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Thread: Inverse/Implicit Function Theorems

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by frillth View Post
    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 $\displaystyle y\in (1-s)B$ is the image of some point in $\displaystyle x\in B$. 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 $\displaystyle B$.

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

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

    3) apply the global inverse function theorem.

    4) ...
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