Questions tagged [contraction-mapping]

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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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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, ...
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 ...
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 ...
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 ...
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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 ...
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 ...
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: \...