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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.

This paper presents some Perron-type results for the nonuniform exponential stability of evolution families on the real line with nonuniform exponential growth. We will mention the notion of the admissibility of the pair $$( \mathcal {L}^p(X), \mathcal {L}^q(X))$$ , where $$ (p,q) \ne (1, \infty )$$ and the admissibility of the pair of Schäffer spaces to an evolution family. This notion will be used to obtain new results for the nonuniform exponential stability of an evolution family on $$ \mathbb {R}.$$ [ABSTRACT FROM AUTHOR]

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup $$\{T(t)\}_{t\geq 0}$$ , in terms of the admissibility of the pair $$(L^{p}({\mathbb{R}}_{+}, X), L^{q}({\mathbb{R}}_{+}, X))$$ . It is already known the equivalence between the $$(L^{p}({\mathbb{R}}_{+}, X), L^{q}({\mathbb{R}}_{+}, X))$$ -admissibility condition $$(1 \leq p \leq q \leq \infty$$ and $$(p, q) \neq (1,\infty))$$ and the hyperbolicity of a C 0-semigroup $$\{T(t)\}_{t\geq 0}$$ , when we assume a priori that the kernel of the dichotomic projector (denoted here by X 2) is T( t)-invariant and $$T(t)|_{X_2}$$ is an invertible operator. We succeed to prove in this paper that the admissibility of the pair $$(L^{p}({\mathbb{R}}_{+}, X), L^{q}({\mathbb{R}}_{+},X))$$ still implies the existence of an exponential dichotomy for a C 0-semigroup $${\bf T} = \{T(t)\}_{t\leq 0}$$ even in the general case where the kernel of the dichotomic projector, X 2, is not assumed to be T( t)-invariant. [ABSTRACT FROM AUTHOR]