Periodic Solutions For Higher-Order Differential Equations With Delays

In this paper,by adopting monotone iterative technique of upper and lower solutions,fixed point theorem of completely continuous operators and fixed point index theory in cones,we research the existence of periodic solutions for several kinds of higher-order differential equation with delays.The following work has been carried out:1.With the help of the known results of periodic solutions for higher-order linear differential equation,we apply the perturbation method of positive operator,and acquire the existence and uniqueness of ?-periodic solutions for corresponding higher-order linear differential equation u(n)(t)+a(t)u(t)=h(t),t?R,where n?2,a:R?(0,+?)is a continuous function which is ?-periodic,h:R?R is a continuous function which is ?-periodic.2.We construct a new maximum principle and acquire the existence and u-niqueness of ?-periodic solutions by the technique of monotone iterative of upper and lower solutions for the higher-order differential equation with delays u(n)(t)+a(t)u(t)=f(t,u(t-?1),u(t-?2),…,u(t-?k)),t ? R,where n?2,a:R?(0,+?)is a continuous function which is ?-periodic,f:R×Rk?R is a continuous function which is ?-periodic on t,and ?i,?2,…,?k?0 are constants.3.Under relatively weak conditions,we use the fixed point theorem of complete-ly continuous operators and acquire the existence and uniqueness of nonnegative?-periodic solutions of the above equation.4.By selecting a cone and applying the fixed point index theory in cone map-ping,we acquire the existence of positive ?-periodic solutions for the higher-order differential equations with delayed derivative terms u(n)(t)+a(t)u(t)=f(t,u(t),u'(t-?1(t)),…,u(n-1)(t-?n-1(t))),t ? R,where n? 2,a:R?(0,+?)is a continuous function which is ?-periodic,f:R ×[0,+?)× Rn-1 ?[0,+?)is a continuous function which is ?-periodic on t,and ?k:R?[0,+?)is a continuous function which is ?-periodic,k=1,…,n-1.