STABILITIES IN DIFFERENTIAL SYSTEMS

We consider the nonlinear nonautonomous differential system $$(1) x' = f(t,x), x(t_0) = x_0,$$ where $f \in C(R^+ \times R^n, R^n)$ and $R^+ = [0, \infty}$. We assume that the Jacobian matrix $f_x = \partail f/\partial x$ exists and is continuous on $R^+ \times R^n$ and that $f(t,0) \equiv 0$. The symbol $$\mid$\cdot$\mid$$ denotes arbitary norm in $R^n$.