Asymptotic Property Reseach For Impulsive Functional Differential Equations And Oscillation For Solutions Of A Neutral Differential Equations

Asymptotic behavior of dynamic is an important conception with rich connota-tion, which mainly includes existance and uniqueness of the solutions, stability, oscillation andperiodicity and so on. These properties reveal the long-term behavior of the dynamical system, sothey have important effect in the area of biotechnology, population dynamics and economics and soon. We know that the study of the co-exist, stability, oscillation of the spices has very importantpractical meaning to keep ecological equilibration and proteco ecological environment, even to savevaluable and rave biologics which are on the verge of becoming extinct.The theory of impulsive differential equations is now being recognized to be not only richerthan the correspondinging theory of differential equations withnot impulsives but also provids amore adequate mathematical model for numerous processes and phenomena studied in physics,biology, economy and so on. However, the theory of impulsive functional differential caused isdeveloping comparatively slowly due to numerous theoretical and technical difficulties caused bytheir peculiarities. In particular, there is little in the way of results for the oscillaton of impulsivedelay differential equation of neutral type despite the extensive development of the oscillatory andnonoscillatory properties of differential equations without impulses. In chapter 2, we study thefirst-order neutral impulsive functional differential equations with multiple delaysUsing a fixed point theorem of Schaefer and a nonlinear alternative of Leray-Schauder, local andglobal existance and uniqueness results are estabished. Meanwhile the examples were givern toillustrate the usefulness of theorem.In recent years, the theory of impulsive differential equations is now being recognized as beingnot only richer than the corresponding theory of differential equations withnot impulses but alsorepresenting a more natural framework for mathematical modelling of many real-world phenom-ena. There has been increasing interest in the stability and oscillation of impulsive delay differentialequations and there are many important results. In chapter 3, the asymptoticy of solutions of non- linear impulsive functional differential equations with positive and negative coefficientswere discussed by using Lyapunov functional. The sufficient conditions are obtained for everysolution of equation tending to a constant. Meanwhile the examples were givern to illustrate theusefulness of theorem.In the study of ecological models, the existence of positive solution has a close relation withoscillation. There are many results about the oscillation of constant coefficient linear delay differ-ential equation, but the studying about the nonantonomous equation is much less. In chapter 4,generalized characteristic equationα(t)+sum from i=1 to m(b_i(t)α(t-Ï„_i(t))+(?)_i(t))(φ(h_i(t)))/(φ(t₀))exp(-integral from H_i(t) to tα(s)ds)+sum from j=1 to r pj(t)(φ((?)_j(t)))/(φ(t₀)exp(-integral from (?)_j(t) to tα(s)ds)=0,and existence of positive solution for a class first-order non-antonomous neutral delay differentialequations d/dt[x(t)+sum form i=1 to m b_i(t)x(t-Ï„_i(t))]+sum from j=1 to r pj(t)x(t-(?)_j(t))=0,t₀≤t＜Tare investigated. Sufficient and necessary condition of existence of positive solutions are obtained bythe mean value theorem and the Picard successive approximation theorem. Meanwhile sufficientconditions of oscillation about the neutral delay differential equations were given. The conclusioncontains the general characteristic equation's.