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

摘要

We prove that the system θ ˙ ( t ) = Λ ( t ) θ ( t ) $dot{heta}(t) =Lambda(t)heta(t)$ , t ∈ R + $tinmathbb{R}_{+}$ , 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 ϕ ˙ ( t ) = Λ ( t ) ϕ ( t ) + e i γ t ξ ( t ) $dot{phi}(t)=Lambda(t)phi(t)+e^{igamma t}xi(t)$ , t ∈ R + $tinmathbb{R}_{+}$ , ϕ ( 0 ) = θ 0 $phi(0)=heta_{0}$ as the approximate solutions of θ ˙ ( t ) = Λ ( t ) θ ( t ) $dot{heta}(t)=Lambda(t)heta(t)$ , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with ξ ( 0 ) = 0 $xi(0)=0$ , and