DSpace at VNU: Some properties almost - periodic solution of linear differential equations

V N U J O U R N A L OF SC IEN CE, M a th e m a tics - Physics T X X IỈ, N - 2006 S O M E P R O P E R T I E S A L M O S T -P E R I O D I C S O L U T I O N O F L I N E A R D I F F E R E N T I A L E Q U A T IO N S T a Q uang H D epartm ent o f Mathematics, Vinh University A b s t r a c t Let R be the set of all real numbers, E be a Banach space and B bethe space of all almost-periodic functions / : R -> E We consider the equation x = Ar + /(t), (1) where A is a bounded linear operator on E As well-known that (see [1]), the equation (1) has an unique solution ip(t) € B for each / GỔ if and only if there existsR(iuj) = (iu>I - A)~l for all UJ IR, further this solution has the form J G(t - s)f(s)dsi oo v>(t)= (2) —oo where G(u) is the Green function defined by ~ -*-J- for u < “ ) = \i euAP: T + z “ >