Smoothness of solutions of nonlinear ODE
A Carleman class \(C^\infty_M\) of \(C^\infty\)-functions is defined as a set of functions \(f\) such that for some \(C\), \(R> 0\), we have \[ |f^{(k)}_{(t)}|\leq C\cdot R^K\cdot M_K,\quad k= 0,1,\dots, t\in I= [0,1], \] where \(M= \{M_K\}^\infty_0\) is a fixed sequence such that \(\log M_K\) is convex. Here, the following statement is proved: Let \(u\) be \(C^\infty\)-function on \([0,1]\) such that \(u'= \Phi(u,t)\), where \(\Phi(z,t)\) is an entire function of order 1, finite type in \(z\) and of a nonanalytic Carleman class in \(t\). Then \(u\) is in the same Carleman class. This is the extension of the authors' results in [Smoothness of solutions of a nonlinear ODE, Int. Equs. Operator Theory 44, No. 2, 149-171 (2002)].
nonlinear ordinary differential equations
scientific article; zbMATH DE number 1837396