- Show that if y
_{1},,y_{r}in ℝ^{p}∖ f, then ifhas the property that_{∞}<min_{i≤r}dist, thenfor each y

_{i}. Hint: Consider for t ∈,f+ t− y_{i}and homotopy invariance. - Show the Brouwer fixed point theorem is equivalent to the nonexistence of a continuous
retraction onto the boundary of B.
- 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 16.6.5 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 2.11.3. - 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 10 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 11 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 8 using this notion of fixed point property. What about a solid ball and a donut? Could these be homeomorphic? - There are many different proofs of the Brouwer fixed point theorem. Let l be
a line segment. Label one end with A and the other end B. Now partition
the segment into n little pieces and label each of these partition points with
either A or B. Show there is an odd number of little segments with one end
labeled with A and the other labeled with B. If f :l → l is continuous, use
the fact it is uniformly continuous and this little labeling result to give a
proof for the Brouwer fixed point theorem for a one dimensional segment.
Next consider a triangle. Label the vertices with A,B,C and subdivide this
triangle into little triangles, T
_{1},,T_{m}in such a way that any pair of these little triangles intersects either along an entire edge or a vertex. Now label the unlabeled vertices of these little triangles with either A,B, or C in any way. Show there is an odd number of little triangles having their vertices labeled as A,B,C. Use this to show the Brouwer fixed point theorem for any triangle. This approach generalizes to higher dimensions and you will see how this would take place if you are successful in going this far. This is an outline of the Sperner’s lemma approach to the Brouwer fixed point theorem. Are there other sets besides compact convex sets which have the fixed point property? - Using the definition of the derivative and the Vitali covering theorem, show that
if f ∈ C
^{1}and ∂U has n dimensional measure zero then falso has measure zero. (This problem has little to do with this chapter. It is a review.) - 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,

(16.11) 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 16.11 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 19, 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 18 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.

Download PDFView PDF