0
votes
2answers
53 views

$|f(x)-f(y)|<|x-y|$ on a non-empty closed bounded set of real numbers

Let $A$ be a non-empty closed bounded set of real numbers and $f: A \to A $ be a function such that $|f(x)-f(y)|<|x-y| , \forall x,y\in A$ , then how to show that $f$ has a unique fixed point ?
2
votes
0answers
43 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
1
vote
1answer
42 views

Conditions yielding a unique fixed point of a continuous differentiable function.

Let $f$ be a function defined on $[0,1]$ which is continuous for each point in $[0,1]$ and differentiable for each point in $(0,1)$. Suppose that $f^\prime (x) \neq 1$ for every $x \in (0,1)$. ...
1
vote
1answer
56 views

Definition of upper hemicontinuity of a correspondence.

When using and examining Kakutani's fixed-point theorem, I've got a question about upper hemicontinuity. A correspondence $f:X\rightarrow2^Y$ is a point-to-set mapping. One way to define upper ...
1
vote
1answer
101 views

Fixed point in plane transformation.

Some one give me a idea to solve this one. It's a problem from Vladimir Zorich mathematical analysis I. Problem 9.c from 1.3.5: A point $p \in X$ is a fixed point of a mapping $f:X \to X$ if ...
0
votes
1answer
67 views

Fixed point and non-fixed point function

For constructing another proof I need two functions explicitly and therefore I was wondering whether there exists a function that has nowhere a fixed point and a function that (maybe depending on the ...
0
votes
1answer
49 views

Numerical Analysis, build a contractive function

I have a question regarding Numerical Analysis. I've never been asked these sorts of questions before and don't even know where to begin. The goal of this exercise is to find a value alpha such that: ...
0
votes
1answer
82 views

A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
0
votes
2answers
60 views

A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer

Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
3
votes
1answer
337 views

Application of Banach fixed-point theorem

I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$. The first thing I do is to define a function ...
1
vote
1answer
45 views

Name a stable output of a function taking 2 arguments

$\mathbb{C}$ is a fixed finite set, a fair chaotic sequence $(c_n \in \mathbb{C})$ is defined such that $\forall c \in \mathbb{C}, \exists n_0 \in \mathbb{N}, n > n_0 \wedge c_n = c$. That means ...
1
vote
2answers
120 views

Show that a function $f(x)$ maps to a set of points.Fixed point theorem

Show that the function $f(x)=\frac{1+x^2}{2}$ maps the set of points $0\leqslant x\leqslant 1$ into itself and has a fixed point in that interval even though there does not exists a positive ...
5
votes
3answers
682 views

Continuous function on unit circle has fixed point

The question I have is: Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.
42
votes
1answer
1k views

What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...