Total stability of oscillating differential equations (Q2770834)

Here, the authors study the total stability of the zero solution to the differential equation \(x' = f(t,x)\). The total stability of the zero solution to the differential equation means that small perturbations of both the initial condition as well as of the right-hand side in the differential equation imply that the solutions to the perturbed equation are small in the future. Such cases as \(f(t,x)\) does oscillate in \(t\) for each \(x\). It is also required that \(\lim\limits_{t\to\infty} = \frac{1 }{t}\int\limits_0^t f(s) ds<0\). Then, under some technical conditions, it is proved that the zero solution is totally stable. Examples are provided.