- Show the Brouwer fixed point theorem is equivalent to the nonexistence of a continuous
retraction onto the boundary of B. That is, a function f : B→ ∂Bsuch that f is the identity on ∂B and continuous.
- Using the Jordan separation theorem, prove the invariance of domain theorem. Hint: You
might consider Band show f maps the inside to one of two components of ℝ
^{n}∖ f. Thus an open ball goes to some open set. - Give a version of Proposition 11.6.5 which is valid for the case where n = 1.
- It was shown that if f is locally one to one and continuous, f : ℝ
^{n}→ ℝ^{n}, andthen f maps ℝ

^{n}onto ℝ^{n}. Suppose you have f : ℝ^{m}→ ℝ^{n}where f is one to one and lim_{}→∞= ∞. Show that f cannot be onto. - Can there exist a one to one onto continuous map, f which takes the unit interval to the unit disk?
- Let m < n and let B
_{m}be the ball in ℝ^{m}and B_{n}be the ball in ℝ^{n}. Show that there is no one to one continuous map from B_{m}to B_{n}. Hint: It is like the above problem. - Consider the unit disk,
and the annulus

Is it possible there exists a one to one onto continuous map f such that f

= A? Thus D has no holes and A is really like D but with one hole punched out. Can you generalize to different numbers of holes? Hint: Consider the invariance of domain theorem. The interior of D would need to be mapped to the interior of A. Where do the points of the boundary of A come from? Consider Theorem 2.7.5. - Suppose C is a compact set in ℝ
^{n}which has empty interior and f : C → Γ ⊆ ℝ^{n}is one to one onto and continuous with continuous inverse. Could Γ have nonempty interior? Show also that if f is one to one and onto Γ then if it is continuous, so is f^{−1}. - Let K be a nonempty closed and convex subset of ℝ
^{n}. Recall K is convex means that if x,y ∈ K, then for all t ∈,tx +y ∈ K. Show that if x ∈ ℝ^{n}there exists a unique z ∈ K such thatThis z will be denoted as Px. Hint: First note you do not know K is compact. Establish the parallelogram identity if you have not already done so,

Then let

be a minimizing sequence,Now using convexity, explain why

and then use this to argue

is a Cauchy sequence. Then if z_{i}works for i = 1,2, consider∕2 to get a contradiction. - In Problem 9 show that Px satisfies the following variational inequality.
for all y ∈ K. Then show that

≤. Hint: For the first part note that if y ∈ K, the function t →^{2}achieves its minimum onat t = 0. For the second part,Explain why

and then use a some manipulations and the Cauchy Schwarz inequality to get the desired inequality.

- Establish the Brouwer fixed point theorem for any convex compact set in ℝ
^{n}. Hint: If K is a compact and convex set, let R be large enough that the closed ball, D⊇ K. Let P be the projection onto K as in Problem 10 above. If f is a continuous map from K to K, consider f∘P. You want to show f has a fixed point in K. - Suppose D is a set which is homeomorphic to B. This means there exists a continuous one to one map, h such that h= D such that h
^{−1}is also one to one. Show that if f is a continuous function which maps D to D then f has a fixed point. Now show that it suffices to say that h is one to one and continuous. In this case the continuity of h^{−1}is automatic. Sets which have the property that continuous functions taking the set to itself have at least one fixed point are said to have the fixed point property. Work Problem 7 using this notion of fixed point property. What about a solid ball and a donut? Could these be homeomorphic? - Suppose Ω is any open bounded subset of ℝ
^{n}which contains 0 and that f : Ω → ℝ^{n}is continuous with the property thatfor all x ∈ ∂Ω. Show that then there exists x ∈ Ω such that f

= 0. Give a similar result in the case where the above inequality is replaced with ≤. Hint: You might consider the function - Suppose Ω is an open set in ℝ
^{n}containing 0 and suppose that f : Ω → ℝ^{n}is continuous and≤for all x ∈ ∂Ω. Show f has a fixed point in Ω. Hint: Consider h≡ t+x for t ∈. If t = 1 and some x ∈ ∂Ω is sent to 0, then you are done. Suppose therefore, that no fixed point exists on ∂Ω. Consider t < 1 and use the given inequality. - Let Ω be an open bounded subset of ℝ
^{n}and let f,g : Ω → ℝ^{n}both be continuous such thatfor all x ∈ ∂Ω. Show that then

Show that if there exists x ∈ f

^{−1}, then there exists x ∈^{−1}. Hint: You might consider h≡f+ tand argue 0hfor t ∈. - Let f : ℂ → ℂ where ℂ is the field of complex numbers. Thus f has a real and imaginary part.
Letting z = x + iy,
Recall that the norm in ℂ is given by

=and this is the usual norm in ℝ^{2}for the ordered pair. Thus complex valued functions defined on ℂ can be considered as ℝ^{2}valued functions defined on some subset of ℝ^{2}. Such a complex function is said to be analytic if the usual definition holds. That isIn other words,

(11.10) at a point z where the derivative exists. Let f

= z^{n}where n is a positive integer. Thus z^{n}= p+ iqfor p,q suitable polynomials in x and y. Show this function is analytic. Next show that for an analytic function and u and v the real and imaginary parts, the Cauchy Riemann equations hold.In terms of mappings show 11.10 has the form

^{T}and h is given by h_{1}+ ih_{2}. Thus the determinant of the above matrix is always nonnegative. Letting B_{r}denote the ball B= Bshowwhere f

= z^{n}. In terms of mappings on ℝ^{2},Thus show

Hint: You might consider

where the a

_{j}are small real distinct numbers and argue that both this function and f are analytic but that 0 is a regular value for g although it is not so for f. However, for each a_{j}small but distinct d= d. - Using Problem 16, prove the fundamental theorem of algebra as follows. Let pbe a nonconstant polynomial of degree n,
Show that for large enough r,

>for all z ∈ ∂B. Now from Problem 15 you can conclude d= d= n where f= a_{n}z^{n}. - Suppose f : ℝ
^{n}→ ℝ^{n}satisfiesShow that f must map ℝ

^{n}onto ℝ^{n}. Hint: First show f is one to one. Then use invariance of domain. Next show, using the inequality, that the points not in fmust form an open set because if y is such a point, then there can be no sequenceconverging to it. Finally recall that ℝ^{n}is connected. - Suppose f : ℝ
^{n}→ ℝ^{n}is locally one to one and continuous and satisfies a coercivity conditionShow that then f must map onto ℝ

^{n}. Hint: By invariance of domain, you have fis open. Is this also closed? Now use connectedness considerations. - Suppose f : ℝ
^{2}→ ℝ^{2}is given byShow that det

≠0. By the inverse function theorem, this function is locally one to one. However, it misses the pointbecause= e^{x}. Does this contradict the above problem? - Suppose f : ℝ
^{n}→ ℝ^{n}satisfiesShow that f must map ℝ

^{n}onto ℝ^{n}. Hint: First show f is one to one. Then use invariance of domain. Next show, using the inequality, that the points not in fmust form an open set because if y is such a point, then there can be no sequenceconverging to it. Finally recall that ℝ^{n}is connected.

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