Stability implications on the asymptotic behavior of nonlinear systems

Abstract

In this paper we generalize Bownds' Theorems (1) to the systems dY(t)dt=A(t)Y(t) and dX(t)dt=A(t)X(t)+F(t,X(t)). Moreover we also show that there always exists a solution X(t) of dXdt=A(t)X+B(t) for which limt→∞sup‖X(t)‖>o(=∞) if there exists a solution Y(t) for which limt→∞sup‖Y(t)‖>o(=∞).