Nonuniform exponential stability for evolution families on the real line

Abstract

The purpose of the present paper is to investigate the problem of nonuniform exponential stability of evolution families on the real line using the input-output technique known in the literature as the Perron method for the study of exponential stability. In this manuscript we describe an evolution family on the real line and we present sufficient conditions for the nonuniform exponential stability of an evolution family on the real line that does not have exponential growth. In 1974 M.G. Krein and J.L. Daleckij in [5, Theorem 4.1, p. 81] shows that if A ∈ B(X) then σ(A) ∩ iR = ∅ if and only if the differential equatioṅ x(t) = Ax(t) + f (t) has an unique solution x ∈ C, for all f ∈ C, where C represents the Banach space of the continuous and bounded functions on R and σ(A) represent the spectrum of the operator A.