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I can think of many situations in which the assumptions of the Implicit Function Theorem do not hold but most of the results of the theorem can be recovered. The example of f(x,a) = a + x^3 comes to mind. At (x,a) = (0,0) the derivative is zero and therefore the implicit function theorem cannot be used. On the other hand, there is a unique continuous solution given by x = (-a)^1/3 for all real a.

I am looking for a review of some literature in which the authors have either developed a theory or have worked with a specific cases in which they side-step the Implicit Function Theorem to recover similar results.

My particular situation is as follows:

I have a nonlinear operator between a Banach spaces whose Frechet derivative is injective but is not bounded below and therefore it cannot have a bounded inverse. I am looking to show that a zero of the function can be continued for a small parameter value but because the derivative is not invertible at this specific point I cannot use the Implicit Function Theorem.

I am familiar with Nash-Moser Theory although through quite a bit of work I am convinced it cannot be applied in this situation.