In this article, we refine and slightly strengthen the metric space version of the Borwein–Preiss variational principle due to Li, Shi, J. Math. Anal. Appl. 246, 308–319 (2000), clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein, Zhu, Techniques of Variational Analysis, Springer (2005) and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein, Preiss, Trans. Amer. Math. Soc. 303, 517-527 (1987) along several directions.
J. Math. Anal. Appl. 435 (2016) 1183–1193. http://dx.doi.org/10.1016/j.jmaa.2015.11.009