Linked Questions

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 ...

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.

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 ...

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 ...

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 ...

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 ...

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 ...

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?

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 ...

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)=...

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.

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 ...