Tagged Questions
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
1
vote
1answer
21 views
Is there any space with normal structure but not uniform normal structure?
It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ...
1
vote
2answers
53 views
Showing a contraction without a fixed point
Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that:
\begin{equation}
|f(x)-f(y)| < |x-y|
\end{equation}
except when $x=y$, but ...
3
votes
1answer
79 views
Understanding the Banach fixed point theorem
The Banach fixed point theorem is stated in my book (Applied Asymptotic Analysis by Miller) as
Let $\mathcal B$ be a Banach space with norm $\|\cdot\|$. Let $X$ be a nonempty bounded subset of ...
1
vote
1answer
77 views
The wedge sum of two circles has fixed point property?
The wedge sum of two circles has fixed point property?
I'm trying to find a continuous map from the wedge sum to itself, that this property fails, I couldn't find it, I need help.
Thanks
1
vote
1answer
80 views
about fixed point set?
let $K$ be a closed convex subset of a normed space $V$. For any $f: K \to K$ define the fixed-point set of $f$ as follows: $fix(f)=\{x$ belongs to $K$ $|f(x)=x \}$. I have to show that a nonempty ...
0
votes
1answer
46 views
IVP- Has at most one solution
Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem
$$
\left\{
\begin{array}{l}
x'(t)=f(t,x) \\
x(t_0)=x_0
\end{array}
\right.
...
7
votes
2answers
216 views
Continuous Brouwer's fixed point theorem via Stokes's theorem?
Let $B$ denote the closed unit ball in $\mathbf{R}^n$. Brouwer's fixed point theorem states that every continuous map $f:B\to B$ has a fixed point. There is a simple proof using Stokes's theorem, at ...
1
vote
0answers
44 views
Fixed point of continuous function on compact metric space [duplicate]
Possible Duplicate:
Prove the map has a fixed point
Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ ...
5
votes
3answers
414 views
Continuous function on unit circle has fixed point
The question I have is:
Let $f: S^1 \rightarrow S^1$ be a continuous function, where $S^1$ is the unit circle. Prove that if $f$ is not onto, then $f$ must have a fixed point.
2
votes
2answers
235 views
Analogue to Fixed Point Theorem for Compact metric spaces
If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
7
votes
1answer
217 views
Minimax Theorems V.S. Fixed Point Theorems.
Is there any relationship between the minimax theorems
$$
\mbox{Regularly hypothesis on } f:U\times V\to\mathbb{R} \implies \min_{u}\max_{v}f(u,v)=\max_{v}\min_{u}f(u,v)
$$
and fixed point ...
1
vote
0answers
54 views
Fixed Point Theorem for Set-to-Set Mappings
Is there a fixed-point theorem regarding mappings of the form $T:2^S\to 2^S$, i.e. mappings that map subsets of $S$ to other subsets in $S$:
$$
T:S\supseteq A \mapsto T(A)\subseteq S
$$
Where $S$ ...
1
vote
1answer
161 views
A corollary of Banach's fixed-point theorem
Can someone give me an idea how to generalize Banach's fixed-point theorem for complete metric spaces such that the constant contraction coefficient $c$ (as in $d(Tx,Ty)\leq c \ d(x,y)$ ) may be ...
3
votes
1answer
160 views
Brouwer FPT and solutions to a system of equations
I am trying to solve the following problem:
Let f, g be continuous positive functions $\mathbb{R}^2 \to \mathbb{R}$: show that the system of equations
$$(1-x^2)f^2(x,y) = x^2 g^2(x,y)$$
...
4
votes
1answer
385 views
Continuous bijections from the open unit disc to itself - existence of fixed points
I'm wondering about the following:
Let $f:D \mapsto D$ be a continuous real-valued bijection from the open unit disc $\{(x,y): x^2 + y^2 <1\}$ to itself. Does f necessarily have a fixed point?
I ...
