The following integral operator $K$ is self adjoint, bounded, with closed range on domain
$D(K) = L^2[0, 1]$:
$$Ku = \int_0^1 x^2 y u(y) dy$$
(a) [2pt] Find the nonzero eigenvalue $\lambda_0$ of $K$, and define the null space $N(K - \lambda_0 I)$.
(b) [2pt] Use the Fredholm Alternative to find all constants $\alpha$ such that
$$(K - \lambda_0 I)u = 5x^2 + 2x + \alpha$$
has a solution.
(c) [4pt] Consider the perturbed nonlinear integral equation:
$$(K - \lambda_0 I)u = \epsilon(u^3 - 9), \quad 0 < \epsilon < 1$$
with expansion $u(x) = u_0(x) + \epsilon u_1(x) + O(\epsilon^2)$. Use the Fredholm Alternative to
show that the $O(\epsilon)$ problem for $u_1$ has a solution only for a unique $u_0 \in N(K - \lambda_0 I)$.
