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

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