Study On Solutions Of Some Pseudo Almost Periodic Systems

As a generalization of almost periodic functions,pseudo almost periodic functions not only keep many excellent properties such as the translation invariance and the convergence,but also have more applications in real worlds.In this paper,we mainly discuss two types of pseudo almost periodic and weighted pseudo almost periodic logarithmic multispecies models,and give the existence and the stability of their solutions,which will be illustrated in the following two parts:In the first part,we consider a time-varying delay pseudo almost periodic logarithmic multispecies model with feedback control.In a pseudo almost periodic function space,we construct a mapping according to the theory of dichotomy.With the aid of the inequality technics and the Ascoli-Arzela theorem,we prove that this mapping satisfies the conditions of Krasnoselskii's fixed point theorem,immediately the fixed point is obtained,which yields the existence of the pseudo almost periodic solution.The criteria for exponential stability of the pseudo almost periodic solution will be established by constructing a Lyapunov function.Finally,an example will be shown in this part.In the second part,a weighted pseudo almost periodic logarithmic multispecies model with impulsive control is studied.In the piecewise weighted pseudo almost periodic space,which is a Banach space,the Cauthy solution of the linear system is used to construct a contraction mapping.With the aid of the properties of piecewise weighted pseudo almost periodic functions and the properties of series convergence,we prove that this mapping satisfies the conditions of the Banach fixed point theorem.The existence and the uniqueness of the fixed point,which also means the piecewise weighted pseudo almost periodic solution,are obtained.