Multivariable Advanced Calculus

290 BROUWER DEGREE11. 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 closedball, D (0, R) ⊇ K. Let P be the projection onto K as in Problem 10 above. If fis a continuous map from K to K, consider f◦P . You want to show f has a fixedpoint in K.12. Suppose D is a set which is homeomorphic ( to B (0, 1). ) This means there exists acontinuous one to one map, h such that h B (0, 1) = D such that h −1 is alsoone to one. Show that if f is a continuous function which maps D to D then f hasa 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 thatcontinuous functions taking the set to itself have at least one fixed point are saidto have the fixed point property. Work Problem 6 using this notion of fixed pointproperty. What about a solid ball and a donut?13. There are many different proofs of the Brouwer fixed point theorem. Let l bea line segment. Label one end with A and the other end B. Now partition thesegment into n little pieces and label each of these partition points with either Aor B. Show there is an odd number of little segments with one end labeled with Aand the other labeled with B. If f :l → l is continuous, use the fact it is uniformlycontinuous and this little labeling result to give a proof for the Brouwer fixedpoint theorem for a one dimensional segment. Next consider a triangle. Label thevertices with A, B, C and subdivide this triangle into little triangles, T 1 , · · · , T min such a way that any pair of these little triangles intersects either along an entireedge or a vertex. Now label the unlabeled vertices of these little triangles witheither A, B, or C in any way. Show there is an odd number of little triangleshaving their vertices labeled as A, B, C. Use this to show the Brouwer fixed pointtheorem for any triangle. This approach generalizes to higher dimensions and youwill see how this would take place if you are successful in going this far. This is anoutline of the Sperner’s lemma approach to the Brouwer fixed point theorem. Arethere other sets besides compact convex sets which have the fixed point property?14. Using the definition of the derivative and the Vitali covering theorem, show thatif f ∈ C 1 ( U,R n) and ∂U has n dimensional measure zero then f (∂U) also hasmeasure zero. (This problem has little to do with this chapter. It is a review.)15. Suppose Ω is any open bounded subset of R n which contains 0 and that f : Ω → R nis continuous with the property thatf (x) · x ≥ 0for all x ∈ ∂Ω. Show that then there exists x ∈ Ω such that f (x) = 0. Give asimilar result in the case where the above inequality is replaced with ≤. Hint:You might consider the functionh (t, x) ≡ tf (x) + (1 − t) x.16. Suppose Ω is an open set in R n containing 0 and suppose that f : Ω → R n iscontinuous and |f (x)| ≤ |x| for all x ∈ ∂Ω. Show f has a fixed point in Ω. Hint:Consider h (t, x) ≡ t (x − f (x)) + (1 − t) x for t ∈ [0, 1] . If t = 1 and some x ∈ ∂Ωis sent to 0, then you are done. Suppose therefore, that no fixed point exists on∂Ω. Consider t < 1 and use the given inequality.17. Let Ω be an open bounded subset of R n and let f, g : Ω → R n both be continuoussuch that|f (x)| − |g (x)| > 0