Course Content

Course objective: The aim of this course is to deal with fixed point theory with their applications.

Metric Contraction Principles: Banach’s Contraction Principle, Extensions of Banach’s Principle, The Caristi-Ekeland Principle, Equivalents of the Caristi-Ekeland Principle, Set-valued contractions.

Banach Spaces and Continuous Mappings: Banach Spaces, Brouwer’s Theorem, Comments on Brouwer’s Theorem, Schauder’s Theorem, Banach Algebras: Stone Weierstrass Theorem, Condensing mappings.

Fixed Point Theory: Contraction mappings, Basic theorems for non-expansive mappings, A closer look at l1, Stability results in arbitrary spaces, The Goebel-Karlovitz Lemma, Orthogonal Convexity, Structure of the fixed point set, Asymptotically regular mappings.

Applications of Fixed Point Theory: Application to System of Linear Equations, Differential Equations and Integral Equations.

Course learning outcome(CLO): The student will be able to

1) understand contraction principles such as Banach contraction principle and their extensions.

2) explain fixed point theorems for various abstract spaces.

3) classify contractive and non-expansive mappings.

4) apply fixed point theorems to many problems like system of linear equations, differential and integral equations.

Recommended Books:

1) R. P. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2004.

2) K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990.

3) V. I. Istratescu, Fixed Point Theory: An Introduction, Springer, 2001.

4) M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley & Sons, 2001.

5) E. Zeidler. Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer, 1986