Periodic Solutions And Almost Periodic Solutions Of Nonlinear Differential Equations With Delay

Theory of periodic solutions and almost periodic solutions have always been an important part of nonlinear differential equations vibration. People make slow progress in this problem and there exists a bit of literature in this field at present. We choose upper and lower solutions method to study periodic solutions and almost periodic solutions of nonlinear differential equations with delay, one-dimensional nonlinear parabolic equations with delay and homogeneous space periodic solutions and homogeneous space almost periodic solutions of multidimensional nonlinear parabolic equations with delay, and existence and uniqueness theory of them are built. In the application of several models, it fully demonstrates the feasibility, effectiveness and extensive of our result. The main content of this article are as follow:The first chapter discusses periodic solutions and almost periodic solutions of nonlinear differential equations with delay, we utilize ordered monotonic and bounded features of monotone iterative sequence, combining the upper and lower solution method and the fixed point theorem to prove the existence of periodic solutions and almost periodic solutions, thus avoiding the difficulties to certify the sequence convergence on the infinite interval, this greatly simplifies the certification process. For the uniqueness condition, depth analysis disparity which nonlinear item affects the system, transformed the uniqueness condition to be controlled by corresponding Lipschitz constant inequality, so we get a unified formed uniqueness condition. Then, as application, we specifically analyze two actual models:1. n population cooperating Lotka-Voltterra model:We find status of a population cycles or almost periodic changes in the status, when the initial status is under a pre-control scope, the status of other population will approach the periodic or almost periodic status with time moves forward, increasingly high degree of approximation, and will not be ups and downs. This result is also pointed out in the application of the model, in order to get to the expected result of control status, we need to note the adjustment of model parameters.2. immune reaction Marchuk model:The result shows that there is a periodic or almost periodic status for antibody, antigen and plasma cells, when the initial antibody, antigen and plasma cells are under a pre-control scope, the corresponding state is getting closer and closer to the periodic or almost periodic status with time moves forward, and distortion will not occur. Related to this, previous used numerical simulation method to discuss periodic situation, this maybe the first time to comprehensive theoretical analysis this problem.The second chapter studies periodic solutions and almost periodic solutions for the first boundary value problem of one-dimensional nonlinear parabolic equations with delay. We use analogous method in the first chapter to discuss the existence of periodic solutions and almost periodic solutions, then we make a priori estimate to the solution, using truncation method for smooth function, the boundary problem is reduced to the initial boundary value problem, so as to do the step-by-step estimate, this method is complicated, but universal. Next we use this conclusion to Marchuk model with diffusion as application: a(t)g,(t),hi(t) are ω-periodic or almost periodic. Using the conclusion of this chapter we get existence and uniqueness of periodic solutions and almost periodic solutions for this model. Then we use eigenvalue nature and comparison principle to get the global stability of both kinds of solutions. These conclusions fully reveal the geometric properties of the solutions of the model, and provide effective ways and means for the study of general nonlinear parabolic equations with delay. The third chapter discusses periodic solutions and almost periodic solutions fo multidimensional nonlinear parabolic equations with delay. This is a new problem, there are onl; few literatures discussed homogeneous space periodic solutions of partial differential equation without delay so far at home and abroad, the literatures for homogeneous space periodic solution almost periodic solutions of partial functional differential equations or partial differentia equations with delay are yet to be seen. The definition shows that the research for homogeneou space periodic solutions, almost periodic solutions of partial functional differential equations i due to two types of solutions of the corresponding ordinary differential equations, so we can ge existence and uniqueness of homogeneous space periodic solutions, almost periodic solutions b; using the conclusions in chapter one. This fact further reveals that the geometric structure of th solutions of partial differential equations is decided by homogeneous space periodic solutions almost periodic solutions in some form and some boundary conditions, This is very important ii the analysis of the qualitative theory.We take the ecology Prey-Predator model as application. Di,ai are ω-periodic or almost periodic. By using the fundamental theorem above combine with Liapunov method we get the existence, uniqueness and global stability for homogeneou space periodic solutions, almost periodic solutions of the model. These conclusions suggest tha there is a periodic or almost periodic state between predator and prey, when both the origin stat are under a pre-control scope, the corresponding state of predator and prey is getting closer an closer to the periodic or almost periodic status. This phenomenon shows the nature of th dynamics of this model specifically.