Qualitative Research On Solutions For Several Kinds Of Differential Systems

As we all know, qualitative properties of the solutions is one of the most important problems in differential equation theory. Qualitative properties of differential equation have been extensively gained. In the real world, lots of the development process show the lag and sudden change in status at a moment, such as biont and mechanical response, reproduction of animals and neural networks jump and so on. The lag and the sudden change in status reflected in the mathematical model is delay effect and impulsive effect. Therefore, the intro-duction of delay and impulsive effect appears very natural and necessary in the differential equations. Our discussion of such a system can be more true and accurate simulation for real life phenomena. So the research on qualitative properties of the solutions of delay and impulsive differential equations has more applications. The paper is based on the qualitative properties of the order of solutions, periodic solutions and almost solutions, according to the differential equation introduction of delay and impulsive effect.This dissertation separately discuss several types of differential equations with delays or impulsive effects. By means of different research methods, several classes of the sufficient conditions on the qualitative properties of the systems solutions have been obtained, and simulations have been done to verify the correctness of some results. The full text structures are as follows:In chapter1, we simply introduce the development and basic information of periodic solutions and almost periodic solutions, delay differential equations, impulsive differential equations, and the differential equations which will be investigated in our dissertation, and propose the background and the main work of this paper.In chapter2, we mainly deals with the open problem (7) provided by Berezansky et al in2010. The open problems is to discuss a Nicholson’s model with a nonlinear density-dependent mortality term, i.e. by using techniques of analysis inequality and oscillation basic theory, the boundedness and oscillations of the solutions for this Nicholson’s model with a nonlinear density-dependent mortality term have been investigated. The stability of the solutions for this model has been solved partially.In chapter3, we mainly study a kind of neutral logarithmic population model with delays and impulse, i.e. by using the method of the pulse form change into the non-pulse form and k-set contraction mapping theory, some sufficient conditions for the existence and global attractive of positive periodic solution for the model are established.In chapter4, we pointed out that there are some problems by Wang in2006and by Stamov in2009. And we mainly research a class of Lasota-Wazewska model, i.e. by using of the contraction mapping principle as well as the Gronwall-Bellman’s inequality, some sufficient conditions for the existence and stability of positive almost periodic solutions for a class of Lasota-Wazewska model are given. An example is provided to illustrate the effectiveness of the proposed result.In chapter5, as this paper’s conclusion, we have carried on the subtotal to the paper and proposed several questions which were worth further studying.