Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [contraction-operator]

Use this tag for questions about operators whose norm is at most 1.

A bounded operator T : X → Y between normed vector spaces X and Y is said to be a contraction if the operator norm ||T|| is at most 1. Every bounded operator can be a contraction after suitable scaling. Analysis of contractions provides insight into the structure of operators or a family of operators.

216 questions
2
votes
2 answers

If $x,y$ are distinct real numbers

If $x,y$ are distinct real numbers, prove that $x+y=-2$ if and only if $(x+1)^2=(y+1)^2$ How would I prove this? do I use contrapositive, contradiction or direct proof?
TheGamer
  • 393
2
votes
3 answers

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

This is what I got: $(x-y)(x+y) = 19$ so $x-y \in \mathbb{N}$ and $x+y \in \mathbb{N}$ $ \implies x-y = 19 =x+y$, we know $x+y \geq 8$, not possible $ \implies x-y = -19 =x+y$, Not possible since $x+y \in \mathbb{N}$ Hence, no $x$, $y \in…
TheGamer
  • 393
1
vote
1 answer

Proving that transformation is a contraction mapping

Let's $(X,\rho)$ be a metric space, where $X=L(0,1)$ and $\rho(x,y)$=$\int_{0}^{1}|x(t)-y(t)|$. I need to prove that transformation $A:X\to X$ is a contraction…
Norbert Dąbrowski
  • 121
1
vote
1 answer

Give an example of contraction function that has no fixed points.

So we have a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is a contraction mapping. Can we find any example of $f$ that has no fixed points? We take into account standard Euclidean distance.
Norbert Dąbrowski
  • 121
1
vote
2 answers

contraction mapping coeficient

I have a question concerning the definition of a contraction mapping. I have found in several books different definitions for the interval from which the constant of contraction is chosen. Some say $c \in [0,1)$ and others say $c \in (0,1)$,…
Jovan Markov
  • 13
1
vote
0 answers

Showing a particular $T$ a contraction on continuous functions on unit square

Let $R = [0,1] \times [0,1]$, $M = \{ f(x,y) \in C(R), 0\leq |f(x,y)|\leq 1\}$, and $0<\alpha< \frac{4}{3}$. I need to show that $T$ is a contraction on $M$ where $T$ is given by $$ Tf(x,y) = \frac{xy}{3} + \alpha\int_{R}(x+y) st(f(s,t))^2…
Chriz26
  • 281
0
votes
0 answers

Is third root of x is contraction mapping?

I need some help with proving that $$f(x)=\sqrt[3]{x}+2$$ is a contraction mapping $(f: \mathcal{R} \rightarrow \mathcal{R})$. So I need to show that there exist $\alpha\in (0,1)$ such that for all $x,y$ $$|f(x)-f(y)|\leq \alpha|x-y|.$$ I have…
Norbert Dąbrowski
  • 121
0
votes
1 answer

Is $sin(x)$ a contraction on $[0,1]$?

I see that the derivative of the function is $cos(x)$ and in $[0,1]$ this can take the value $cos(0)=1$ implying that is not a contraction. Is this correct ?
Long Claw
  • 39
  • 7
0
votes
1 answer

Spectral radius of Jacobian and global convergence of function iterations

Suppose we have a continuously differentiable mapping $f:\mathbb{R}^n\rightarrow \mathbb{R^n} $ with a unique fixed point $x^*$. If the spectral radius of the Jacobian $\rho(\nabla f(x))$ is less than unity at $x^*$, then function iterations…
Asco
  • 23