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

We prove that the system θ̇(t) =Λ(t)θ(t) , t₊ , 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 φ̇(t)=Λ(t)φ(t)+eiγ tξ(t) , t₊ , φ(0)=θ₀ as the approximate solutions of θ̇(t)=Λ(t)θ(t) , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with ξ(0)=0 , and Λ(t) is a 2-periodic square matrix of order l.