Some easy to remember abstract forms of Ekeland's variational principle and Caristi's fixed point theorem

Abstract

If X is a set, then an extended real-valued function Φ of X2 is called an ecart on X. Moreover, if d and F are ecarts on X, then for any x, y Є X we write x ≤ y if and only if d(x, y)≤ Φ(x, y). Thus, we call the pair (d, Φ) admissible at a point a Є X if X has a maximal element b with a ≤ b. Here, an important particular case is when d is a certain metric on X and Φ is of the form Φ(x, y)=φ(x)-ψ(y). These definitions allow us to easily state and prove some easily remembered abstract forms of Ekeland's variational principle and Caristi's fixed point theorem. For instance, we show that if F is a relation on X and a Є X such that there exists a pair (d,Φ) of ecarts on X which is admissible at a and satisfies d(x,y)≤ Φ(x,y) for all x Є X and y Є F(x), then there exists b Є X, with d(a,b)≤ Φ(a,b), such that F(b)
{ b }.