Linked Questions

1
vote
0answers
373 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 $x,y\in X$ such ...
-2
votes
1answer
185 views

Existence of unique fixed point in compact Metric space [duplicate]

Let $(X,d)$ be compact. Show: for a map $f$ that when $\forall x, y \in X$ with $x\neq y$ $d(f(x),f(y))<d(x,y)$ is fulfilled. Then $f$ has a unique fixed point.
2
votes
1answer
77 views

Fixed Point Problem. [duplicate]

Suppose $(X,d)$ is compact and we have a mapping $T:X \rightarrow X$ such that $d(Tx,Ty)<d(x,y)$ for every $x,y\in X$ with $x\neq y$. The question is that to show $T$ has a unique fixed point. ...
5
votes
1answer
91 views

How to show there exist a unique $x_o$ in X such that f($x_o$)=$x_o$? [duplicate]

Let $(X, d)$ be a compact metric space. Let $f:X\rightarrow X$ be such that $d(f(x), f(y)) < d(x, y)$ for all $x, y\in X$ with $x$ not equal to $y$. Show that $f$ has a fixed point, that is, there ...
24
votes
6answers
35k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
0
votes
3answers
1k views

Prove there is no contraction mapping from compact metric space onto itself

This question is from Foundations of mathematical analysis by Richard Johnsonbaugh The thing with this question is that there is a question that seems to prove the opposite claim Prove the map has a ...
7
votes
2answers
175 views

How can I prove $x_{n+1} = e^{-x_n}$ is convergent?

I'm doing a practice problem which asks to prove that the sequence defined by $x_{n+1} = e^{-x_n}$ is convergent (or rather "study the convergence of $(x_n)$"). So I'd like to try and find some ...
1
vote
1answer
611 views

Infimum of Distance in Compact Space

If $X$ is a compact metric space, $A: X\to X$, is it true that if $a = \inf d(x,Ax),\space x \in X$, then there exists $y \in X$ such that $d(y,Ay) = \inf d(x,Ax)$? If so, why?
2
votes
3answers
441 views

If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A $ have only one fix point on X.

I have problems with this demostration can anybody help me please? If $X$ is a compact space and $A:X\rightarrow X$ is such that $\rho(Ax,Ay)<\rho(x,y)$, $x\neq y, \Rightarrow A $ have only one ...
2
votes
1answer
126 views

Fixed point without the constant

If $d(Fx,Fy)<d(x,y)$ for all $x,y$ in a closed bounded subset $X$ of Euclidean space and $F\colon X\rightarrow X$ then there is a unique fixed point $x_0$ and $\lim \limits _{n\to\infty} F^n(x)=...
2
votes
1answer
109 views

convergence of continued nested function

Does $$\sin (x + \sin (x + \sin (x + \sin (x + ... ) ) ))) $$ converge to any limit? If so, to what? Recursive plotting appears at times to come to same/similar profiles.
0
votes
1answer
104 views

Question about application of Contraction Principle

If $K$ be a compact metric space and $\Phi:K \to K$ be such that $d(\Phi(x),\Phi(y))<d(x,y)$, Show that $\Phi$ has a unique fixed point. Here is my approach but not sure To show that $\Phi$ has ...