Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 8 / a / Solution

The Green's function is nonpositive, so $f\ge 0$ gives $y\le 0$. If also $f\le m$, thenThe integral solution therefore givesBoth bounds are attained, by the constant forcings $f=m$ and $f=0$, respectively. In particular, boundedness of the solution does not require it to approach zero.