Integral Equations on Function Spaces and Dichotomy on the Real Line

Abstract.

The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted ${\user1{\mathcal{T}}}({\user2{\mathbb{R}}})$ and ${\user1{\mathcal{H}}}({\user2{\mathbb{R}}})$ and we prove that if V,W are Banach function spaces with $V \in {\user1{\mathcal{T}}}({\user2{\mathbb{R}}})$ and $W \in {\user1{\mathcal{H}}}({\user2{\mathbb{R}}})$, then the admissibility of the pair $(W({\user2{\mathbb{R}}},X),V({\user2{\mathbb{R}}},X))$ for an evolution family ${\user1{\mathcal{U}}} = \{ U(t,s)\}_t \geq s$ implies the uniform dichotomy of ${\user1{\mathcal{U}}}$. In addition, we consider a subclass ${\user1{\mathcal{W}}}({\user2{\mathbb{R}}}) \subset {\user1{\mathcal{H}}}({\user2{\mathbb{R}}})$ and we prove that if ${\user1{\mathcal{W}}} \in {\user1{\mathcal{W}}}({\user2{\mathbb{R}}})$, then the admissibility of the pair $(W({\user2{\mathbb{R}}},X),V({\user2{\mathbb{R}}},X))$ implies the uniform exponential dichotomy of the family ${\user1{\mathcal{U}}}$. This condition becomes necessary if $V \subset W$. Finally, we present some applications of the main results.