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Name:
Math 5410 Prelim August, 7, 2009
Choose 5 out of the 6 questions.
(1a) State and prove an existence and uniqueness theorem for the equation d^2 x dt^2 +^ f^ (x) = 0 with initial conditions^ x(0) =^ a^ and^ x
′(0) = b under the as-
sumption that f and its partial derivatives are continuous. (You can assume the Contraction Mapping Theorem).
(1b) Let a = b = f (0) = 0 in part (a). Can x(t) = t^3 be a solution to part (a)? Explain.
(2a) Find the Green’s function G(x, y) for the operator A where
Au = −u′′^ + u
with u′(0) = u′(1) = 0. (2b) Define T : L^2 (0, 1) → L^2 (0, 1) such that for any f ∈ L^2 (0, 1),
(T f )(x) =
∫ (^1)
0
G(x, y)f (y) dy.
Explain what spectral theorem is and why it is applicable. (2c) Show that ‖T ‖ = max{|λ| : λ is an eigenvalue of T}. (2d) Compute ‖T ‖. (hint: find eigenvalues of A).
(3) Let
U (x, y) =
1 , if 0 ≤ x ≤ 1 , y ≥ 0 ,
0 , otherwise,
Compute its distributive derivative DxyU in R^2.
(4) Let H be a Hilbert space and K : H → H is a linear, bounded, compact operator. Define A = I + K. Show that if A is injective, then it is surjective.
(5) Let H be a Hilbert space and A : H → H is compact. Show that (a) xn ⇀ x weakly implies Axn → Ax. (b) The operator norm of A is attained.
(6a) Let K be a closed convex set in a Hilbert space X. Let x ∈ X and let y be the point of K closest to x. Prove that Re〈x − y, v − y〉 ≤ 0 for all v ∈ K, where Re denotes the real part. (6b) For each x in X, we use P x to denote the point of K closest to x. Using part (a) or otherwise, prove that
‖P x − P z‖ ≤ ‖x − z‖.
