Past exam of the differential equations 1 course of the University of Oxford / 2023 / 1 / c / ii

Existence was proved by Picard iteration. If $y$ and $v$ are any two continuous solutions with the same $b\ne 0$, the previous part makes both bounded by $2B$. Subtracting their Volterra integral equations givesSince $\gamma <1$, this forces $y=v$.

For $b=0$, zero is a solution. On any finite interval $[0,T]$, another continuous solution has a finite bound $K_T={\max }_{[0,T]}|y|$. Since $K(t,s)\le 1$, its integral equation impliesThe Gronwall inequality gives $y=0$ on every such interval. Hence