Are several recent generalizations of Ekeland’s variational principle more general than the original principle?

We show that several recent results proposed as generalizations of Ekeland’s variational principle are in fact equivalent to the original principle. In 1974, Ekeland [2] introduced a variational principle, which appears to be one of the most important results in nonlinear analysis during the last three decades and has numerous applications. Recently several generalizations of this principle were proposed (see [11, 12]). The purpose of the present note is to show that these seemingly more general results can be derived from the original Ekeland variational principle. Let us begin with the original principle. Theorem 1 [2]. Let (X, d) be a complete metric space, f : X → R ∪ {+∞} be a lower semicontinuous functional, not identically +∞ and bounded from below. Then, for every e > 0, λ > 0 and x0 ∈ X such that f(x0) < inf x∈X f(x) + e, ————– 1991 Mathematics Subject Classification. 49J40, 46N10, 90C99.