Total Stability of Scalar Differential Equations Determined from Their Limiting Functions

Stability properties of the zero solution of a scalar ordinary differential equation are studied using functions, called fences, which act as bounds to the solutions of the differential equation. It is shown, under simple restrictions, that these fences are bounds for the solutions of the perturbed differential equations as well. Hence one finds that total stability of the zero solution of a scalar ordinary differential equation is equivalent to the existence of these functions.