Constructive Solutions of Ordinary Differential Equations

Abstract

For Helmut Schwichtenberg, with thanks for his friendship and for his hospitality in Munich on numerous occasions over the past 15 years.A Bishop-style constructive analysis is given for the Peano existence theorem for solutions of the differential equation y = f (x, y) with specified initial condition. In particular, it is shown that the existence of a solution in the general case implies the omniscience principle LLPO, but that introducing an a priori hypothesis of uniqueness of a solution can enable the construction of the solution.

http://logicandanalysis.org/index.php/jla/article/viewFile/114/41

“…Although Picard's Theorem has thus a constructive core, the same cannot be said for Peano's Theorem. This theorem is inherently nonconstructive: it is equivalent to the nonconstructive Lesser Limited Principle of Omniscience [4,10]:…”

10.4115/jla.v3i0.114

“…This theorem is inherently nonconstructive: it is equivalent to the nonconstructive Lesser Limited Principle of Omniscience [4,10]:…”

10.4115/jla.v4i0.157

“…With this lemma at hand we can weaken the standard hypothesis of the approximate Brouwer fixed point theorem; we only require that f : [0, 1] n → [0, 1] n be uniformly sequentially continuous 6 : for all sequences (…”

“…We give an application of the approximate Schauder fixed point theorem for uniformly convex spaces (Corollary 10). A standard application of Schauder's fixed point theorem is in proving Peano's Theorem asserting the existence of solutions to particular differential equations: However, since the exact version of Peano's Theorem is constructively equivalent to LLPO (see [6], which also gives an alternative constructive proof of an approximate Peano's Theorem, [2] gives a proof that Peono's Theorem implies LLPO), we can only hope to prove an approximate version of Peano's Theorem.…”