The Fenchel duality formula, The Ekeland variational principle, and Rockafellar's Theorem for maximal monotonicity of subdifferentials

Abstract

In this paper, we first give two versions of the Fenchel duality formula and Brøndsted-Rockafellar's theorem. The versions are useful and the proofs are not so difficult. Next, we present a theorem in a Banach space to reconsider the Ekeland variational principle and Takahashi's nonconvex minimization theorem proved in a complete metric space. Finally, using the version of the Fenchel duality formula, the Ekeland variational principle and the version of Brøndsted-Rockafellar's theorem, we prove a minimization theorem in a Banach space and then present a short proof of well–known Rockafellar's theorem for maximal monotonicity of subdifferentials.

Keywords:

• Rockafellar's theorem for maximal monotonicity of subdifferentials
• the Fenchel duality formula
• the Ekeland variational principle
• Takahashi's nonconvex minimization thorem
• minimization problems

2010 Mathematics Subject Classifications:

• 47H14
• 47H07
• 47H11

Acknowledgments

The authors would like to express their sincere gratitude to all who have helped to publish this paper. Sad to say, the first author, Prof. Takahashi, passed away after submitting this paper. The second author thinks that Prof. Takahashi rejoices the publication of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).