10 votes

Fastest way showing the limit exists without finding the limit?

Write your formulas in a matrix form: $$\begin{bmatrix}a_n\\b_n\end{bmatrix} = \begin{bmatrix}1&1\\r&1\end{bmatrix}\begin{bmatrix}a_{n-1}\\b_{n-1}\end{bmatrix}$$ i.e. $$w_n=Aw_{n-1}$$ ...
7 votes
Accepted

Does there exist $x$, $y \in \mathbb{N}$ such that $x^2 − y^2 = 19$

Notice that $19$ is prime. because of this, one of $x-y$ and $x+y$ must be $1$, and the other $19$. Therefore, if we solve the system $x-y = 1, x+y=19$, we can find a solution to this problem. You ...
  • 226
6 votes
Accepted

Fastest way showing the limit exists without finding the limit?

Not sure if the following fits the requirement to avoid finding the limit, but anyway... Consider the change of variable $$x_n=c_n-\frac{r}{c_n}$$ then $$x_{n+1}=-f(c_n)x_n$$ with $$f(c)=\frac{(r-1)c}{...
  • 272k
6 votes

Does there exist $x$, $y \in \mathbb{N}$ such that $x^2 − y^2 = 19$

You've came to $(x+y)(x-y) = 19$ The only possible natural numbers that multiply to 19 are 1 and 19. And sum/difference of natural numbers are also natural numbers. Since $x-y < x+y$, you have: $$...
6 votes
Accepted

Does proper contraction on Hilbert space necessarily lead to convergence in norm to zero?

Let $\{a_n\}$ be a sequence of positive real numbers that increases to $1$, with the property that the sequence of products $$ a_1,\ a_1a_2,\ a_1a_2a_3,\ a_1a_2a_3a_4,\ \ldots$$ converges to a ...
  • 50.5k
5 votes
Accepted

Prove that $f(x)=x^2$ is a contraction on each interval in $[0,0.5]$

Remark that $$ |f(x)-f(y)| = |(x+y)(x-y)| \leq (|x|+|y|)|x-y| < 2a |x-y| $$ if $x$, $y \in [0,a]$. Now $2a<1$ by assumption.
  • 34k
5 votes
Accepted

an example of a complete space $X$ and a such mapping $T$ without fixed points & Show that if $X$ is compact then such $T$ has a unique fixed point.

Hint for (a): Apply the Mean Value Theorem to that function you were given. Hint for (b): Since $d(T(x),T(T(x))) < d(x, T(x))$ if $x\ne T(x)$, you must have $\inf d(x,T(x))=0$. (You should ...
5 votes
Accepted

Using connectedness to prove surjectivity...

Let $(x,y)\in\mathbb R^2$ and let $$g: u \mapsto x - f\left(y-f(u)\right)$$ \begin{align} \left|g(u) - g(v)\right|& = \left|x-f(y-f(u)) - x + f(y-f(v))\right|\\ &= \left|f(y-f(u))-f(y-f(v))\...
  • 6,068
4 votes

Fastest way showing the limit exists without finding the limit?

Again, not sure if this satisfies your criteria, but if we let $f : [0,\infty) \to \mathbb{R}$ by $$ f(z) = \frac{r+z}{1+z} $$ then $f$ is decreasing, $0\leq f \leq r$, and $c_{n+1} = f(c_n)$. So if ...
4 votes
Accepted

prove there is no integer solution to x^2 -6 = 0 by contradiction

Assume an integer solution $x$ such that $x^2 = 6$ exists. An integer is either odd or even. The square of odd $x$ is odd (whereas $6$ is even), so $x$ must be even. Let $x = 2k$, where $k$ is an ...
  • 26.1k
4 votes
Accepted

Topological transitivity+contraction

If you assume $X$ to be a compact metric space, then transitivity implies the existence (in fact, denseness) of transitive points. (A transitive point for $T$ is a point $x$ whose orbit is dense in $...
  • 1,885
4 votes
Accepted

Looking for function $g$ such that $|g'(x)|\leq k<1\ \forall\ x \in X$ and $g$ is not a contraction

Let $X=(-\infty,-1]\cup[1,\infty)$ and $g(x)=\frac12x+\operatorname{sgn}(x)$.
4 votes

Contraction, doubt on definition

It seems an odd definition, as for one thing the CMT will no longer work. As an example, let $X$ be $\mathbb R$ with the usual distance function, and let $Y$ also be $\mathbb R$ but with the ...
4 votes

Examples of contraction homeomorphism

There is no such map. Let $x,x'\in X$ such that $d(x,x')$ is maximal. Take, if possible, $y,y'\in X$ such that $f(y)=x$ and that $f(y')=x'$. Then$$d(x,x')=d\bigl(f(y),f(y')\bigr)\leqslant cd(y,y')<...
4 votes
Accepted

Is log-sum-exp a contraction

You cannot do better than $L=1$. To see this, let us first get rid of the $\tau$. Using the fact that $\tau\neq 0$ and scaling the coordinates by $\tau$, it clearly follows that proving the displayed ...
  • 3,505
4 votes
Accepted

contraction in five dimensional Euclidean space

First, $T$ and $S$ cannot be strict contraction mappings with respect to the same norm. To see this, suppose for contradiction that there exists a norm $\lVert \cdot \rVert$ on $\mathbb{R}^5$ (or $\...
4 votes
Accepted

Contraction of a tensor over all indexes?

$\newcommand\Tr{\operatorname{Tr}}$The most simple contraction operation is the trace : $$\Tr : V\otimes V^* \to \mathbb K$$ Given a general tensor product $T^k_l(V) = V^{\otimes k} \otimes (V^*)^{\...
  • 7,406
4 votes

Trace and Lie derivative of a $(1,1)$-tensor commute (Direct proof)

$\newcommand\tr{\operatorname{tr}}$By the linearity of trace, it suffices to prove the identity for $S = \theta\otimes v$, where $\theta$ is a $1$-form and $v$ is a vector field. First, observe that \...
  • 3,415
3 votes

Fastest way showing the limit exists without finding the limit?

From the first relation $b_{n-1}=a_n-a_{n-1}$, then substituting into the second one: $$ a_{n+1} = 2 a_n +(r-1)a_{n-1} \tag{1} $$ From the second relation $ra_{n-1}=b_n-b_{n-1}$, then substituting ...
  • 70.8k
3 votes

Proving that the mapping is a contraction

Consider the function $$f(x) = \sqrt{1+x^2}, \,\,\,\,\, x \in \mathbb R.$$ We note that $$\lvert f'(x) \rvert = \frac{\lvert x\rvert}{\sqrt{1+x^2}} \le 1, \,\,\,\,\, x \in \mathbb R.$$ Now for any $x,...
  • 14.9k
3 votes

Fastest way showing the limit exists without finding the limit?

Note that $$c_{n+1}=\frac{ra_n+b_n}{a_n+b_n}=1+\frac{r-1}{\frac{a_n+b_n}{a_n}} =1+\frac{r-1}{1+c_n}.$$ Wanting to show that the map $$f(x)=1+\frac{r-1}{1+x} $$ is a contraction, is a bit problematic ...
3 votes

The image of every strictly non-expanding map between spheres is contained in an open hemisphere?

Here's a proof of the main statement. I will use induction on the dimension $n$ so, suppose that the result holds in $\mathbb S^m$ all $0\le m < n$. Set $B=f(A)$ and, by uniform continuity, extend ...
3 votes
Accepted

Under what condition is a linear mapping contractive in the sup-norm?

Proof: (of your guess) Note that $$ \|Ax-Ay\|_\infty \leq l\|x-y\|_\infty \quad \forall x,y \in \Bbb R^n \quad \iff\\ \|A(x-y)\|_\infty \leq l\|x-y\|_\infty \quad \forall x,y \in \Bbb R^n \quad \iff\\...
3 votes
Accepted

Let $T:C[0,1] \to C[0,1]:Tx(t) = \int_0^t x(s)ds$. Is it a contraction? Is any power of T a contraction?

No, $T$ is not a contraction; an example has already been given. But it is a "weak contraction", in that $d(Tx,Ty)\le d(x,y)$. And $T^2$ is a contraction; hence so is $T^k$ for any $k\ge2$, by ...
3 votes

Let $T:C[0,1] \to C[0,1]:Tx(t) = \int_0^t x(s)ds$. Is it a contraction? Is any power of T a contraction?

Since $\|x\|=d(0,x)$ and $T$ is linear (so that $T0=0$), we can talk about norms instead of distances. Now, $\|Tx\| = \sup\limits_{t\in[0,1]}|(Tx)(t)| = \sup\limits_{t\in[0,1]}\left|\int\limits_0^t x(...
  • 15.3k
3 votes

Continuous function from set to proper subset is a contraction

Let $f:[0,1]\to [0,1)$ be given by $f(x)=\frac12\sqrt{x}$. Then the ratio $|f(x)-f(0)|/|x-0|$ grows without bound as $x\to0$.
  • 59.5k
3 votes

Contraction, doubt on definition

Here they call it a 'contractive mapping', which I think makes no difference with the word 'contraction'. Well, you do not necessarily have fixed points.
  • 7,279
3 votes
Accepted

Solve differential equation via successive approximations (contraction mapping principle)

Contractractivity on a rectangular domain To apply the fixed-point theorem one needs to fix a domain for the ODE and determine the Lipschitz constant there. In the most simple case the domain is ...
3 votes

Picard's method does not solve first order differential equation?

You should have gotten a plot like this which gives a visible idea of convergence for $|t|\le 0.5$. Let's consider the interval $|x-1|<1$ in state space in the proof construction of the Picard ...

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