Asymptotic and locally asymptotic funnels for systems from the Osgood class for the inner product (Q1119799)

Let \(x\in R_+\), \(w=colon(y,z)\in R^ k\times R^{n-k}\), \(n>k\), \[ F(x,w)=colon(f(x,y,z),g(x,y,z)): R_+\times R^ k\times R^{n-k}\to R^ k\times R^{n-k\quad}. \] The author considers the nonlinear n dimensional system (1) \(w'=F(x,w)\) defined in the domain \[ S\{(x,w):\quad x\in (0,a),\quad \| y\| \in [0,b),\quad \| z\| \in [0,d)\}, \] (a, b, d, are fixed real numbers, \(\| \cdot \|\) is the Euclidean norm). The solution \(w(x;x^ 0,w^ 0)\) of (1) satisfying \(w(x^ 0;x^ 0,w^ 0)=w^ 0\) is called an 0-solution if there holds \(\lim \| w(x;x^ 0,w^ 0)\| =0,\) \(x\to +0\). The set M of such solutions is studied, its dimension, asymptotics and a representation is determined under the assumption that the scalar product for components of F(x,w) belongs to the Osgood class as defined by the author. This extends previous results of the author, A. F. Andreev and of N. A. Bodunov, where the mentioned scalar product belongs to the Lipschitz class.