On the reducibility of nonlinear differential equations (Q1589389)

The author investigates sufficient conditions for the reducibility of equations of the form \[ \frac{dx}{dt} = f(t,x), \] with \( f \in X\), \(f(t,0) \equiv 0, \) and \( X = C^{(p,q)}([T,+\infty) \times\mathbb R^n,\mathbb R^n) \) is the space of all vector functions of dimension \(n,\) which are defined on the set \([T,+\infty) \times\mathbb R^n\), \(p\) times continuously differentiable with respect to variable \(t\), \(p \geq 0,\) and \(q\) times continuously differentiable with respect to the components of the vector \( x\), \(q \geq 1.\) In the first Lyapunov method there is the notion of the Lyapunov transformation and the Erugin theorem (see \textit{B. P. Demidovich} [Lectures of the mathematical theory of stability. Moscow: Nauka (1967; Zbl 0155.41601)] gives the criteria for the reducibility of two linear homogeneous differential equations in the context of the Lyapunov transformation. The author extends the concept of Lyapunov transformation and defines the notion of Lyapunov group of transformations. The definition is used here for the proof of two theorems on the reducibility and one theorem on the stability. The author's theorem 2 contains the Erugin theorem as a special case.