Buse, ConstantinJitianu, Oprea2020-09-142020-09-142003-02-11Buse, C., & Jitianu, O. (2003). A new theorem on exponential stability of periodic evolution families on Banach spaces. <i>Electronic Journal of Differential Equations, 2003</i>(14), pp. 1-10.1072-6691https://hdl.handle.net/10877/12605We consider a mild solution v_f (·, 0) of a well-posed inhomogeneous Cauchy problem v̇(t) = A(t)v(t) + ƒ(t), v(0) = 0 on a complex Banach space X, where A(·) is a 1-periodic operator-valued function. We prove that if vƒ (·, 0) belongs to AP0 (ℝ₊, X) for each ƒ ∈ AP0(ℝ₊, X) then for each x ∈ X the solution of the well-posed Cauchy problem u̇(t) = A(t)v(t), u(0) = x is uniformly exponentially stable. The converse statement is also true. Details about the space AP0(ℝ₊, X) are given in the section 1, below. Our approach is based on the spectral theory of evolution semigroups.Text10 pages1 file (.pdf)enAttribution 4.0 InternationalAlmost periodic functionsExponential stabilityPeriodic evolution families of operatorsIntegral inequalityDifferential inequality on Banach spacesArticle