Admissibility and nonuniform exponential dichotomy on the half-line

The aim of this paper is to deduce new conditions for the existence of the nonuniform exponential dichotomy of evolution families on the half-line. We consider an evolution family having a nonuniform exponential growth and we associate to it an input–output equation. We prove that the admissibility of the pair (Cb(R+,X),Lp(R+,X)) with respect to this equation implies the existence of a nonuniform exponential dichotomy. We also present an illustrative example which shows that, generally, the converse implication is not valid in the nonuniform case. Finally, we give an application to the case of uniform exponential dichotomy.