Questions tagged [contraction-mapping]

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2 votes
1 answer
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Differentiability of the fixed points of a family of contraction maps

Given a general Banach space $B$ and a one-parameter family of contractions $F_t:B\to B$ which is defined for all $t \in (a,b)$. $F_t$ depends continuously on $t$ (in the sense $\lim_{\varepsilon\to 0}...
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0 votes
0 answers
83 views

Existence of distance-preserving mappings for general norm in vector space

We say a mapping $f:\mathbb R^n\to \mathbb R^n$ be 1-Lipschitz with respect to a norm $\|\cdot\|$ if $\|f(x)-f(z)\|\le\|x-z\|$ holds for all $x,z\in\mathbb R^n$. Such a mapping are sometimes called a ...
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0 votes
1 answer
188 views

Is this a contraction mapping for small $T$?

Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$ $$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$ For $T>0$, let $\mathcal H_T:=\{h:[0,T]\...
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2 votes
0 answers
95 views

Multivariate extensions of Ledoux--Talagrand contraction principle

Let $\{\varepsilon_i\}_{i=1}^n$ be a sequence of independent Radecmacher (i.e., symmetric Bernoulli) variables, and let $\phi_i :\mathbb R \to \mathbb R$ be contraction (i.e., 1-Lipschitz) mappings ...
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5 votes
0 answers
151 views

Steps of the MMP "in family"

Let $\pi\colon X\to Y$ be a morphism between irreducible varieties with terminal singularities (let us say smooth if you want). I suppose that I have an open subset $U$ of $Y$ over which the fibres ...
2 votes
0 answers
151 views

The contraction principle in quasi metric spaces

I am researching contractive mappings and I need the article of I. A. Bakhtin "The contraction principle in quasi metric spaces"(1989) or at least part where explanation is given for ...
1 vote
0 answers
86 views

Determining the behavior of a contraction mapping with undefined points

Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
1 vote
2 answers
454 views

Gradient flows: convex potential vs. contractive flow?

Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$). Consider the autonomous gradient-flow $$ \dot ...
2 votes
1 answer
214 views

Set operations over iterated function systems

An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically, $$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$ is an IFS if each $...
3 votes
0 answers
74 views

Solving for a monotone function - contraction operator for functions?

I want to solve a problem for an increasing function $g(x)$, for $x \in [0,1]$ and with $g(0) = 0$ and $g(1) = 1$. The solution will be solution to the following equation $\forall x$, $f_1(x) = f_2(g(...
4 votes
0 answers
62 views

Can a nonlinear dynamical system be rewritten in terms of constraints?

My question is based on thoughts after reading to a specific section in the paper "On Contraction Analysis for Nonlinear Systems" by W. Lohmiller and JJ. Slotine, Section 4.2 Constrained Systems. ...
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2 votes
1 answer
222 views

Generation of strict contraction semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dissipative operator then it generates a $C_0$-...
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1 vote
1 answer
173 views

Small contraction for Hyperkähler Varieties

I have the following basic question. Everything is over $\mathbb{C}$. Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow ...
1 vote
0 answers
37 views

A exemple of a strongly-continuous contraction semigroup : how to prove the contraction?

I am trying to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$, $P$ being a probability kernel), is ...
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2 votes
0 answers
53 views

Notion of contractive map for certain set-valued functions

A map $f:\mathbb{C} \to \mathbb{C}$ is contractive (and Lipschitz) if for every $z_1,z_2$, we have $|f(z_1)-f(z_2)|\le K|z_1-z_2|$ for some positive $K<1$. This implies the existence of a unique, ...
4 votes
2 answers
480 views

On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces

I am presently doing research concerned with operator algebras and operator theory and I thought to write here in the hopes of seeking expert advice on an idea I had here. The classic Sz.-Nagy ...
4 votes
0 answers
221 views

On contractive properties of a nonlinear matrix algorithm

I’m stuck in a problem that concerns a nonlinear iterative matrix algorithm. Although the problem is quite complicated to explain I’ll try to describe it in a simple way, neglecting unnecessary ...
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3 votes
1 answer
280 views

Convergence of trajectories and asymptotic stability

Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are ...
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1 vote
0 answers
62 views

Unfolding subspace algebraic space

Let $ Y \subset X$ be a closed subspace of an algebraic space of finite type over $\mathbb{C}$. Let $p : Y' \rightarrow Y$ be a proper map of algebraic spaces. Artin proved that there exists a ...
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2 votes
1 answer
203 views

Does a particular iteration produce a weak solution to a non linear pde?

Consider the following non linear pde in the unknown $v(x,y)$: $$ \frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$ where $t$ is some fixed small ...
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5 votes
0 answers
373 views

Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function $\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted $L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions $f: \...
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