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Questions tagged [contraction-operator]

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

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1answer
32 views

Is log-sum-exp a contraction

For $x \in \mathbb{R}^n$, the log-sum-exp (lse) function $\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $lse(x) = \tau \log \sum_{i=1}^n \exp(\frac{x_i}{\tau})$, where $\tau > 0$ is a ...
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28 views

On the invariance of ball $B(a,r)$.

Let $X$ be a complete metric space and a continuous map $ f: X \to X $. A condition sufficient for a ball $ B(a, r) \subseteq X $ to be invariant by $ f $, that is $ f \left (B (a, r) \right) \...
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0answers
16 views

Is the generator of a uniformly continuous contraction semigroup contractive?

Let $E$ be a $\mathbb R$-Banach space and $A$ be the generator of a uniformly continuous contraction semigroup $(T(t))_{t\ge0}$ on $E$. Are we able to derive some bound on $\left\|A\right\|_{\mathfrak ...
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1answer
18 views

I want to understand the triangle inequality about contraction operator

I read the Chow's paper, Multigrid algorithms and complexity results, https://dspace.mit.edu/handle/1721.1/14254 I have a question on page 42. Let me write the Lemma 2.4.3 on the page. Lemma 2.4.3 ...
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1answer
31 views

Variant of the Contraction Mapping Theorem

Let (C, ||.||) be a closed subset of a Banach space, and let f: C -> C be a mapping such that ||f(x) - f(y)|| < ||x - y|| for all x, y in C. Must there exist a fixed point in C that f maps to ...
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44 views

Using the contraction mapping principle for the solution of an ODE

Suppose we want to solve a simple ODE $$\dot{x}=2t(1-x)=f(t,x)\qquad (1)$$ with initial condition $x_0=2$ using the contraction mapping principle. I showed that $f$ is uniformly Lipschitz continuous ...
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0answers
14 views

Using Banach fixed-point theorem to show existence and uniqueness of the following system of linear equations

Consider the following system of linear equations : $$ \begin{cases}2x_1 + x_2 = 5, \\ x_1 - 5x_2 = -3.\end{cases}$$ Using basic linear algebra, it is easy to show that $x_1 = 2$ and $x_2 = 1$ is the ...
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1answer
28 views

Contraction and $\max$ function

$f: \Bbb R \mapsto \Bbb R$ $g: \Bbb R \mapsto \Bbb R$ $h: \Bbb R \mapsto \Bbb R$ $h:=\max\{f(x), g(x)\}$ Is $h$ a contraction on $ \Bbb R$ if $f$ and $g$ are both so? First attempts of ...
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1answer
40 views

Tensor contraction computation

I am currently trying to implement tensors as multidimensional arrays in C++, which is why i am asking this with regards to algorithmic computability. I would like to implement the contraction of two ...
3
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1answer
42 views

Example of a nowhere differentiable contraction mapping

The Weierstrass function https://en.wikipedia.org/wiki/Weierstrass_function is a pathological example of a continuous nowhere differentiable function.. Since a conttaction mapping is necessarily ...
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91 views

Prove that this function is a contraction

Let $f \in C^2([a,b])$ be a function with the following properties: (i) $f(a)f(b) <0 $ (ii) $f'(x) \neq 0$ $\forall x$ (iii) $\frac{|f(a)|} {|f'(a)|} <(b-a)$ and $\frac{|f(b)|} {|f'(b)|} &...
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1answer
33 views

Examples of contraction homeomorphism

Let $X$ be a compact metric space with metric $d$ and let $f:X \rightarrow X$ be a homeomorphism. We say that $f$ is a contraction, if there exists $0 < c < 1$ such that $d(f(x), f(y)) \leq c d(...
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2answers
104 views

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

I have the following first order differential equation $$x^\prime(t)=-(x(t))^2+2x(t),\quad t\geq 0,\quad x(0)=1$$ Now I want to obtain an approximation of $x(t)$ by using Picard's method. Then the ...
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1answer
36 views

can someone explain me in simpler way of what this paper has conveyed

I do not have a thorough knowledge of measure-theoretic probability and Markov chain but I would start to learn by myself soon, but for few research-related works, I have to understand the theme of ...
1
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1answer
52 views

Solve differential equation via successive approximations (contraction mapping principle)

I'm working on a problem (self study) to solve the differential equation $$ \frac{dx}{dt} = tx^2 + x^3 $$ using successive approximations within a neighborhood interval of $\delta < 1/5$. I'm given ...
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2answers
176 views

Proving system of equations has unique solution using contraction mapping principle

I am trying to prove that the linear system $x = Ax + b$ has a unique solution using the contraction mapping principle, where $$ A = \begin{pmatrix} \frac{1}{4} & -\frac{1}{4} & \frac{2}{15}...
3
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2answers
59 views

Contraction, doubt on definition

In a handout I've found this definition of contractions: Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $\varphi : X \to Y$ is called a contraction if there exists a positive number $c <...
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1answer
14 views

Proving the error estimate of a contraction in Banach space via induction

My task is to prove the following statement: if $T:V$—>$V$ is an $\alpha$-contraction and $V$ is a Banach space, then T has a globally attracting fixed point $\bar v$ in $V$, and for any initial point ...
2
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1answer
47 views

Is the function is contraction? How to find fixed point?

Task 1) Show that function $T:C[0, 1] \to C[0, 1]$ is a contraction, then $$T(f)(x)=\int_{0}^x (x-t) f(t) dt,$$ $$x\in [0,1], f\in C[0, 1].$$ 2) Find a fixed point of $T(f)(x)$. My progress 1) I ...
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1answer
38 views

nonlinear contraction

Let $H$ be a Hilbert space, and let $T: H\to H$ be a nonlinear contraction, i.e. $$||Tu-Tv||\le||u-v||, \forall u,v\in C$$ Let $(u_n)$ be a sequence in $H$ s.t. $$u_n\rightharpoonup u\text{ weakly ...
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0answers
23 views

Convergence of Matrix product (not memoryless)

I try to analyze the following iterative matrix product: $x_{i+1} = A_{i} \cdot x_{i}$. The matrices $A_i$ are defined as follows: $A_i = \begin{bmatrix} \frac{1}{2\sqrt{x_{i,1}^2 + y_{i,1}^2}} &...
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1answer
20 views

Questions from An Introduction to Semilinear Evolution Equations Corollary 4.1.9

Before, proceeding to the problem, I will mention some notations on this book to be easier to read. 0. $(X, ||\, \cdot\,||)$ is Banach space 1. $L^{1}((0,T),X)$ : the space of measurable functions $u$ ...
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2answers
56 views

Contraction operator.

Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ a function derivable so that, for all $t \in \mathbb{R}$, $\vert {\psi'(t)} \vert \leq \alpha < 1$. How do I prove that $\psi$ is a contraction? How ...
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2answers
49 views

Banach fixed point.

Let $T \in B(\mathcal{B})$. If $\Vert{T}\Vert < 1$ then, given $\eta \in \mathcal{B}$, the equation $\xi-T\xi=\eta$ has a unique solution. In effect, defining the operator $S:\mathcal{B} \...
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0answers
29 views

Finding a contraction of a transformation by Contraction Mapping Theorem

I want to demonstrate that this transformation is a contraction: $$Tx(t)=\frac{1}{2}\left (x^2(t) + 1 -t^2 \right )$$ In the closed set $S$: $$S=\{x\in C([-1,1; \mathbb{R} ]):\Vert x\Vert_0 \leq 1\}$$ ...
1
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1answer
30 views

Prove that the operator $Tv = (v_2,\frac{1}{2} v_1) $ is a contraction

Consider the complete metric space $(R^2,d_2) $ where $d_2$ is the Euclidean distance. I have to check whether the following operator $T:R^2 \rightarrow R^2$ is a contraction: $ T(v) = (v_2,\frac{1}{...
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1answer
42 views

Show that a complex Differentiable function f with $|f'|\leq 1$ is a contraction.

I am working on the proof of the following statement; Suppose $f$ is analytic on a a rectangle $R$ and $|f'(z)|\leq 1$ for all $z \in R$. Then $f$ is a contraction on $R$, that is $$ |f(b)-f(a)|\leq |...
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0answers
56 views

Products of Positive Matrices

Is it true that every infinite product of positive matrices (every element is larger than 0) converges to a rank one matrix? I learned that Birkhoff's Contraction Coefficient is smaller than one for ...
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1answer
63 views

Show that a mapping is a contraction in $ \mathbb{R}^{2} $ [closed]

Consider $ \mathbb{R}^{2} $ with the usual Euclidean distance $ d$. I have to show that the equation $ v = M(v)$ where $$ M(v) = (1,1) + \beta(x,y), \quad \beta \in [0,1) $$ for all $ v=(x,y)$, has ...
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0answers
76 views

Lowering index on $\Gamma^\beta_{\mu\gamma}$

I was doing an exercise in Schutz (A First course in General Relativity). The exercise wanted the double covariant derivative calculated for a vector $V^\mu$ i.e. $\nabla_\alpha \nabla_\beta V^\mu$ i....
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2answers
61 views

Confusion about proving $f$ is not a contraction

Let $f:[0,2]\to\mathbb R, f(x)=\frac{1}{3}(4-x^{2})$ I know that $f$ is a contraction $\iff$ $\exists L \in ]0,1[$ such that for all $x, y \in [0,2]: |f(x)-f(y)|\leq L|x-y|$ I am having trouble ...
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2answers
47 views

How can the distance metric of two different metric spaces be comparable?

I come from a machine learning background and recently was working on understanding self normalizing networks. It requires me to know the proof of banach fixed-point theorem which needs me to know ...
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0answers
76 views

Fixed-Point theorem, Banach Spaces

My assumptions are: My space of interest is $\mathbb{R}$ as Banach space. My sequence of interest $X_{n+1}=\beta+\alpha X_n$ converges in my space $\mathbb{R}$. (It is already proved). Here $\beta\...
2
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1answer
37 views

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

I'm looking for such function g defined in some closed $X\subset\mathbb{R}$ such that $g'$ exists on $X$ and $g: X \rightarrow X$. Some examples that I tried that does not work $g(x)=\frac{|x|^3+x^2+...
2
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1answer
40 views

How do you know if a function is a contraction or a shrinking map and how do you find a unique point?

Let $f(x) = (1/2)|x|$ a). What is the value of $f′(x)$ on $(−∞,0)$? b). What is the value of $f′(x)$ on $(0,∞)$? c). What is the value of $|f′(x)|$ on $(−∞, 0) ∪ (0, ∞)$ d). By part c). we can ...
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2answers
161 views

Is a Contraction also a Contraction under equivalent metrics?

Definition of a Contraction. Let $(X, d)$ be a metric space. Then a map $T : X → X$ is called a contraction on $X$ if there exists $q ∈ [0, 1)$ such that $d(T(x),T(y)) \le q d(x,y)$ for all $x, y$ in $...
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0answers
13 views

Help with this proof (k-set contractive mappings)

In this paper, I do not understand the Example 1 (p. 8). The statenment is the following: Let $X$ a Banach space, $0<r\leq 1$ and $B(0,r)$ the open ball centered at 0 (the null vector of $X$) and ...
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1answer
81 views

Topological transitivity+contraction

I'm looking for a hint: Suppose $T:X→X$ is a dynamical system. Assume that $T$ is both topologically transitive and a contraction, i.e. $d(T(x),T(y))≤d(x,y)$ for all $x,y∈X$. Prove that T is minimal. ...
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1answer
20 views

Finding coefficient in contraction mapping

I got a task to find for which $l$ and $m$ this mappings would be contraction mappings ($C[a,b] \rightarrow C[a,b]$) a) $f \rightarrow (lx+m)f$ b) $f \rightarrow \int_m^x f(t)dt$ I came to point ...
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2answers
122 views

Is $\ln(x)+1$ contractive?

I am having trouble understanding the definition of a contractive function. The definition is: Let $(\Bbb R,d)$ be a Metric Space with the standard metric. Let $A\subseteq X$ (with the same metric ...
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1answer
23 views

Shouldn’t squaring the (metric) tensor give a scalar?

What I understand is that $$g^{\mu\nu} g_{\mu\nu} = g^2 = \text{Id}_{4×4}$$ First, is this statement true? Now if it is, as we are contracting two indices here, shouldn’t the result be a 0-rank ...
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2answers
59 views

Continuous function from set to proper subset is a contraction

I want to show for part of a proof that any continuous mapping $f \colon [a,b] \to [a,b)$ must be a contraction. I'm not even sure if this is true, or how to prove it if it is.
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1answer
81 views

Abstract index notation vs Ricci Calculus

I have come accross some comparison between the abstract index notation and Ricci calculus as it pertains to contraction and what I find is: The former (abstract notation) indicates that a basis-...
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1answer
41 views

Contraction mapping theorem generalization

We know that if $T$ is a contraction in a complete metric space $X$, then $T$ has a unique fixed point and it holds $$\lim_{k\to\infty} T^k(z)=x \ \ \ \ \forall z\in X.$$ If for some integer $p>0$ ...
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1answer
52 views

Contraction mapping theorem to solve ODE

I came across this exercise and I am really not sure how to solve it: 1) Prove that for $t_0 > 0$ $A = \{ f \in C[0,t_0]: f(s) \in [0,2] \forall s \in [0,t_0]\}$ is closed in $(C[0,t_0],\|.\|_{\...
0
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1answer
104 views

Contraction mapping theorem

We know that if $T:K\to K$ is a continuous contraction map in a complete metric space $(K,d)$ then it admits an unique fixed point $x$ which is equal to $\lim_{n\to\infty}T^n(z)$ for any $z\in K$. ...
3
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1answer
47 views

Is the operator $Tf = f(x) - f(x) \int_0^1 f(y) dy$ a contraction?

Is the operator $$Tf = f(x) - f(x) \int_0^1 f(y) dy$$ in $C[0,1]$ (with the uniform norm) a contraction and what is the possible fixed point?
0
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1answer
54 views

Union of contractions is a contraction

Can someone give me a proof of the following statement: Let $(X,d)$ be a metric space. If $w_{i} : D_{i} \rightarrow X$ , $D_{i} \subset X$, are contraction mappings with corresponding contractivity ...
1
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1answer
49 views

Contractive sequence together with not contract mapping

Suppose that $(x_{n})$ is a contractive sequence in $\mathbb{R}$ and that $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function with $x_{n+1}=f(x_{n})$ for all $n\in\mathbb{N}$. Is there a example ...
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0answers
43 views

how to show that this non-linear operator is a contraction

Let: $$ Tf(t):=\gamma(t)+k\int_0^t\cos(f(t)-f(s))\,ds. $$ How can we show/discuss whether this linear operator is a contraction for some particular $k?$ I proceed by normal step but do not know ...