Math 618, Assignment 5
Due Wednesday, December 12, 5:00 pm
Remember to provide full reasoning for all answers!
1. Show the equation x3−x−1 = 0 has a root between 1 and 2 by forming a fixed point problem g(x) = x
(there are multiple ways to do this!) where gis a contraction on [0,1].
2. Let Vbe a closed subset of a Banach space Xand let T:V→Vbe a map. If Tnis a contraction for
some positive integer n, show that Thas a unique fixed point in V.
3. Consider the solution of a linear system of equations Ax =b, where Ais an invertible n×nmatrix,
and xand bare n×1 vectors by writing A=N−Mwhere Nand Mare n×nmatrices with N
invertible and then solving
Nx =Mx +b.
Define an iteration to solve this equation and derive a condition that guarantees convergence using the
Banach Fixed Point Theorem.
4. Use Schauder’s Fixed Point Theorem to prove that a linear integral operator Lwith a positive contin-
uous kernel satisfies Lu =λu for some positive λ.
5. Prove Theorem 7.2.2.
6. Let Fbe a Frechet differentiable function from a Hilbert space Xto R. The gradient of Fat xis a
vector v∈Xsuch that F0(x)h= (h, v) for all h∈X. Prove that such a vexists (it depends on x).
Compute the gradient for F(x) = (a, x)2for some a∈X.
7. Let X=C([0,1]). Select {ti}n
i=1, 0 ≤ti≤1, and {vi}n
i=1,vi∈X. For x∈X, define
F(x)(t) =
n
X
i=1
¡x(ti)¢2vi(t),0≤t≤1,
Compute the Frechet derivative of F.
8. Recall the definition of c0, the space of sequences converging to 0 with norm kxk= maxnkxnk. Prove
that the norm is Frechet differentiable at xif and only if there is a unique nsuch that |xn|=kx
.
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