caa a22 4500 467932077 CHVBK 20180406152946.0 cr unu---uuuuu 170328e20060101xx s 000 0 eng 10.1007/s00020-004-1347-z doi (NATIONALLICENCE)springer-10.1007/s00020-004-1347-z Exponential Dichotomy on the Real Line and Admissibility of Function Spaces [Elektronische Daten] [Adina Sasu, Bogdan Sasu] Abstract.: The purpose of this paper is to give characterizations for uniform exponential dichotomy of evolution families on the real line. We consider a general class of Banach function spaces denoted $$\mathcal{T}{\left( \mathbb{R} \right)}$$ and we prove that if $$B \in \mathcal{T}{\left( \mathbb{R} \right)}$$ with $$B\backslash L^{1} {\left( {\mathbb{R},\mathbb{R}} \right)} \ne \emptyset $$ and the pair $${\left( {C_{b} {\left( {\mathbb{R},X} \right)},B{\left( {\mathbb{R},X} \right)}} \right)}$$ is admissible for an evolution family $$\mathcal{U} = {\left\{ {U{\left( {t,s} \right)}} \right\}}_{{t \geqslant s}} ,$$ then $$\mathcal{U}$$ is uniformly exponentially dichotomic. By an example we show that the admissibility of the pair $${\left( {C_{b} {\left( {\mathbb{R},X} \right)},L^{1} {\left( {\mathbb{R},X} \right)}} \right)}$$ for an evolution family is not a sufficient condition for uniform exponential dichotomy. As applications, we deduce necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pairs $${\left( {C_{b} {\left( {\mathbb{R},X} \right)},L^{p} {\left( {\mathbb{R},X} \right)}} \right)},{\left( {C_{b} {\left( {\mathbb{R},X} \right)},C_{b} {\left( {\mathbb{R},X} \right)}} \right)},{\left( {C_{b} {\left( {\mathbb{R},X} \right)},C_{0} {\left( {\mathbb{R},X} \right)}} \right)}$$ and $${\left( {C_{b} {\left( {\mathbb{R},X} \right)},C_{0} {\left( {\mathbb{R},X} \right)} \cap L^{p} {\left( {\mathbb{R},X} \right)}} \right)},$$ with $$p \in \left[ {1,\infty } \right).$$ Birkhäuser Verlag, Basel, 2006 Evolution family nationallicence exponential dichotomy nationallicence admissibility nationallicence Banach function space nationallicence Sasu Adina Department of Mathematics Faculty of Mathematics and Computer Science, West University of Timişoara, Timisoara, Romania aut Sasu Bogdan Department of Mathematics Faculty of Mathematics and Computer Science, West University of Timişoara, Timisoara, Romania aut Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 54/1(2006-01-01), 113-130 0378-620X 54:1<113 2006 54 20 https://doi.org/10.1007/s00020-004-1347-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00020-004-1347-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Sasu Adina Department of Mathematics Faculty of Mathematics and Computer Science, West University of Timişoara, Timisoara, Romania aut NATIONALLICENCE 700 1- Sasu Bogdan Department of Mathematics Faculty of Mathematics and Computer Science, West University of Timişoara, Timisoara, Romania aut NATIONALLICENCE 773 0- Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 54/1(2006-01-01), 113-130 0378-620X 54:1<113 2006 54 20 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer