Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems

We prove that the system $\dot{\theta}(t) =\Lambda(t)\theta(t)$ , $t\in\mathbb{R}_{+}$ , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$ , $t\in\mathbb{R}_{+}$ , $\phi(0)=\theta_{0}$ as the approximate solutions of $\dot{\theta}(t)=\Lambda(t)\theta(t)$ , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with $\xi(0)=0$ , and $\Lambda(t)$ is a 2-periodic square matrix of order l.