Skip to main content
Log in

Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems

  • Original Paper
  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder’s fixed point theorem are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliprantis C.D., Border K.C. (1999). Infinite Dimensional Analysis. Springer Verlag, Berlin, Germany

    Google Scholar 

  2. Ansari Q.H., Lin L.J., Su L.B. (2005). Systems of simultaneous generalized vector quasiequilibrium problems and applications. J. Optim. Theory Appl. 127: 27–44

    Article  Google Scholar 

  3. Aubin J.P., Cellina A. (1994). Differential Inclusion. Springer Verlag, Berlin, Germany

    Google Scholar 

  4. Blum E., Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Students 63: 123–145

    Google Scholar 

  5. Caristi J. (1976). Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215: 241–251

    Article  Google Scholar 

  6. Chen M.P., Lin L.J., Park S. (2003). Remarks on generalized quasi-equilibrium problems. Nonlinear Anal. 52: 433–444

    Article  Google Scholar 

  7. Chen G.Y., Huang X.X., Yang X.Q. (2005). Vector Optimization. Springer-Verlag Berlin Heidelberg,, Germany

    Google Scholar 

  8. Dancš S., Hegedüs M., Medvegyev P. (1983). A general ordering and fixed point principle in complete metric spaces. Acta Sci. Math. 46: 381–388

    Google Scholar 

  9. Deguire P., Tan K.K., Yuan G.X.Z. (1999). The study of maximal elements, fixed point for Ls-majorized mappings and the quasi-variational inequalities in product spaces. Nonlinear Anal. 37: 933–951

    Article  Google Scholar 

  10. Ekeland I. (1972). Remarques sur les problémes variationnels. I, C. R. Acad. Sci. Paris Sér. A-B. 275: 1057–1059

    Google Scholar 

  11. Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324–353

    Article  Google Scholar 

  12. Ekeland I. (1979). Nonconvex minimization problems. Bull. Amer. Math. Soc. 1: 443–474

    Article  Google Scholar 

  13. Fu J.Y., Wan A.H. (2002). Generalized vector equilibria problems with set-valued mappings. Math. Meth. Oper. Res. 56: 259–268

    Article  Google Scholar 

  14. Göpfert A., Tammer Chr., Zălinescu C. (2000). On the vectorial Ekeland’s variational principle and minimal points in product Spaces. Nonlinear Anal. 39: 909–922

    Article  Google Scholar 

  15. Hamel A. (1994). Remarks to an equivalent formulation of Ekeland’s variational principle. Optimization 31(3): 233–238

    Article  Google Scholar 

  16. Hamel A. (2003). Phelps’ lemma, Dancš’ drop theorem and Ekeland’s principle in locally convex spaces. Pro. Amer. Math. Soc. 131: 3025–3038

    Article  Google Scholar 

  17. Hamel A., Löhne A.: A minimal point theorem in uniform spaces, In Agarwal, R.P., O’Regan, D. (eds) Nonlinear Analysis and Applications to V. Lakshmikantham on his 80th Birthday, vol. 1, pp. 577–593. Kluwer Academic Publisher (2003)

  18. Hyers D.H., Isac G., Rassias T.M. (1997). Topics in Nonlinear Analysis and Applications. World Scientific, Singapore

    Google Scholar 

  19. Kada O., Suzuki T., Takahashi W. (1996). Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jan. 44: 381–391

    Google Scholar 

  20. Lin L.J., Ansari Q.H. (2004). Collective fixed points and maximal elements with applications to abstract economies. J. Math. Anal. Appl. 296: 455–472

    Article  Google Scholar 

  21. Lin L.J., Du W.S. (2006). Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 323: 360–370

    Article  Google Scholar 

  22. Lin L.J., Still G. (2006). Mathematical programs with equilibrium constraints: The existence of feasible points. Optimization 55(3): 205–219

    Article  Google Scholar 

  23. Lin, L.J.: Existence theorems for bilevel problems with applications to mathematical programs with equilibrium constraints and semi-infinite problems J. Optim. Theory and Appl. (to appear)

  24. Lin L.J., Yu Z.T., Kassay G. (2002). Existence of equilibria for multivalued mappings and its application to vectorial equilibria. J. Optim. Theory Appl. 114: 189–208

    Article  Google Scholar 

  25. Lin L.J., Yu Z.T. (2001). On some equilibrium problems for multimaps. J. Computional Appl. Math. 129: 171–183

    Article  Google Scholar 

  26. Luc D.T. (1989). Theory of Vector Optimization, Vol. 319. Lecture notes in economics and mathematical systems. Springer Verlag, Berlin, Germany

    Google Scholar 

  27. McLinden L. (1982). An application of Ekeland’s theorem to minimax problems. Nonlinear Anal. 6(2): 189–196

    Article  Google Scholar 

  28. Oettli W., Théra M. (1993). Equivalents of Ekeland’s principle. Bull. Austral. Math. Soc. 48: 385–392

    Article  Google Scholar 

  29. Park S. (2000). On generalizations of the Ekeland-type variational principles. Nonlinear Anal. 39: 881–889

    Article  Google Scholar 

  30. Penot J.-P. (1986). The drop theorem, the petal theorem and Ekeland’s variational principle. Nonlinear Anal. 10(9): 813–822

    Article  Google Scholar 

  31. Stein O., Still G. (2002). On generalized semi-infinite optimization and bilevel optimization. Europ. J. Operat Res 142(3): 442–462

    Google Scholar 

  32. Suzuki T. (2001). Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253: 440–458

    Article  Google Scholar 

  33. Takahashi W. (2000). Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan

    Google Scholar 

  34. Tammer Chr. (1992). A generalization of Ekeland’s variational principle. Optimization 25: 129–141

    Article  Google Scholar 

  35. Tan N.X. (1985). Quasi-variational inequalities in topological linear locally convex Hausdorff spaces. Mathematicsche Nachrichten 122: 231–245

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lai-Jiu Lin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, LJ., Du, WS. Systems of equilibrium problems with applications to new variants of Ekeland’s variational principle, fixed point theorems and parametric optimization problems. J Glob Optim 40, 663–677 (2008). https://doi.org/10.1007/s10898-007-9146-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-007-9146-0

Keywords

Navigation