# MTH604: Fixed Point Theory and Applications

## Course Objectives:

This course is intended as a brief introduction to the subject with a focus on Banach Fixed Point theorems fixed point theorem and its application to nonlinear differential equations, nonlinear integral equations, real and complex implicit functions theorems and system of nonlinear equations. Some generalizations and similar results e. g. Kannan Fixed Point theorems, Banach Fixed Point theorem for multi-valued mappings are also educated.

## Course Contents:

Basic concepts: metric spaces, complete metric spaces, Vector spaces (linear spaces) normed spaces, Banach spaces, Banachs contraction principle, non-expansive mappings and related fixed point theorems. Contractive maps, properties of fixed points set and minimal sets. Multi-valued mappings and related fixed point theorems. Best approximation theorems.

## Notes Handout

## Sample Questions

- Define the followings:

Metric spaces, open ball, closed ball, sphere, open set, limit point of a set, closure of the set, dense set, countable set, separable space, neighbourhood of a point, interior point, continuity, convergence of sequences, Cauchy sequence, sub-sequence, complete space, subspace, nested sequences, compact space, vector space, norm space. - Examples related to above notions.
- Let $d$ be usual metric on $\mathbb{R}$. Then find open ball, closed ball and sphere with radius $\frac{1}{2}$ and center $1$.
- Define fixed point with example.
- Find fixed point of the function $f(x)=x^2-3x+4$.
- Define Lipschitzian.
- Define Contraction and give its example.
- Define non-expensive.
- State and prove Banach's contraction principle.
- Let $(X,d)$ be a compact metric space with $F:X\to X$ satisfying $d(F(x),F(y)<d(x,y)$ for $x,y \in X$ and $x\neq y$. Then $F$ has a unique fixed point in $X$.
- Let $(X,d)$ be a complete metric space and let $B(x_0, r) = \{x \in X : d(x, x_0) < r\}$, where $x_0 \in X$ and $r>0$. Suppose $F:B(x_0,r) \to X$ is a contraction (that is, $d(F(x), F(y)) \leq L d(x, y)$ for all $x, y \in B(x_0, r)$ with $0<L<1$) with $d(F(x_0,x_0)<(1-L)r$. Then $F$ has a unique fixed point in $B(x_0,r)$.
- Let $\overline{B_r}$ be the closed ball of radius $r>0$, centred at zero, in a Banach space $E$ with $F : \overline{Br} \to E$ a contraction and $F(\partial \overline{B_r}) \subseteq \overline{B_r}$. Then $F$ has a unique fixed point in $\overline{B_r}$.
- Continuity on open and closed interval.
- State intermediate value theorem.
- Give an example of a function which don't satisfy intermediate value theorem.
- State and prove the fixed point theorem.
- What are attracting, repelling and neutral fixed point.
- Find the nature of the fixed point of the function $F(x)=\cos x$ in the interval $[0,\frac{\pi}{2}]$.
- Find fixed point of the function $F(x)=x-x^2$ and determine its nature.
- State mean value theorem and give its geometric interpretation.
- State and prove attracting fixed point theorem.
- State and prove repelling fixed point theorem.

## Assignments

## Other Resources

## Books:

- B. Bollobas: W. Fulton, A. Katok, F. Kirwan and P. Sarnak: Fixed Point Theory and Applications, Cambridge University Press, 2001.
- K. Geoble and W.A Kirk: Topics in Metric Fixed Theory, Cambridge university Press, 1990
- M.C. Joshi And R.K. Bose: Some Topics in Nonlinear Functional Analysis, John Wiley and Sons, 1985
- James Dugundji and A. Granas: Fixed Point Theory, Vol. 1, Polish Scientific Publishers, 1982
- W.A. Kirk and B. Sims: Handbook of Metric Fixed Point Theory, Klawer Academic Publishers 2001
- M. Aslam Noor, Principles of Variational Inequalities Lapt-Lambert Academic Publishing AG & Co. Saarbrucken, Germany 2009.