sequence of solutions to an ODE

Hi all,

The premise is that I am rather ignorant about differential equations.
I have to solve the following problem.

I consider an initial value problem

x'= f(x, t, k),
x(0) = b(k),

with x, t and k reals, b(k) a function and k a parameter. Function f
is continuously differentiable on an open set D of R x R x (k_min,
k_max). Hence, for any value of the parameter k, a solution exists and
is unique by standard theorems. Let's call it x(t; k)
Now I consider a sequence {k_n} converging to k_min. I do not know
anything about function f when k=k_min. What can I say about the
induced sequence of solution {x(t; k_n)} ? Does it converge? If so,
what type of convergence?

Any suggestion will be very much appreciated.

Thx,

NaxLeo

.