Equivalent formulations of Ekeland's variational principle

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 inf_Xf < f(x) < inf_Xf + 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.