Heyting valued considerations on some fundamental existence theorems in modern analysis
\textit{C. Rousseau} [Lect. Notes Math. 753, 623-659 (1979; Zbl 0433.32003)] has shown that an intuitionistic division theorem in one variable, interpreted in the Heyting valued set theory \(V^{(\Omega)}\) with \(\Omega\) being the topology \({\mathfrak O}({\mathbb{C}}^{n-1})\) of \({\mathbb{C}}^{n-1}\), gives rise to the classical Weierstrass division theorem in n variables. Following these lines, we will show that the inverse function theorem, demonstrated intuitionistically and interpreted in the Heyting valued set theory \(V^{(\Omega)}\) with \(\Omega\) being the topology of some appropriately chosen space of parameters, is nearly the classical implicit function theorem. Similarly it will be demonstrated that Cauchy's celebrated local existence theorem for ordinary differential equations, established intuitionistically and interpreted in the Heyting valued set theory \(V^{(\Omega)}\) with \(\Omega\) being the topology of some appropriately chosen space of parameters, gives rise to the classical theorems concerning the smooth dependence of local solutions on parameters and on initial conditions.
Cauchy's local existence theorem for ordinary differential equations
scientific article; zbMATH DE number 4085638
scientific article