0
votes
1answer
33 views

Homogeneous topological space with the fixed-point property

Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point ...
1
vote
1answer
65 views

How to use Markov-Kakutani fixed point theorem to show that abelian groups are amenable?

Recall that a group $\Gamma$ is amenable if there exists a state $\mu$ on $l^{\infty}(\Gamma)$ which is invariant under the left translation action: for all $s\in \Gamma$ and $f\in ...
7
votes
0answers
157 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
2
votes
1answer
48 views

Does Particular Point Topology has Fixed Point Property?

A Particular Point Topology is not compact, is path-connected but what about Fixed Point Property? Does it have fixed point property? If so how? I have been told that it should have Fixed Point ...
30
votes
2answers
658 views

Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
1
vote
0answers
76 views

Importance of Banach fixed point in mathematics

I know the proof of Banach fixed point theorem in complete metric space and two example of it. It is useful to prove Picard's theorem on existence and uniqueness of ordinary differential equation. In ...
0
votes
1answer
39 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
4
votes
1answer
559 views

Does a compact connected complete linear order have the fixed point property?

Would the same arguments used for showing $[0,1]$ has the fixed point property hold in this general case? What could go wrong? EDIT: The fixed point property can be interpreted in two ways that I ...
0
votes
1answer
35 views

Compactness of a subset of a specific bounded $L^2$ space

For my research, I am working with the set $$S = [0,1] \times [0,\delta] \times[0,\delta^2] \times \cdots $$ where $S\subset \mathbb{R}^\infty$. I am using the $\|\cdot\|_2$ norm. I was hoping to ...
1
vote
1answer
117 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
2
votes
1answer
99 views

Quesion on a detail of the proof of Schauder-Tychonoff fixed point theorem

I'm trying to understand the proof of Schauder-Tychonoff fixed point theorem on page $96-97$, in Fixed Point Theory and Applications, Ravi P. Agarwal,Maria Meehan,Donal O'Regan, which can be found ...
1
vote
2answers
51 views

Brouwer fixed poin in 1-D

In $\mathbb R_{+}$ (non-negative) I realize that any nonnegative valued continuous function $f\colon X \rightarrow Y$ where $X, Y \subseteq \mathbb R_{+}$ with a negative finite gradient has a fixed ...
1
vote
2answers
326 views

Brouwer's fixed point theorem (for unit balls) and retractions

Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$. I want to prove that the following proposition $B$ is a fixed-point space if ...
1
vote
3answers
308 views

Prove that $f$ has a fixed point . [duplicate]

For $f:[a,b]\rightarrow [a,b]$ is a continuous . Prove that $f$ has a fixed point . Is that true if we change $[a,b]$ by $[a,b)$ or $(a,b)$.
3
votes
1answer
38 views

Brouwer transformation plane theorem

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 ...
6
votes
1answer
95 views

periodic point of homeomorphism of plane

Let $h$ be a homeomorphism of $\mathbb{R}^2$ onto itself such that $h(K)=K$ for some compact subset $K$ of $\mathbb{R}^2$. Show that $h$ has a periodic point in $\mathbb{R}^2$.
3
votes
1answer
190 views

Generalization of Banach's fixed point theorem

I wanted to show that if $f:X\to X$ is a function from a complete metric space to itself and if $f^k$ is a contraction, then $f$ has a unique fixed point (say $p$) and for any $x$ in $X$ ...
4
votes
1answer
90 views

Unique solution to $x^4 + 7x -1 = 0 $ on $[0,1]$ (Banach's fixed point theorem)

I want to show that $x^4 + 7x -1 = 0 $ has a unique solution on $[0,1]$. The idea is to use Banach's fixed point theorem. However, I see a problem with this as the statement of the theorem says that ...
1
vote
1answer
111 views

Functions $f:X\to X$ with no fixed points, for $X$ a punctured disk or a sphere.

$X$ is the punctured closed unit disc $D^2-\{0\} = \{(x, y) \in \mathbb{R}^2: 0 \lt x^2+y^2 \le 1\}$ Is the answer that $f$ maps all $(x,y)$ to $0$ which is not included in the unit disk and so ...
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
146 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
0
votes
1answer
90 views

a fixed point theorem for rectangle in $\mathbb{R}^2$

Well, Could any one tell me how to prove this one or any reference? Let $f$ be a continuos map on $\mathbb{R}^2$, and $S$ be a rectangular region such that as the boundary of $S$ is traversed, the ...
-3
votes
4answers
174 views

Three questions about fixed points

Pick out the true statements. Let $f : [0, 2] \to [0, 1]$ be a continuous function. Then, there always exists $x \in [0, 1]$ such that $f(x) = x$. Let $f : [0, 1] \to [0, 1]$ be a continuous ...
1
vote
1answer
102 views

Topology Fixed Point Theorem

suppose the wind is blowing on the surface of the earth in a constant and continuous fashion. Suppose also that at every point on the equator, the wind is blowing directly east, so the wind doesnt ...
0
votes
2answers
56 views

Applying a contraction to balls' centers increases the size of the balls' intersection?

The following statement seems clearly true, but I'm having a hard time proving it: Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix ...
2
votes
2answers
454 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
282 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 ...
22
votes
1answer
325 views

Does every continuous map from $\mathbb{H}P^{2n+1}$ to itself have a fixed point?

This question is motivated by Fixed point property of Cayley plane and idle curiosity. In the link, it is shown that every continuous map from the Cayley plane to itself has a fixed point and the ...
11
votes
2answers
649 views

Generalization of “easy” 1-D proof of Brouwer fixed point theorem

In topology class, before we learned the homotopy-based proof of Brouwer's fixed point theorem, the professor mentioned the easy proof of the one-dimensional case: just draw the graph $f(x) = x$ in ...
2
votes
2answers
268 views

Brouwer's fixed point theorem in a practical setting

If we assume that a fluid is a continuum then if we have for example a cup of tea and we stir the fluid then there will be a point in the fluid that is on the same location before and after the ...