Borwein-Preiss Variational Principle Revisited

TL;DR

This paper refines and strengthens the metric space version of the Borwein-Preiss variational principle, clarifies assumptions, streamlines proofs, and extends results to Banach spaces, improving upon previous variational principles.

Contribution

It provides a refined, strengthened metric space variational principle with clearer assumptions and streamlined proofs, extending classical results to broader settings.

Findings

Refined metric space variational principle established

Extended and strengthened results in Banach spaces

Streamlined proofs and clarified assumptions

Abstract

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.