Abstract

This paper is related to the works [\textit{C.-K. Zhong}, “A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P. S. condition and coercivity”, Nonlinear Anal., Theory Methods Appl. 29, No. 12, 1421–1431 (1997; Zbl 0912.49021)] and [\textit{J. Zhu, C.-K. Zhong} and \textit{Y. J. Cho}, “Generalized variational principle and vector optimization”, J. Optimization Theory Appl. 106, No. 1, 201–217 (2000; Zbl 0971.90080)]. The main purpose of the present paper is to prove that the generalizations of the Ekeland variational principle stated in the above mentioned works can be deduced from the original Ekeland variational principle. The proofs are based on elementary properties in complete metric spaces.