I have dug into all the posts in the forum on this topic but haven't quite yet found a satisfying answer to my problem. So my current problem is part of a big assignment at my college, I just need someone to shed some light on this. The problem is as follows: Given f and g are 2 real-valued functions defined over $\mathbb{R}$. Consider this system of equations: \begin{cases} R' & = & f(t,R,J)\\ J' & = & g(t,R,J) \end{cases} The problem asks me to find a condition on $f$ and $g$ such that the above system have a solution (not necessarily unique).
My guess is that $f$ and $g$ being continuous is sufficient. And my approach is to construct a sequence of functions $h_1$, $h_2$, ..., which are approximations using the Euler method. Then I want to prove that if the step size in the Euler method is sufficiently small, the sequence will converge to some kind of function and that function would be the solution.
However, I haven't quite figured it out yet. I hope someone would clarify 2 things:
- Is $f$ and $g$ being continuous sufficient?
- Would my approach work?
Thank you very much!