For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, but I don’t even know what the second part of the problem is asking for or even what to do. Can someone […]

Let a sequence $(a_n)_{n=0}^\infty$ be defined recursively $a_{n+1} = (1-a_n)^{\frac1p}$, where $p>1$, $0<a_0<(1-a_0)^{\frac1p}$. Let $a$ be the unique real root of $a=(1-a)^{\frac1p}$, $0<a<1$. It is clear $0<a_0<(1-a_0)^{\frac1p}\Leftrightarrow 0<a_0<a$. Prove 1) $a_{2k-2}<a_{2k}<a<a_{2k+1}<a_{2k-1}$ and $a_{2k+1}-a<a-a_{2k}$. 2) $\lim\limits_{n\to\infty}a_n=a$. Define $f(x):=(1-x)^{\frac1p}$. Consider $f^2$. When $p=2$, $a_{n+2}=f^2(a_n)=\big(1-(1-a_n)^{\frac12}\big)^{\frac12}$. $a_{n+2}>a_n\Leftrightarrow (1-a_n)(1+a_n)^2>1\Leftrightarrow a_n<(1-a_n)^{\frac12}$, and the conclusion is proved. But I am having difficulty […]

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a polynomial. Define property $B$ for an entire function $f(z)$ as $1)$ $f(z)=f_1(z)$ with property $A$. $2)$ $f_i(z)= ln(f_{i-1}(z)/z)$ for every positive integer […]

In one dimension, Brouwer’s fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from the BFPT?

This question already has an answer here: Convergence of the fixed point iteration for sin(x) 1 answer

I want to know whether the Cayley plane has fixed point property or not. I think it is so but I am not able to prove this. It certainly does not admit maps of period two without fixed points. By fixed point property, I mean that any continuous self map has a fixed point.

Can somebody show that BPTT, version 2 is deduced from BPTT, version 1 [BPTT, version 1] Let $h$ be a fixed point free orientation preserving homeomorphism of $\mathbb{R}^2$ Every point $m\in\mathbb{R}^2$ belongs to a properly embedded topological line $\bigtriangleup $ which separates $h^{−1}(\bigtriangleup )$ and $h(\bigtriangleup)$. [BPTT, version 2] Let $h$ be a fixed point […]

I’m stuck with the following. I need to prove that in $D:=[0,1]\times[0,1]$ the function $F$ is contractive, where $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ is defined as: \begin{align} F(x,y):=(\frac{1}{2} e^{-x}+\frac{y}{2},\frac{1}{3} e^{-y}+\frac{x}{3}) \end{align} Actually the problem is to prove: $\exists !_{(x^*,y^*)\in D}: F(x^*,y^*)=(x^*,y^*)$ using Banach fixed point theorem. First of all $D$ must be closed and $F(D)\subset D$, and both are […]

Why does $\chi(SL_n(R))=0$? I’m going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper right corner entry. So $\det(BA)=1$, and the map is smooth. I also know $f$ has no fixed points, since $A=BA$ implies […]

I am stuck in following homework question. Let $f : \mathbb R^n \to \mathbb R^n$ be a uniform contraction and $g(x) = x – f(x)$. Investigate whether $g : \mathbb R^n \to \mathbb R^n$ is a homeomorphism or not. The definition of uniform contraction is as follows: $(X,d)$ is metric space. $f: X \to X$ […]

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