Thread: Inverse/Implicit Function Theorems

Hey guys, I need to solve the following problem:

Let B=B(0,r) be an open ball of radius r centered at the origin in R^n. Suppose U is an open subset of R^n containing the closed ball of radius r centered at the origin, f is a function from U to R^n that is differentiable, f(0) = 0, and ||Df(x) - I|| <= s < 1 for all x in the open ball. Prove that if ||y|| < r(1-s), then there is an x in the open ball such that f(x) = y.

I'm pretty sure that this problem uses the inverse and implicit function theorems, but I'm not sure how to start it, and I don't really have any idea what to do with the I that is being subtracted from the derivative matrix. Can somebody please get me pointed in the right direction?

Originally Posted by frillth

Hey guys, I need to solve the following problem:

Let B=B(0,r) be an open ball of radius r centered at the origin in R^n. Suppose U is an open subset of R^n containing the closed ball of radius r centered at the origin, f is a function from U to R^n that is differentiable, f(0) = 0, and ||Df(x) - I|| <= s < 1 for all x in the open ball. Prove that if ||y|| < r(1-s), then there is an x in the open ball such that f(x) = y.

I'm pretty sure that this problem uses the inverse and implicit function theorems, but I'm not sure how to start it, and I don't really have any idea what to do with the I that is being subtracted from the derivative matrix. Can somebody please get me pointed in the right direction?

You wish to prove that any point is the image of some point in . For that, you need the global inverse function theorem. The inverse function theorem would indeed only give local inversion: "there is an open set..." and we don't know about inversion on the whole ball .

This theorem requires two hypotheses: the function is one-to-one and its differential is invertible. Here are the first steps for a proof:

1) apply the mean value theorem to to prove that is one-to-one on (hint: the differential of is ...)

2) prove that the differential is not singular. (This is standard. If you've never seen this, justify the following: if is a linear map such that , then the series converges and its sum satisfies so that is invertible)

3) apply the global inverse function theorem.

4) ...