Applications Of Monotonicity Methods To Delay Differential Equtions

Monotonicity methods are powerful tools for the treatment of differential equations and enable us to obtain information about the existence, stability and asymptotic behavior of solutions of differential equations. The main objective of this dissertation is to present results on the employment of the monotonicity methods for investigating delay differential equations and further develop the idea of combining monotonicity methods with dynamical systems ideas.After briefly reviewing the development of monotone dynamical systems theory which is a synthesis of monotonicity methods and dynamical systems ideas, we particularly focus on some important theoretical results concerning the non-standard order cones due to their theoretical importance and potential applications. By introducing the non-standard order cones to neutral delay differential equations, we then establish the generic convergence results for such neutral equations, which improve and extend some important related results in the literature. Meanwhile, we also outlines some research results on exponential orderings. In this dissertation, a class of generalized exponential orderings are introduced and shown to allow a weakening of the generalized quasi-monotonicity condition, and some more general conditions are given for the solution semiflows generated by a class of systems of delay differential equations to be strongly order preserving. By applying the obtained results together with the abstract results on strongly order preserving semiflows in the literature to autonomous systems of delay differential equations, we establish the generic convergence result for such systems, which extends some previous results.Then this dissertation introduces a class of pseudo monotone semiflows on product ordered topological spaces of n ordered topological spaces. Such semiflows are generalizations of monotone semiflows and only partly order-preserving, but the order-preserving properties allow us to utilize monotonicity methods and dynamical systems ideas to carry out research. It should be pointed out that the global convergence principles for continuous- and discrete-time monotone semiflows fail to apply to many differential equations without enjoying a comparison principle. Fortunately, we can still combine monotonicity methods and dynamical systems ideas to obtain the convergence in some partly order-preserving dynamical systems. For this class of pseudo monotone semiflows, several convergence principles are established, which enables us to extend the convergence results for a class of well-knowndifferential equations to a higher dimensional case and overcome the drawbacks of previous results. Furthermore, a combination of monotonicity methods with dynamical systems ideas and a combination of monotonicity methods with functional methods are respectively used to investigate the convergence of solutions to two systems of delay differential equations having important practical applications, and some novel and more elaborate results are obtained and also confirm the Bernfeld-Haddock conjecture. Additionally, we point out the errors in one existing paper on the convergence of solutions to a class of neutral delay differential equations and then tackle this problem under weaker conditions by combining monotonicity methods and dynamical systems ideas. Our result corrects and improves the existing ones and also confirms the Haddock's conjecture.Finally, the permanence of a generalized quasi-monotone system of delay differential equations is discussed by appealing to the theory of monotone dynamical systems, and an application to some multi-species competition model with delay is made. The sufficient conditions for the permanence and the existence and global attractivity of a positive equilibrium of the multi-species competition model are then obtained by embedding the system into a larger quasi-monotone system. Our results extend and improve some previous ones in the literature.