| Although the oscillation properties of functional equations are now studied extens-ively, when both impulsive and delays are involved, additional care has to be taken si-nce the meaning of a solution may be quite different from the ones without delays andhence the concept of oscillation may also be altered. In this paper, we firstly definite ahigher order differential equation with impulsive and delays under the classical soluti-on and give the proof of the existence of the solution, which corrected the errors or i-mprecise in the existing literatures with delays. Secondly, we note that the classic im-pulsive differential inequality without delays of V.lakshmikantham and make the res-ults can be applied to study the classic impulsive differential equations with delays. B-ased on the above, using the impulsive differential inequality and analytical skills,andovercomed the difficulties caused by the N triple integral order and complex calculati-on, we study the oscillatory and asymptotic behaviors of a class of higher order nonli-near impulsive differential equations with delays and get some criteria which are disti-nct from the recent results and provide more precise control of the qualitative propert-ies of these equations. This essay is composed of three parts. In the first part, we defi-nite a higher order differential equation with impulsive and delays as well as its solut-ion, and then give the theorem of existence of the solutions. Besides, we provided annotation about the impulsive differential inequities. In the second part, we study theoscillatory behavior of even order nonlinear delay differential equations with impulse-s, and derive four oscillation criteria and two corollaries. Moreover, two examples aregiven to illustrate our theorems. Finally, we investigate the oscillatory and asymptoticbehaviors of odd order nonlinear differential equations with impulses, and get threecriteria. All of the results are new. |
PDF Full Text Request