Linked Questions

1
vote
0answers
305 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
120 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.
0
votes
0answers
23 views

A problem with a compact metric space and $ f \colon X \to X$ a uniformly continuous function [duplicate]

Let (X,d) be a compact metric space and $ f \colon X \to X $ a uniformly continuous function so that $ d( f(x),(f(y)) < d(x,y) , \forall x,y \in X , x \neq y $ . Prove that there exists x' in X so ...
19
votes
6answers
26k 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
914 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
154 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 ...
6
votes
0answers
2k 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 ...
1
vote
1answer
509 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
356 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
108 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
95 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
93 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 ...