Studies Of Some Aspects On The Asymptotic Behavior Of Solutions Of Functional Differential Equations

This paper is divided into two parts. The first part is concerned with the stability and Hopf bifurcation from an equilibrium for two classes of retarded differential difference systems, while the second part deals with the existence of positive periodic solutions of retarded functional differential equations with infinite delay.The stability and Hopf bifurcation of functional differential equations have great attention to mathematicians (especially applied mathematicians) and engineers, both of its theory and applications to the objective phenomena has broad prospects. The first part of this paper (chapters 2 and 3) discusses the stability and Hopf bifurcation for two classes of retarded differential difference systems. By analyzing the eigenvalues of linear variation equations, some algebraic criteria for the absolute stability (independent of the delays) are established for two special 3-dimensional systems. Also, the sufficient conditions under which the systems hold a Hopf bifurcation are obtained, which are easy to verify.Recently, the theory of monotone semiflows generated by functional differential equations has great attention to mathematicians. In the second part of this paper (chapter 4), by constructing the monotone semiflows for retarded functional differential equations with infinite delay, the existence of positive periodic solutions of is given by combining the fixed point theorems of abstract operators. A nontrivial application to periodic ecological systems is presented to illustrate our results.