# Questions tagged [contraction-mapping]

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### A question about the existence of surjective contractions

A few years ago I was doing some research in origami, and was motivated to as the following questions: Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a ...
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### 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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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 $$\dot ... • 5,163 2 votes 1 answer 270 views ### Set operations over iterated function systems An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$is an IFS if each ... • 21 3 votes 0 answers 104 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(... • 51 4 votes 0 answers 66 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. ... • 141 2 votes 1 answer 290 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-... • 561 1 vote 1 answer 213 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 ... • 175 1 vote 0 answers 40 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 ... • 111 2 votes 0 answers 54 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, ... • 15.6k 4 votes 2 answers 576 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 231 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 ... • 2,682 3 votes 1 answer 318 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 ... • 175 1 vote 0 answers 64 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 ... • 7,200 2 votes 1 answer 206 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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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: \...