Equivalent formulations of Ekeland's variational principle

Zili Wu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


We prove that for some 0 < α and 0 < ε ≤ + ∞ a proper lower semicontinuous and bounded below function f on a metric space (X,d) satisfies that for each x ∈ X with infXf < f(x) < infXf + ε there exists y ∈ X such that 0 < αd(x,y) ≤ f(x) - f(y) iff for each such x this inequality holds for some minimizer z of f. Similar conditions are shown to be sufficient for f to possess minimizers, weak sharp minima and error bounds. A fixed point theorem is also established. Moreover, these results all turn out to be equivalent to the Ekeland variational principle, the Caristi-Kirk fixed point theorem and the Takahashi theorem.

Original languageEnglish
Pages (from-to)609-615
Number of pages7
JournalNonlinear Analysis, Theory, Methods and Applications
Issue number5
Publication statusPublished - Nov 2003
Externally publishedYes


  • Ekeland's variational principle
  • Error bounds
  • Fixed point theorem
  • Weak sharp minima
  • ε-condition of Hamel
  • ε-condition of Takahashi

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