Consider the operator
$$
T(f)=\sin (2 \pi x)+\lambda \int_{-1}^1 \frac{f(y)}{1+(x-y)^2} d y
$$
on the space of functions $C^0[-1,1]$ equipped with the sup-norm (3.3).
(a) Show that if $f \in C^0[-1,1]$, then so is $T(f)$.
(b) Find a $\lambda_o>0$ such that if $|\lambda|<\lambda_o$, then $T(f)$ is a contraction mapping, and if $|\lambda|>\lambda_0$, then it is not. (Hint: To show the second part it is sufficient to find a pair of functions, $f, g$, for which $\rho(T(f), T(g))>\rho(f, g)$.)
(c) Investigate the fixed point numerically. Start with $f(x)=0$, and then try several other initial states. Try values of $\lambda$ both smaller and larger than $\lambda_o$. (Hint: Numerical integration may be necessary.)