Peano’s 1886 existence theorem on first-order scalar differential equations: a review

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• 作者：Gabriele H. Greco ; Sonia Mazzucchi
• 关键词：Peano ; History of mathematics ; Ordinary differential equations ; Existence of solutions
• 刊名：Bollettino dell'Unione Matematica Italiana
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：9
• 期：3
• 页码：375-389
• 全文大小：481 KB
• 刊物类别：Mathematics, general;
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：2198-2759
• 卷排序：9

文摘

In 1886 Giuseppe Peano presents the first proof of the existence of a solution of an initial value problem \(y^\prime =f(x,y)\), \(y(a)=b\), 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).