The originality of Peano's 1886 existence theorem on scalar differential equations (Q2821968)

This paper revisits Giuseppe Peano's 1886 proof of the result that the continuity of a function \(f:[a,b]\times\mathbb R\rightarrow\mathbb R\) is a sufficient condition for an initial value problem \(y'=f(x,y)\), \(y(a)=c\), to have one or more local solutions. The authors provide a proof that is historically sensitive to Peano's work: it uses only those techniques and concepts that were available to Peano himself. The authors' goal is to emphasise the now-largely-forgotten methods employed by Peano, and also to give a proof that is more complete than the original, which has been criticised for lack of rigour. The paper begins with a detailed statement of the theorem in question, together with a brief outline of its historical development. A series of preparatory lemmas is proved, and then the theorem itself.