Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 3 / c

Taylor's theorem and $g(0)=g^{\prime }(0)=0$ giveCombining these with the product estimate in a Hölder space yieldsWritingand using the same product estimate gives

Let $G=L^{-1}$ and use the bound from part (b),On the closed ball ${\mathcal{B}}_{\epsilon }\subset C_0^{2,\alpha }(B)$ defineThe Hölder product estimate and part (i) giveandChoose ${\epsilon }_0$ so that $2M{\epsilon }_0<1/2$, then choose ${\delta }_0$ so that the remaining terms make $T$ preserve ${\mathcal{B}}_{{\epsilon }_0}$ and have contraction constant less than one. The Banach fixed-point theorem gives a unique $u$ in that ball satisfying