Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems

Abstract

In this paper, the concepts of Hyers–Ulam stability are generalized for non-autonomous linear differential systems. We prove that the k-periodic linear differential matrix system Z˙(t)=A(t)Z(t),t∈R is Hyers–Ulam stable if and only if the matrix family L=E(k,0) has no eigenvalues on the unit circle, i.e. we study the Hyers–Ulam stability in terms of dichotomy of the differential matrix system Z˙(t)=A(t)Z(t),t∈R. Furthermore, we relate Hyers–Ulam stability of the system Z˙(t)=A(t)Z(t),t∈R to the boundedness of solution of the following Cauchy problem:{Y˙(t)=A(t)Y(t)+ρ(t),t≥0Y(0)=x−x0,where A(t) is a square matrix for any t∈R,ρ(t) is a bounded function and x,x0∈Cm.