# Uniqueness of solution for a system of differential equations

A friend of mine working on Auction Theory needs to establish uniqueness of solution (up to initial and boundary conditions) of a system of differential equations of the form

$$F(y_1,y_2,y_3,\dot{y}_1,\dot{y}_2,\dot{y_3},x) = 0 \\ y_1(y_2) = y_3$$

where $F:\mathbb{R}^6\times X\to\mathbb{R}^n$ is a polynomial, $X\subseteq\mathbb{R}$ is a compact set, and $y_1,y_2,y_3:X\to X$ are the functions of interest.

What throws us off is the equation $y_1(y_2) = y_3$. We don't know how to deal with it.

Are there any standard tools that may help him ot establish uniqueness for this sort of problems?

• By $y_1(y_2)$ do you mean $y_1 \circ y_2$? But $y_1$ is only defined on $X$, and $y_2$ maps $X$ into $\mathbb R$, not into $X$. Aug 15 '14 at 21:24
• Yes, I do mean composition. I made a change on the image of the functions to solve that problem. Aug 15 '14 at 23:24
• @brunosalcedo: Similar question as Robert, Do you mean $y_1(y_2(x))=y_3(x)$ ? If so, you have $y_1(x)$ and $y_1(y_2(x))$ in your system... If so, this is not an ODE sys. If your friend want to talk about this problem, feel free to email me. Aug 16 '14 at 14:07