My understanding is that for what is being asked here is much more elementary and does not need the Lowasiewicz inequality. Rather, it is a simple application of the implicit function theorem in the analytic category. See 
http://mathoverflow.net/questions/73388/analytic-implicit-function-theorem
for a discussion in the finite dimensional setting. One needs to generalize that to maps between Banach manifolds and that is done in an old Springer lecture notes volume of Lowasiewicz for example (but perhaps also in Lang or some other more elementary reference?)