1
vote
1answer
64 views

An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, ...
3
votes
2answers
194 views

Banach fixed point theorem and Picard-Lindelöf applied to this equation (explanation needed)

Consider the following equation which holds for all $w$ in some space, $$\langle v(t), w \rangle = \langle v(0), w \rangle - \int_0^t \langle F(s,v(s)), w \rangle$$ where $\langle F(s,v),w \rangle$ is ...
4
votes
2answers
151 views

Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some ...
1
vote
0answers
27 views

How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
1
vote
1answer
293 views

Lipschitz condition on nonlinear ODE

Suppose we have the ODE $$x''=-a\sin{x}.$$ Then let $$x'=y$$ and $$y'=-a\sin{x}.$$ So $$\mathbf X = \begin{pmatrix} y \\ -a\sin{x} \end{pmatrix}.$$ Im confused about how to show a Lipschitz ...
2
votes
2answers
144 views

does it have unique fixed point?

$p:C[0,1]\rightarrow C[0,1]$ defined by $p(f(x))=\int_{0}^{x} (x-t)f(t)dt$, well, I am getting all constant functions are fixed points, but the answer says that it has unique fixed point. I got ...
1
vote
1answer
98 views

Fixed Points of a system of diff.equations

I have $f'_1(t)=-af_1(t)f_2(t)+bf_3(t)$ $f'_2(t)=f_1(t)$ $2f'_3(t)=-f_1(t)$ How is it possible to evaluate fixed points of this system of equations and afterwards the stability of these points. I ...
0
votes
1answer
62 views

IVP- Has at most one solution

Suppose $f(t, x)$ is nonincreasing with respect to $x$ for all $t \geq 0$ and $x \in \mathbb{R}$. Prove that the IVP problem $$ \left\{ \begin{array}{l} x'(t)=f(t,x) \\ x(t_0)=x_0 \end{array} \right. ...
3
votes
1answer
155 views

Fixed point: a consequence of symmetry?

I'm studying a dynamical system with $\mathbf{D}_{3}$ symmetry (the symmetry group of an equilateral triangle), which is given by: $\begin{align*} d\mathbf{x}_{0}/dt &= ...