Total stability of scalar differential equations determined from their limiting functions (Q5935541)

Stability properties of the zero solution to the scalar ordinary differential equation (1) \(\frac{dx}{dt}=f(t,x)\), where \(f\) does not satisfy the Lipschitz condition in \(x\) uniformly in \(t\), are studied using functions, called fences, which act as bounds to the solutions to the differential equation. It is shown, under simple restrictions, that these fences are bounds for the solutions to the perturbed differential equations as well. Hence, one finds that the total stability of the zero solution to equation (1) is equivalent to the existence of these functions. The paper is well written and contains a number of interesting examples.