Equivalent formulations of Ekelandʹs variational principle Original Research Article

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 View the MathML source 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.