The Asymptotic Bounds of Solutions of Linear Delay Systems

Our aim is to obtain conditions under which every solution y of the vector functional differential equation y9 x.sA x.y t x..qB x.y x., xg w x0 , `. can be estimated in terms of a solution v of a scalar functional equation a x.v t x..qbv x.s0, xg w x0 , `., where a x. is a suitable real positive function and b is a suitable real negative constant. Moreover, if B x. is diagonal matrix, we derive more precise estimates of y in terms of a solution n of a vector functional equation Aq x.n t x..qB x.n x.s0, xg w x0 , `., where the elements of Aq x. are formed by the absolute values of the correspond- ing elements of A x.. These asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems.