Positive Periodic Solutions For Delay Differential Equations With Applications In Population Models

Ordinary differential equation is an important branch of modern mathematics. Withthe development of modern society, it is widely used in engineering, astronavigation, ecol-ogy, economics, finance etc. However, in the real world, the future state of the systemdepends on not only the present state but also the past states. Therefore, it is only aapproximation to the truth that ordinary differential equations have been used to describethe development of the system. In order to describe appropriately the real problems, amore realistic model would be delay differential equations. Consequently, the researchs ondelay differential equations are very meaningful in the theory and applications.In addition, it is also a meaningful subject of research to consider the periodicity andstability of solutions to ordinary differential equations, which represent the balance andstability in construction. Furthermore, it is very important in nuclear physics, electronicsignal system, ecological system, epidemiology and control theory etc. Similarly, it is veryimportant to research the existence and stability of periodic solutions to delay differentialequations. In particular, the effects of a periodically varying environment are importantfor evolutionary theory as the selective forces on systems in a fluctuating environmentdiffer from those in a stable environment. Thus, the assumption of periodicity of theparameters in the system (in a way) incorporates the periodicity of the environment (e.g., seasonal effects of weather, food supplies, etc.). Naturaly, the existence and stabilityof periodic solutions to these models have been studied by many author.In this thesis, we shall study exhaustively the existence and stability of positiveperiodic solutions for delay differential equations with applications in population models.Chapter 1 introduces first the research significance and development of delay differ-ential equations. In particular, we summarize the methods for studying the existence ofperiodic solutions to delay differential equations and neutral delay differential equations,and the limitations of the corresponding known results. Then, this chapter also presentsmain results, organization and preliminaries of this thesis.Using a fixed point theorem of decreasing operator, Chapter 2 shows the existence ofuniqueω-periodic positive solution (?) to Lasota-Wazewska model:In particular, this chapter not only gives the conclusion of convergence of x_n to (?), where{x_n} is a successive sequence, but also shows that (?) is a global attractor of all other positive solutions. That is, approximations to (?) are given. Further, the results in thischapter are more valuable in the applications. Compared to the known results, the resultsin this chapter are more easily verifiable.Applying a fixed point theorem of decreasing operator, Chapter 3 obtains sufficientconditions for the existence of uniqueω-periodic positive solution (?) to hematopoiesismodel:In addition, this chapter not only gives the conclusion of convergence of x_n to (?), where{x_n} is a successive sequence, but also shows that (?) is a global attractor of all otherpositive solutions. That is, approximations to (?) are given. Further, the results in thischapter are more valuable in the applications. Compared to the known results, the resultsin this chapter are more easily verifiable.Chapter 4 is devoted to a fixed point theorem of strict-set-contraction that improvesthe corresponding known results. Then, applying this fixed point theorem, we study themore general neutral delay differential equation:and obtains more verifiable sufficient conditions for the existence ofω-periodic posi-tive solutions. In addition, employing the above results to the following neutral delaydifferential equation:the corresponding known results are improved. Furthermore, Open problem 9.2[19] isgiven an answer. In addition, we can see it is one of good methods to use the fixed pointtheorem of strict-set-contraction in this chapter to study the existence of periodic solutionsfor neutral delay differential equations.Using a fixed point theorem of strict-set-contraction, Chapter 5 deals with the fol-lowing neutral delay Lotka-Volterra system: and obtains sufficient conditions for the existence ofω-periodic positive solutions. Thischapter extends and improves the corresponding known results. Moreover, the results inthis chapter are more verifiable.Chapter 6 considers the following neutral delay differential system with feedbackcontrol:Applying a fixed point theorem of strict-set-contraction, this chapter obtains more verifi-able sufficient conditions for the existence ofω-periodic positive solutions to the abovesystem. Furthermore, some known results are extended and improved.In addition, some results in this thesis have been published in Nonlinear Analysis,Journal of Mathematical Analysis and Applications, Computers and Mathematics withApplications etc.