Almost Periodic Solutions For Differential Equations With Delays

This dissertation deals with the following two kinds of differential equations with delays(?)(t)=F(t,x(t),{x([t-j])}_j=0 ~N),(1)(?)=r(t)N(t)(a(t)-b(t)N(t)-l(t,x_t))．(2)In the first part, by the proof of an another form of Kronecker's theorem, we obtain a new property for the module containment of the almost periodic functions as follows:Assume that f(t) and g(t) are almost periodic. If (?)α(?)Z such that T_αf=f, there existsα′(?)αsuch that T_α′g = g, then mod(g) (?) span(mod(f)∪{2π}).By employing the above property, we study the module containment of almost periodic solution for almost periodic differential equations with piecewise constant delays. Besides, for two kinds of special almost periodic differential equations of (1), we obtain sufficient conditions for the unique existence of almost periodic solution and its module containment. Our results improve the corresponding ones in some literature. In the second part, we first prove the uniform persistence for the solutions of Eq. (2) under some reasonable assumptions, then we obtain some sufficient conditions for the global attractivity of positive solutions of Eq.(2) as b(t)≠0 and b(t)≡0, respectively. Moreover, we prove the unique existence and global attractivity of positive almost periodic solution for almost periodic differential equation (2) under the conditions similar to the above conditions. The results impove some in literature.