# Inverse/Implicit Function Theorems

• Jan 29th 2009, 07:09 AM
frillth
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?
• Jan 29th 2009, 02:43 PM
Laurent
Quote:

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 $y\in (1-s)B$ is the image of some point in $x\in B$. 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 $B$.

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 $F:x\mapsto f(x)-x$ to prove that $f$ is one-to-one on $B$ (hint: the differential of $F$ is $df-I$...)

2) prove that the differential is not singular. (This is standard. If you've never seen this, justify the following: if $A$ is a linear map such that $\|I-A\|<1$, then the series $\sum_{n=0}^\infty (I-A)^n$ converges and its sum $B$ satisfies $AB=I$ so that $A$ is invertible)

3) apply the global inverse function theorem.

4) ...