Pregunta: Este problema tiene que ver con EL TEOREMA DEL MAPEO DE CONTRACCIONES. Una SUGERENCIA más extensa: reformule el problema como una ecuación integral no lineal multiplicando por una función de Green e integrando. Además,

Exercise 3.7 Prove that there is a unique solution of the following nonlinear BVP when the constant $\lambda$ is sufficiently small, $\begin{array}{l}-u^{\prime \prime }+\lambda \sin u=f(x),\\ u(0)=0,u(1)=0.\end{array}$ Here, $f:[0,1]\to \mathbb{R}$ is a given continuous function. Write out the first few iterates of a uniformly convergent sequence of approximations, beginning with $u_0=0$. HinT. Reformulate the problem as a nonlinear integral equation.