Elements of Functional Analysis and Applications
Ph.D. Program in Mathematics
University of Messina
10 hours
This doctoral course focuses on Ekeland's Variational Principle and some of its applications to nonlinear analysis, with particular emphasis on Caristi's Fixed Point Theorem, Gâteaux differentiable functionals on Banach spaces, and a complement to Ekeland's principle.
Course Contents
1. Ekeland's Variational Principle
Review of the theory of complete metric spaces. Lower semicontinuous functions and their characterizations. Ekeland's Variational Principle: statement and proof. Geometric interpretation of Ekeland's Variational Principle. First applications to nonlinear analysis. Caristi's Fixed Point Theorem.
2. Gâteaux Differentiable Functionals on Banach Spaces
Gâteaux differentiable functionals defined on Banach spaces. Gâteaux derivative and its fundamental properties. Necessary conditions for minima and existence results. Applications of Ekeland's Variational Principle to Gâteaux differentiable functionals.
3. A Complement to Ekeland's Variational Principle
A complement to Ekeland's Variational Principle. Applications to Caristi's Fixed Point Theorem and to Gâteaux differentiable functionals. Extensions and possible developments.
References
- I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, SIAM, 1987.
- D. G. de Figueiredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research, Bombay, 1989.
- F. Cammaroto, A. Chinnì, A Complement to Ekeland's Variational Principle in Banach Spaces, Bulletin of the Polish Academy of Sciences. Mathematics 44 (1), 29–33, 1996.