Equivalent formulations of Ekeland's variational principle

Abstract

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 X f<f(x)< inf X f+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.