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

For $s>t$, the coefficient bound gives $A(s)-A(t)\ge C_1(s-t)$. Thus for $0\le s\le t$ the positive kernel satisfiesThe starting Picard iterate is continuous and has supremum norm $M_0=|b|$. If the $n$th Picard iterate is bounded and continuous, its integral defines another continuous function, andTaking the supremum norm and applying mathematical induction proves boundedness of every iterate and

Put $B=|b|>0$ and $\gamma =4B/C_1<1$. Initially $M_0=B<2B$. If $M_n<2B$, the preceding supremum norm estimate givesHence mathematical induction gives

The difference of two successive Picard iterates factors asThe preceding bound and the kernel estimate therefore giveMoreover $D_1:=\Vert y_1-y_0{\Vert }_{\infty }\le B^2/C_1$. Summing the resulting geometric series, for $m>n$,Thus there is a uniform limit $y$ on $[0,\infty )$, continuous by the uniform limit theorem, with $\Vert y{\Vert }_{\infty }\le 2B$. The integral passes to this limit becauseTherefore $y$ satisfies the Volterra integral equation, and hence the initial value problem. This proves global existence by Global Picard iteration with a decaying kernel. The same estimate gives the useful error boundIf $b=0$, every iterate is zero and the convergence assertion is immediate.