Monotone Dynamical Systems Theory And Its Applications

The content of this dissertation consists of two main parts.In the first part, the development of monotone dynamical systems theory, in particular some important theoretical results on the generic (quasi) convergence for strongly order-preserving seiniflows (SOP serniflows for short), are briefly reviewed. Meanwhile, it is also emphasized that when these results arc applied to systems of delay differential equations and reaction-diffusion equations with delay, there are some drawbacks not to be ignored such as: the requirements of a suitable choice of state space and the technical "ignition" assumption. To overcome these drawbacks, we introduce a class of generalized SOP serniflows and prove that the analogs of the limit set dichtomy and the sequential limit set trichotomy still hold for such a class of semiflows, which enables us to establish quasi convergence and stability for the semiflows. The aforementioned drawbacks are exactly overcome when the established results are applied to systems of reaction-diffusion equations with delay. A stronger monotonicity and smoothness of the semiflow with an additional spectral hypothesis are further assumed in order that some more advantageous results are derived. Additionally, we give several generalizations of the classical Krein — Rutman theorem and show that monotonicity and these generalized theorems are sufficient to guarantee the additional spectral hypothesis, which supplies us with many conveniences in the applications. As above, generic convergence and stability results are established, and the aforementioned drawbacks are also overcome when the established results are applied to a generalized cooperative and irreducible system of delay differential equations (which does not require the "ignition" assumption). Finally, we give and prove the generalized Perron — Frobenius theorem, which is then used to prove that the linearized stability of an equilibrium of a generalized cooperative and irreducible system of delay differential equations is the same as for the corresponding system without delays.In the second part, we introduce the notion of pseudo monotone semiflow, which i.s a generalization of monotone semiflow and a synthesis of monotonicity methods and dynamical systems ideas as well. It should point out that pseudo monotone semiflows are defined on ordered topological spaces and preserve some extent of order relation in some ordered pair points. The order-preserving properties allow us to utilize monotonicity methods and dynamical systems ideas to carry out research. Apparently, a precise definition of this class of semiflows isdifficult to be given mathematically and therefore, the above general terms seem to be more meaningful. We also provide several class of pseudo monotone semiffows, employ monotonicity methods and dynamical systems ideas to investigate the relation between the positive limit set and some phase points, and then establish several convergence principles. Besides significant theoretical improvement of the existing related results, these convergence principles are shown to be distinctive in the applications. As applications of some of the obtained results in this part, several more general equations than that considered in a conjecture of Bernfeld and Haddock are discussed and our results substantially improve some important existing ones.