Proof Lipschitz continuous of complicated function

by Bram K.   Last Updated September 22, 2018 07:20 AM

I have the following equations:

\begin{equation*}\label{transformedformula} \begin{aligned} & \frac{d}{dt}x^i_t=\frac{1}{N} \sum_{j\neq i} K(x^i_t-x^j_t) +\frac{1}{M} \sum_{l=1}^{M} \Phi(x^i_t-u^l_t)\\ & \frac{d}{dt}u^l_t=w^l_{T-t}\\ & \frac{d}{dt}p^i_t=\frac{1}{N} \sum_{j\neq i} \nabla K(x^i_{T-t}-x^j_{T-t})(p^i_t-p^j_t) +\frac{1}{M} \sum_{l=1}^{M} \nabla \Phi(x^i_{T-t}-u^l_{T-t})p^i_t\\ & \frac{d}{dt}w^l_t=-\sum_{i=1}^{N} \nabla \Phi(x^i_{T-t}-u^l_{T-t})p^i_t \end{aligned} \end{equation*}

$$X=(x^1,...,x^N),U=(u^1,...,u^M),P=(p^1,...,p^N),W=(w^1,...,w^M)$$

Let $$Y =\begin{cases} X\\ U\\ P\\ W \end{cases}$$, so I can rewrite the above equations into the form $$\frac{d}{dt}Y_t=G(t,Y)$$.

I want to show that G(t,Y) is Lipschitz continuous with respect to Y and I want to determine the Lipschitz constant. So I need that $$|G(t,Y_1)-G(t,Y_2)|\leq L|Y_1-Y_2|$$ with $$L>0$$.

I have split up G to prove this: $$|G(t,Y_1)-G(t,Y_2)|^2=|G_1(t,Y_1)-G_1(t,Y_2)|^2+|G_2(t,Y_1)-G_2(t,Y_2)|^2+...$$

However, I have no idea how to proceed, since the formulas are very complicated. For example, the first part becomes: $$|G_1(t,Y_1)-G_1(t,Y_2)|^2=|\frac{1}{N} \sum_{j\neq i} K(x^i_1(t)-x^j_1(t)) +\frac{1}{M} \sum_{l=1}^{M} \Phi(x^i_1(t)-u^l_1(t))-\frac{1}{N} \sum_{j\neq i} K(x^i_2(t)-x^j_2(t)) -\frac{1}{M} \sum_{l=1}^{M} \Phi(x^i_2(t)-u^l_2(t))|^2$$

I tried using the triangle inequality and the mean value theorem, but it doesn't give me a nice expression.

I'm allowed to make assumptions on the functions (for example on $$\Phi$$ and $$K$$). Do you guys have any suggestions how to proceed my proof?

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