THE UNIVERSITY OF QUEENSLAND
Sample Examination, May, 1999
MP479
ADVANCED ANALYSIS
(UNIT COURSES)
Time:     Three (3) Hours for working
Ten minutes for perusal before examination begins
Candidates may attempt ALL questions.
The values of parts of each question are shown in square brackets.
Pocket calculators allowed.

NOTATION: If X is a Hilbert or a Banach space then W Ì X denotes a bounded open set and B_r the sphere {x Î X  : ||x|| < r}. If S Í X then ¶S denotes its boundary and `S denotes its closure.

If X is a Hilbert space and x,y Î X let áx, yñ denote the usual inner product. Thus ||x||² = áx, xñ. Moreover, if X = Rⁿ, and x = (x₁, ..., x_n), y = (y₁, ..., y_n) where x_i, y_i Î R, i £ i £ n, then áx, yñ = å_{i = 1}ⁿ x_iy_i.

 

1.

(a)

Let W Í Rⁿ be a bounded open set, f Î C¹( Rⁿ; Rⁿ) and p Î Rⁿ satisfy p Ï f(¶W) and det f¢(x) ¹ 0 for all x with f(x) = p.
Give the weighted sum formula for the Brouwer degree d(f, W, p).
If d(f, W, p) = m show that f(x) = p has at least m solutions x Î W.
Give an example of a function g satisfying g(0) = 0, det g¢(0) = 0,
g(x) ¹ 0 for all x ¹ 0 and d(g, W, 0) = 2. [9]

(b)

Let W Í Rⁿ be a bounded, open, symmetric set with 0 Î W. Let f Î C¹( Rⁿ; Rⁿ) satisfy f is an odd function(that is, f(x) = - f(-x) for all x), 0 Ï f(¶W), and det f¢(x) ¹ 0 for all x.
Show that d(f, W, 0) is an odd number. [7]

(c)

Let F  : R²® R² be given by

If W = {(x, y) Î R² : x²+y² < 4} find d(F, W, (0, 1)). [5]

 

2

(a)

List the basic axioms of Brouwer degree. If f Î C(`B<sub>1</sub>;`B₁) use these properties to show f has a fixed point. [10]

Questions 2 continued next page COPYRIGHT RESERVED TURN OVER

2
Sample Examination - MP479 - Advanced Analysis
continued.

 

2.

(b)

Let W Í Rⁿ be a bounded open set 0 Î W and f Î C(`W ; Rⁿ). If f(x) ¹ lx for all x Î ¶W and l < 1, show that f has a fixed point in W. [7]

(Hint: Consider the homotopy H(t,x) = (2t-1)x-tf(x).)

(c)

Let f Î C( Rⁿ; Rⁿ). If <x-y, f(x)-f(y)ñ ³ ||x-y||² for all x,y Î Rⁿ show that f is one-to-one and onto Rⁿ. [10]

3.

(a)

Define what is meant by the following:
A Banach space (B,||·||);
A continuous function f : B® B;
A bounded linear mapping L : B® B;
Sequential compactness. [7]

(b)

Prove that a linear mapping L : B® B is continuous if and only if it is bounded. [7]

(c)

Prove that a compact set is bounded. [5]

(d)

Let (B,||·||) be a Banach space and K Í B be a compact subset of B. If f  : K® B is continuous show that f(K) is compact. [10]

4.

(a)

Let B be a Banach space, W Í B be bounded open and f Î C(`W; B) satisfy f(`W) Í K for some compact set K. If there is x₀ Î W such that f(x) ¹ lx+(1-l)x₀ for all x Î ¶W and l > 1, show that f has a fixed point in W. [9]

(Hint: You may assume without proof that d(I-x₀, W, 0) = 1, where I(x) = x for all x is the identity mapping.)

(b)

Let H be a Hilbert space, W Í H be a bounded open subset, 0 Î W and
f Î C(`W ; H) satisfy f(`W) Í K for some compact set K. If

for all x Î ¶W, show that f has a fixed point in W. [9]

(Hint: You may assume without proof that d(I,W,0) = 1, where I(x) = x for all x is the identity mapping.)

5.

(a)

State Schauder's Fixed Point Theorem. [3]

(b)

Let f [0,1]× R® R be continuous and assume that |f(x,y)| £ k for all (x,y) Î [0,1]× R and some constant k. Show that there is a solution y Î C([0,1]) of

[10]

Questions 5(c)-7 see next page TURN OVER

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Sample Examination - MP479 - Advanced Analysis
continued.

5. Continued

(c)

Let B be a Banach Space and F  : B®B be completely continuous. Show that either there is x Î B such that

or the set

is unbounded. [9]

6.

(a)

Let W Ì Rⁿ be bounded open, f,g  : `W -> Rⁿ be continuous, and

Show that f(x) = lg(x), has solutions (l₁, x₁) and (l₂, x₂) with
l₁ < 0 < l₂ and x₁, x₂ Î ¶W. [8]

(b)

Let K Ì Rⁿ be a cone and W Ì Rⁿ be bounded open with 0 Î W. Let
f  : Rⁿ -> K\{0} be continuous. Show there is l > 0 and x Î ¶WÇK such that f(x) = lx. [8]

(Hint: Compute d(f, W, 0) and d(g, W, 0), where g(x) = x for all x.)

(c)

Let W Í Rⁿ be bounded open, 0 Î W and V Í Rⁿ be a proper subspace. If
f  : Rⁿ -> V is continuous and 0 Ï f(¶W), show that there is x Î ¶WÇV and l ³ 0 such that f(x) = lx. [9]

7.

(a)

State and prove the Banach Contraction Mapping Principle. [12]

(b)

Show that there is a unique continuous function f on [0,1] such that

for all x in [0,1]. [10]

(Hint: You may assume without proof that 8s/(4+s²)² £ 1/2.)