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I have a system of ODEs: $$f_1'(t) + \sum_{j=1}^n f_i(t)a_{j1}(t)=d_1(t)$$ $$...$$ $$f_n'(t) + \sum_{j=1}^n f_i(t)a_{jn}(t)=d_n(t)$$

How can I show that there exists a unique solution $(f_1$, ..., $f_n)$?

According to wiki, if I put this equation into the form $y'(t)= F(y(t),t)$ I need $F$ to be Lipschitz wrt. $y$, which in my case works if I look at each individual equation but as a system I don'tknow. What assumptions do I need on the functions $a$, $c$ and $d$?

I don't necessarily have continuity of $a$ or $d$. They may lie in some Lebesgue or Sobolev space.

Thanks for any help

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It's a linear system of odes. Let $A_{ij}:=-a_{ji}$, then you have that

$$f'(t)=A(t)f(t)+d(t).$$

Suppose that $A$ and $d$ are defined on some interval $\mathcal{I}\subseteq\mathbb{R}$. Then sufficient conditions for the existence and uniqueness of are that $A$ and $d$ are continuous functions. That is, if $A:J\rightarrow\mathbb{R}^{n\times n}$ and $d:J\rightarrow\mathbb{R}^n$ are continuous, then for every $x\in\mathbb{R}^n$ and $t_0\in J$ there exists a unique continuous function $f:J\rightarrow\mathbb{R}^n$ such that $f(t_0)=x$ and

$$f(t)=x_0+\int_{t_0}^t[A(s)f(s)+d(s)]ds$$

holds for all $t\in J$. A proof for this is can be found page 54 of this book (and, I suspect, also just googling "existence and uniqueness of solutions, linear systems" for a bit).

EDIT: Ok, so apparently $d$ and $A$ being measurable and locally integrable is sufficient to guarantee the conclusion above. By locally integrable I mean that for all $a,b\in J$,

$$\int_a^b||A(t)||_\infty dt<\infty,\quad\quad, \int_a^b||d(t)||dt<\infty.$$

For details see Proposition C.3.8. of the appendix of this book.

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  • $\begingroup$ Thanks, but yeah as you wrote we don't have continuity.. $\endgroup$ – matt.w Apr 25 '13 at 15:30
  • $\begingroup$ See the edit to the answer. $\endgroup$ – jkn Apr 25 '13 at 16:59

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