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.