Research On The Positive Periodic Solutions Of Delay Differential Equation And The Boundary Value Problem Of Fractional Differential Equation

Functional differential equations describe the mathematical models with delay phe-nomenon. Functional differential equation with distributed delays and cycle delay in eco-nomics, power systems, biology,ecology and practical problems of population has a wide range of applications. For example, ecosystem dynamics of feedback control,Fuzzy neu-ral networks,Animal model of red blood cells exist, and the population dynamic system model and so on.Therefor,studying the existence of periodic solution of functional differ-ential equation with feedback,delay,distribute delays is even more relevant. In addition, the significance of stability,whether small to a specific control system, to a social system, ecological system,it is always in various accidental or continued interference run under, bear the interference, whether can remain after the operation or work. Reservation stabil-ity problem comes from applied mathematics, physics, biology, etc, are nonlinear analysis is one of the most active area. Thus, the study functional differential equation stability problem becomes very important. Therefore, the study functional differential equation periodic solution and stability problem, not only have great application value, but also enriched the functional differential equation theory system.In this paper,we study the periodic solutions and stability of impulsive biological systems with s-type distributed delays and feedback controls,positive periodic solution of second-order neutral functional differential equation with mixed delay,nonlinear fractional four boundary value problems, We obtain existence of positive periodic solutions of a sufficient condition and some related results.The organization of this thesis is as follows.In Part 1, the backgroud of the subjects relevant to this dissertation is introduced.Main results in this dissertation are summarized.In Part 2, by using the continuation theorem of coincidence degree theory and Lya-punov's approach,some results ensuring the existence and stability of positive periodic solutions of the system on a type of impulsive Lotka-Volterra biological systems with s-type distributed delays and Feedback Controls are obtained, which improve some recent work.In Part 3, by applying Krasnoselskii's fixed-point theorem, some results about the existence of positive periodic solution to the systems are obtained. In Part 4, by using Krasnoselskii fixed-point theorem and some analytical skills,we studing four-point boundary value problems for a kind of fractional differential equation with alternating coefficient.