I have a nonlinear, two point boundary value problem of the form $F(x, y(x), y'(x); \Omega ) = y''$ along with some boundary conditions of the form $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. The set $\Omega$ is a set of parameters which both alters the dynamics and changes the locations of the boundary points.

Now, suppose to I have a solution to this BVP (call it $y_\Omega(x))$, and I change the parameters in $\Omega$ a small bit (resulting in new boundary points) - am I guaranteed to again have a solution $Y_{\Omega'}$ that is "close" to the original function in some sense?

Really, I am asking if there is an implicit function theorem for boundary value problems.

Alternatively, since my BVP is derived via the Euler-Lagrange equations, I would be fine if there was an implicit function theorem for the minimum of an integral, as well.