A Survey of Ekeland's Variational Principle and Related Theorems and Applications

Abstract

A Study of Ekeland’s Variational Principle and Related Theorems and Applications by Jessica Robinson ???David Costa ???, Examination Committee Chair Professor of Mathematical Sciences University of Nevada, Las Vegas Ekeland’s Variational Principle has been a key result used in various areas of analysis such as fixed point analysis, optimization, and optimal control theory. In this paper, the application of Ekeland’s Variational Principle to Caristi’s Fixed Point Theorem, Clarke’s Fixed Point Theorem, and Takahashi’s Minimization theorem is the focus. In addition, Ekeland produced a version of the classical Pontryagin Minimum Principle where his variational principle can be applied. A further look at this proof and discussion of his approach will be contrasted with the classical method of Pontryagin. With an understanding of how Ekeland’s Variational Princple is used in these settings, I am motivated to explore a multi-valued version of the principle and investigate its equivalence with a multi-valued version of Caristi’s Fixed Point Theorem and Takahashi’s Minimization theorem. iii