scholar
  13. [13]
    P.Q. Khanh, N.H. Quan: Intersection theorems, coincidence theorems and maximal-element theorems in GFC-spaces. Optimization 59 (2010), 115–124.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    G. Köthe: Topological Vector Spaces I. Springer, Berlin, 1969.MATHCrossRefGoogle Scholar
  15. [15]
    K.Q. Lan: An intersection theorem for multivalued maps and applications. Comput. Math. Appl. 48 (2004), 725–729.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    B. S. Lee, A.P. Farajzadeh: Generalized vector implicit complementarity problems with corresponding variational inequality problems. Appl. Math. Lett. 21 (2008), 1095–1100.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    H. Lu, D. Tang: An intersection theorem in L-convex spaces with applications. J. Math. Anal. Appl. 312 (2005), 343–356.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
  • Mircea Balaj
    • 2
  • Donal O’Regan
    • 3
  1. 1.Department of MathematicsTexas A&M UniversityKingsvilleUSA
  2. 2.Department of MathematicsUniversity of OradeaOradeaRomania
  3. 3.Department of MathematicsNational University of IrelandGalwayIreland

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