Starting with the difference equation:
$$f(x(t+dt),t+dt)= (1-a)f(x(t),t) + a f(x(t+dt),t)$$
where $x(0)$ is given and positive, $a\in(0,1)$, $f(0,t)=0$, and $f$ is increasing in both arguments. This clearly defines a positive, decreasing sequence.
I would like to know conditions such that as I decrease the step size (i.e. $dt\to 0$), that this sequence converges to the solution of the corresponding differential equation:
$$x'(t) = -\frac{f_t(x,t)}{(1-a)f_x(x,t)}$$
The domain can be taken to be $[0,x_0]×[0,1]$. $f$ is nonnegative, bounded, and strictly increasing in both arguments, differentiable, etc.
Ideally would also like to handle the case where $x_0 = \infty$.
Thanks for any suggestions on how to approach this.
Any references or pointers to resources (for a relative beginner!) would be appreciated.