Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

Abstract

We establish conditions for the existence of an invariant set of the system of differential equations $$\frac{dφ}{dt} = a(φ),\quad \frac{dx}{dt} = P(φ)x + F(φ,x),$$ where $a: Φ → Φ, P: Φ → L(X, X)$, and $F: Φ × X→X$ are continuous mappings and $Φ$ and $X$ are finite-dimensional Banach spaces.