Skip to main content
Log in

On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings

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

Abstract

We propose a very weak type of generalized distances called a weak τ-function and use it to weaken the assumptions about lower semicontinuity in existing versions of Ekeland’s variational principle and equivalent formulations.

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. Arutyunov A., Bobylev N., Korovin S.: One remark to Ekeland’s variational principle. Comp. Math. Appl. 34, 267–271 (1997)

    Article  Google Scholar 

  2. Attouch H., Riahi H.: Stability result for Ekeland’s ε-variational principle and cone extremal solutions. Math. Oper. Res. 18, 173–201 (1993)

    Article  Google Scholar 

  3. Bao T.Q., Khanh P.Q.: Are several recent generalizations of Ekeland’s variational principle more general than the original principle?. Acta Math. Vietnam. 28, 345–350 (2003)

    Google Scholar 

  4. Bao T.Q., Mordukhovich B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cyber. 36, 531–562 (2007)

    Google Scholar 

  5. Bianchi M., Kassay G., Pini R.: Ekeland’s principle for vector equilibrium problems. Nonlin. Anal. 66, 1454–1464 (2007)

    Article  Google Scholar 

  6. Borwein J.M., Preiss D.: A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. Amer. Math. Soc. 303, 517–527 (1987)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Chen Y., Cho Y.J., Yang L.: Note on the results with lower semicontinuity. Bull. Korean Math. Soc. 39, 535–541 (2002)

    Article  Google Scholar 

  9. Daneš J.A.: A geometric theorem useful in nonlinear analysis. Bull. U.M.I. 6, 369–375 (1972)

    Google Scholar 

  10. Daneš S., Hegedus M., Medvegyev P.: A general ordering and fixed-point principle in completed metric space. Acta Sci. Math. (Szeged). 46, 381–388 (1983)

    Google Scholar 

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

    Article  Google Scholar 

  12. El Amrouss A.R.: Variantes du principle variationnel d’Ekeland et applications. Revista Colombiana de Matemáticas. 40, 1–14 (2006)

    Google Scholar 

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

    Article  Google Scholar 

  14. Ha T.X.D.: The Ekeland variational principle for set-valued maps involving coderivatives. J. Math. Anal. Appl. 286, 509–523 (2003)

    Article  Google Scholar 

  15. Ha T.X.D.: Some variants of Ekeland variational principle for a set-valued map. J. Optim. Theory Appl. 124, 187–206 (2005)

    Article  Google Scholar 

  16. Ha T.X.D.: Variants of the Ekeland variational principle for a set-valued map involving the Clarke normal cone. J. Math. Anal. Appl. 316, 346–356 (2006)

    Article  Google Scholar 

  17. Hamel A.H.: Phelp’s lemma, Danes’s drop theorem and Ekeland’s principle in locally convex topological vector spaces. Proc. Amer. Math. Soc. 10, 3025–3038 (2003)

    Article  Google Scholar 

  18. Hamel A.H.: Equivalents to Ekeland’s variational principle in uniform spaces. Nonlin. Anal. 62, 913–924 (2005)

    Article  Google Scholar 

  19. Hamel A.H., Löhne A.: Minimal element theorems and Ekeland’s principle with set relations. J. Nonlin. Convex Anal. 7, 19–37 (2006)

    Google Scholar 

  20. Huang X.X.: New stability results for Ekeland’s ε-variational principle for vector-valued and set-valued maps. J. Math. Anal. Appl. 262, 12–23 (2001)

    Article  Google Scholar 

  21. Huang X.X.: Stability results for Ekeland’s ε-variational principle for set-valued mappings. Optim. 51, 31–45 (2002)

    Article  Google Scholar 

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

    Google Scholar 

  23. Khanh P.Q.: On Caristi-Kirk’s Theorem and Ekeland’s variational principle for Pareto extrema. Bull. Polish Acad. Sci. Math. 37, 1–6 (1989)

    Google Scholar 

  24. Khanh, P.Q., Quy, D.N.: On Ekeland’s variational principle for Pareto minima of set-valued mappings. J. Optim. Theory Appl., to appear (2010)

  25. Khanh, P.Q., Quy, D.N.: A generalized distance and Ekeland’s variational principle for vector functions. Nonlinear Anal. Onlinefirst (2010)

  26. Kuroiwa D.: On set-valued optimization. Nonlinear Anal. 47, 1395–1400 (2001)

    Article  Google Scholar 

  27. Li Y., Shi S.: A generalization of Ekeland’s \({\varepsilon}\) -variational principle and its Borwein-Preiss smooth variant. J. Math. Anal. Appl. 246, 308–319 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Lin L.J., Du W.S.: Some equivalent formulations of generalized Ekeland’s variational principle and their applications. Nonlin. Anal. 67, 187–199 (2007)

    Article  Google Scholar 

  30. Loridan P.: \({\varepsilon}\) -Solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)

    Article  Google Scholar 

  31. Oetli W., Théra M.: Equivalents of Ekeland’s principle. Bull. Austr. Math. Soc. 48, 385–392 (1993)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  34. Phelps R.R.: Support cones in Banach spaces and their applications. Adv. Math. 13, 1–19 (1974)

    Article  Google Scholar 

  35. Qui J.H.: Local completeness, drop theorem and Ekeland’s variational principle. J. Math. Anal. Appl. 311, 23–39 (2005)

    Article  Google Scholar 

  36. Suzuki T.: Generalized distance and existence theorems in the complete metric space. J. Math. Anal. Appl. 253, 440–458 (2001)

    Article  Google Scholar 

  37. Suzuki T.: Generalized Caristi’s fixed point theorems by Bae and others. J. Math. Anal. Appl. 302, 502–508 (2005)

    Article  Google Scholar 

  38. Takahashi W.: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Théra, M.A., Baillon, J.B. (eds) Fixed Point Theory and Applications, pp. 397–406. Longman Scientific and Technical, Essex (1991)

    Google Scholar 

  39. Tataru D.: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 163, 345–392 (1992)

    Article  Google Scholar 

  40. Valyi I.: A general maximality principle and a fixed-point theorem in uniform spaces. Periodica Math. Hung. 16, 127–134 (1985)

    Article  Google Scholar 

  41. Zabreiko P.P., Krasnoselski M.A.: Solvability of nonlinear operator equations. Functional Anal. Appl. 5, 206–208 (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. Q. Khanh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khanh, P.Q., Quy, D.N. On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings. J Glob Optim 49, 381–396 (2011). https://doi.org/10.1007/s10898-010-9565-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-010-9565-1

Keywords

Navigation