Peano's 1886 existence theorem on first-order scalar differential equations: A review

In 1886 Giuseppe Peano presents the first proof of the existence of a solution of an initial value problem y′=f(x,y), y(a)=b, under the assumption of the continuity of the function f. The present paper gives a detailed description of Peano’s original statements and proofs, filling gaps, clarifying obscure points and avoiding ambiguous use of mathematical symbols. Peano’s 1886 work is compared with later papers of Peano himself as well as of Mie (Math Ann 43:553–568, 1893), Osgood (Monatsh Math Phys 9:331–345, 1898) and Perron (Math. Ann. 76:471–484, 1915).