Questions tagged [contraction-mapping]

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0answers
51 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
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0answers
53 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. ...
3
votes
1answer
98 views

Generation of strictly 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-dessipative operator then it generates a $C_0$-...
1
vote
1answer
117 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
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0answers
32 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 ...
2
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0answers
52 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, ...
3
votes
2answers
341 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
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0answers
202 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 ...
3
votes
1answer
215 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 ...
1
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0answers
56 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 ...
2
votes
1answer
193 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 ...
5
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0answers
298 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: \...