Unit-I

Contraction Principle, and its variants and applications.

Unit-II

Fixed points of non-expansive maps and set valued maps, Brouwer -Schauder fixed point theorems.

Unit-III

Ky Fan Best Approximation Theorem, Principle and Applications of KKM -maps, their variants and applications.

Unit-IV

Fixed Point Theorems in partially ordered spaces and other abstract spaces.

Unit-V

Application of fixed point theory to Game theory and Mathematical Economics.

Course Outcomes:
CO-1: Understand and apply the concepts of fixed point theorems to prove the existence and uniqueness of solution to certain ordinary differential equations.
CO-2: To understand the existence and uniqueness of fixed point for non expansive and set valued mappings.
CO-3: To understand the existence of best approximation point for non expansive mapping and its applications.
CO-4: To understand the existence and uniqueness of fixed point for partially ordered metric space.
As an application, to prove the existence and uniqueness of solution for a periodic boundary value problem.
CO-5: Applying the fixed point theorems of multivalued mappings to demonstrate the conditions for existence of Nash equilibria in strategic games.

Textbooks/ References

1. M.A.Khamsi and W.A.Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Wiley – Inter Sci. (2001).
2. Sankatha Singh, Bruce Watson and Pramila Srivastava, Fixed Point Theory and Best Approximation: The KKM – map Principle, Kluwer Academic Publishers, 1997.
3. Kim C. Border, Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, 1985.