Question

suppose I have a system of three differential equations of the form: \cdot{x}_1=F(t_1,x_1,x_2,x_3); \cdot{x}_2=F(t_2,x_1,x_2,x_3); \cdot{x}_3=F(t_3,x_1,x_2,x_3)
where t_j is a scalar. Can I show existence appealing to usual existence theorems if I show that each equation (say eq 1) admits a solution for any array of equations other than x_1 and then extend this argument by substituing this solution in the remainign equations?

Answer 1

No, the first step of your bootstrap argument fails.

However, provided F is Lipschitz, then this system will have a solution.

By the way, do you want the same F for all three equations?

Answer 2

In other words, you need to consider the system in its entirely; writing for instance
v= (x1, x2, x3), t = (t1, t2, t3) then this is a system in R^3 where
\[ v' = \hat{F}(t, v), \ \ \ \ \ \hat{F} = (F(t_1, .), F(t_2, .), F(t_3, .)) \]

Answer 3

Thanks James. No, F need not be the same in all equations. I though about writing the system such that F=(F_1,F_2,...,F_m) and \cdot{v}=(\cdot(x_1},...,\cdot{x}_m) and then appeal to standard existence theorems. I can show that F (as defined above) is Lipschitz but I am still wondering if this is enough to use standard theorems (all versions I have read assume that the variable t belongs to R^1 and this made me wonder whether I can simply extend it to vectors).

Thanks for your help! I truly appreciate it.

Answer 4

keep digging in your source materials/text book. The truly standard proof is in R^n. it is by and large a straight adaptation of the theorem from R.

Answer 5

Hi James, thanks a lot...the thing is that I learnt this on my own (coz' I needed for another proof) and I lack of a good reference book. Could you please lee me know of so0me good book/reference where I can take a look? Once again, thanks a lot!

Answer 6

I think you'll find it in Rudin, "Principles of Mathematical Analysis"

Answer 7

great. I'll take a look at it...thanks!