Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T11:20:37.256Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalents of Ekeland's principle

Published online by Cambridge University Press:  17 April 2009

W. Oettli  and
M. Théra
W. Oettli
Affiliation:
Universität Mannheim Lehrstuhl für Mathematik, VII Schloss 68131 Mannheim, Germany
M. Théra
Affiliation:
URA 1586, Université de Limoges, Département de Mathématiques 123 Avenue A. Thomas 87060 Limoges Cedex, France
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note we present a new result which is equivalent to the celebrated Ekeland's variational principle, and a set of implications which includes a new non-convex minimisation principle due to Takahashi.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 3 , December 1993 , pp. 385 - 392
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Brøndsted, A., ‘On a lemma of Bishop and Phelps’, Pacific J. Math. 55 (1974), 335–341.CrossRefGoogle Scholar
[2]Caristi, J., Kirk, W.A., ‘Geometric fixed point theory and inwardness conditions’, in The geometry of metric and linear spaces, Lecture Notes in Math. 490 (Springer-Verlag, Berlin, Heidelberg, New York, 1975), pp. 74–83.CrossRefGoogle Scholar
[3]Danes, J., ‘A geometric theorem useful in nonlinear functional analysis’, Boll. Un. Mat. Ital. (4) 6 (1972), 369–375.Google Scholar
[4]Dugundji, J., Granas, A., Fixed point theory 1 (PWN, Warszawa, 1982).Google Scholar
[5]Ekeland, I., ‘On the variational principle’, J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
[6]de Figueiredo, D.G., The Ekeland variational principle with applications and detours (Springer-Verlag, Berlin, Heidelberg, New York, 1989).Google Scholar
[7]Georgiev, P.G., ‘The strong Ekeland variational principle, the strong drop theorem and applications’, J. Math. Anal. Appl. 131 (1988), 1–21.CrossRefGoogle Scholar
[8]Guillerme, J., ‘On the drop theorem in the nonconvex case’, (Preprint, Université de Limoges, 1990).Google Scholar
[9]Oettli, W., ‘Approximate solutions of variational inequalities’, in Quantitative Wirtschafts-forschung, (Albach, H., Helmstädter, E. and Henn, R., Editors) (Verlag J. C. B. Mohr, Tübingen, 1977), pp. 535–538.Google Scholar
[10]Penot, J.P., ‘The drop theorem, the petal flower theorem and Ekeland's variational principle’, Nonlinear Anal. 10 (1986), 813–822.CrossRefGoogle Scholar
[11]Takahashi, W., ‘Existence theorems generalizing fixed point theorems for multivalued mappings’, in Fixed point theory and applications, (Baillon, J.-B. and Théra, M., Editors), Pitman Research Notes in Mathematics 252 (Longman, Harlow, 1991), pp. 397–406.Google Scholar
[12]Théra, M., ‘A survey on equivalent forms of Ekeland's variational principle’, presented at the Conference on Operations Research, Vienna, 1990, and the Workshop on Applied Analysis and Related Topics, Santa Barbara, 1990.Google Scholar