~~DISCUSSION~~

# MTH604: Fixed Point Theory and Applications (Spring 2020)

## 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.

## Sample questions

- State and prove intermediate value theorem.
- State and prove the fixed point theorem.
- Define attracting, repelling and neutral fixed point theorem.
- Consider a function $f(x)=x^2-1$. Find its fixed points and also find the nature of fixed point.
- Define orbit of the point $x_0$ under function $f$.
- Consider $C(x)=\cos (x)$. Draw the cobweb representation of the orbit of $-2$ under $C$.
- Consider $C(x)=\cos (x)$. Draw the cobweb representation of the orbit of $0$ under $C$.
- Let $L(x)=-1.4x$. Draw the cobweb representation of the orbit of $0.1$ under $L$.
- Consider a function $L(x)=mx$, where $m\in \mathbb{R}$. Find the value of $m$ for which $x=0$ is attracting fixed point.
- Let $F(x)=2(\sin x)^2$. Find the interval in which $F$ has attracting fixed point.
- State the mean value theorem.
- State and prove attracting fixed point theorem.
- State and prove repelling fixed point theorem.
- Find the fixed point of $f(x)=2x-2x^2$ by iteration method by taking initial guess $x_0=0.1$.
- Define metric spaces, open ball, closed ball, sphere, open set, limit point of a set, closure of a set, dense set, countable set, separable space, neighbourbood of a point, interior point, continuity, convergence of sequence, Cauchy sequence, subsequence, complete space, subspace, complete space, bounded metric, diameter of the metric.
- Let $B(x;r)$ represents open ball with centre $x$ and radius $r$ in some metric space. Find $B\left(0;5 \right)$ in discrete metric on $\mathbb{R}$.
- Let $B(x;r)$ represents open ball with centre $x$ and radius $r$ in some metric space. Find $B\left(1;0.7 \right)$ in discrete metric on $\mathbb{R}$.
- Let $B(x;r)$ represents open ball with centre $x$ and radius $r$ in some metric space. Find $B\left(2;5 \right)$ in usual metric on $\mathbb{R}$.
- Define Lipschtiz continuous.
- Define contraction map.
- Define contractive map.
- Define nonexpansive map.
- Prove that a contraction map is contractive but converse is not true.
- Prove that every Lipschtiz continuous is continuous.
- Prove that every contractive map is continuous.
- Prove that every contraction is contrinuous.
- State and prove Banach contraction principle.
- Let $(X,d)$ be a complete metric space and $T:X\to X$ be a mapping such that for some integer $m$, $T^m=\underbrace {T \circ T \circ ... \circ T}_{m\,times}$ is contraction mapping. Then $T$ has a unique fixed point.
- Define cover and open cover of metric space.
- Define sub cover and finite subcover in metric space.
- Define compact metric space.
- State Heine-Borel theorem.
- State extreme value theorem.
- Let $(X,d)$ be a compact metric space and $T:X\to X$ be a contractive mapping. Then prove that $T$ has unique fixed point.
- Let $X=[a,b]$ be a metric space with the usual metric and $T:[a,b]\to[a,b]$ be a differentiable function. Then $T$ is a contractive mapping on $X$ if and only if there exists a real number $\alpha <1$ such that $|T'(x)|\leq \alpha$ for all $x\in(a,b)$.
- Let $X=[1,\infty)$ be a metric space with the usual metric and $T:X\to X$ be a mapping defined by $T(x)=\frac{10}{11}\left(x+\frac{1}{x} \right)$ for all $x\in X$. Prove that $T$ is a contraction mapping with Lipschitz constant $\alpha=\frac{10}{11}$.
- Let $X=[0,1]$ be a metric space with usual metric and $T:X\to X$ be a mapping defined by $T(x)=\frac{1}{7}(x^3+x^2+1)$ for all $x\in X$. Prove that $T$ is contraction mapping with Lipschtiz constant $\alpha=\frac{5}{7}$.
- Let $X=\{x\in\mathbb{Q}:x\geq 1\}$ be a metric space with usual metric and $T:X\to X$ be a mapping defined by $T(x)=\frac{x}{2}+\frac{1}{x}$ for all $x\in X$. Prove that $T$ is a contraction mapping with Lipschtiz constant $\alpha=\frac{1}{2}$.
- Let $X=[0,\infty)$ be a usual metric space and $T:X\to X$ be a mapping defined as $T(x)=\frac{1}{1+x^2}$, $x\in[0,\infty)$. Then prove that $T$ is contractive map.
- Let $C$ be a nonempty, closed, convex subset of a normed linear space $E$ with $F:C\to C$ nonexpansive and $F(C)$ a subset of a compact set of $C$. Then $F$ has a fixed point.
- State and prove Schauder theorem for non-expansive map.
- Define convex set.
- Let $(X,d)$ be a complete metric space and let $B(x_0,r)$ be open ball with center $x_0\in X$ and $r>0$. Suppose $F: B(x_0,r)\to X$ is a contraction with Lipschitz constant $L$ and also $d(F(x_0),x_0)<(1-L)r$. Then $F$ has a unique fixed point in $B(x_0,r)$.
- Prove that $\frac{1}{2}e^x \cos(x)$ has attracting fixed point in the interval $(-\infty,0]$.

## Resources

## Videos:

Notes and videos can be downloaded from the following university official page:

## 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.