Ekeland’s variational principle and its extensions with applications

In 1972, Ekeland [35] (see also, [36, 37]) established a theorem on the existence of an approximate minimizer of a bounded below and lower semicontinuous function. This theorem is known as Ekeland’s variational principle (in short, EVP). It is one of the most applicable results from nonlinear analysis and used as a tool to study the problems from fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, etc; see, for example, [7–9, 19, 20, 34–38, 46, 55, 60, 67, 72, 85] and the references therein. Later, it was found that several well-known results, namely, Caristi–Kirk fixed point theorem [24, 25], Takahashi’s minimization theorem [84], the Petal theorem [72], and the Daneš drop theorem [32] from nonlinear analysis are equivalent to the Ekeland’s variational principle.