Ordinary differential equations with a non-Lipschitz right-hand side
Consider the initial value problem \[ u_t(t, x)=f(t, u(t, x)), u(0, x)=x\in F,\tag{1} \] where \(f(t, x)=(f^1, f^2, \dotsc, f^m)(t, x)\in {\mathcal C}(I_{\tau}\times \overline{Q}, \mathbb R^m)\), \(\sup_{(t, x)\in I_{\tau}\times \overline{Q}}||f(t, x)||<\infty\), \(Q\) is an open domain in \(\mathbb R^m\), \(F\subset Q\) is a compact set, \(I_{\tau}=[0, \tau)\). The author proves that problem (1) has a general solution \(u(t, x)\) such that the functions \(x\mapsto u(t, x)\) and \(u_t(t, x)\) lie in \({\mathcal B}(F, {\mathcal C}(I_T, \mathbb R^m))\), and, if \(u\in {\mathcal B}(F, {\mathcal C}(I_T, \mathbb R^m))\) is a general solution of (1), then the map \(t\mapsto u(t, x)\) is in \({\mathcal C}^1(I_T, B(F))\). Here, \({\mathcal B}(F, {\mathcal C}(I_T, \mathbb R^m))\) is the set of Borel measurable functions of \(F\) to \({\mathcal C}(I_T, \mathbb R^m)\) and \(B(F)\) is the space of \(F\)-bounded and Borel measurable mappings \(v:F\to \mathbb R^m\), \(T\in (0, \tau]\).
scientific article; zbMATH DE number 6106511
scientific article