Periodic Solutions For Third-order Differential Equations With Delays

In this paper,by using monotone iterative method of upper and lower solutions,fixed point theorem of completely continuous operators and fixed point index theory in cones,we deal with the existence of solutions for some third-order differential equation with delays.The main results of this paper are as follows:1.we build a new maximum principle by using the method of positive operator perturbation,and obtain the existence and uniqueness of ω-periodic solutions by the method of monotone iterative of upper and lower solutions for the third-order differential equation with multiple delays u’’’(t)＝f(t,u(t),u(t-T1),u(t-T2),…u(t-Tn)),t∈R where f:R x Rn-1 R is a continuous function which is ω-periodic in t and T1,T2,…,Tn≤0 are constants.2.Under the linear growth conditions,we obtain the existence and uniqueness of ω-periodic solutions by using the Schauder fixed point theorem of completely con-tinuous operators for the third-order functional differential equation with multiple delays.3.With the aid of the existence and uniqueness of solutions for corresponding third-order linear differential equation,we use the fixed point theorem of completely continuous ope.rators under some weak conditions,and we obtain the existence and uniqueness of nonnegative ω-periodic solutions of the third-order functional differ-ential equation with multiple delays u’’’(t)=f(t,u(),u(t-T1),u(t-T2),…,u(t-Tn)),t∈R where/:R x Rn+1 → R is a continuous function which is ω-periodic in t and T1,T2,·…,Tn>0 are constants.4.By choosing a special cone and applying the fixed point index theory in cones,we obtain the existence of positive ω-periodic solutions for the third-order differential equations with delayed derivative terms u’’’(t)=f(t,u(),u(t-T1),u(t-T2),…,u(t-Tn)),t∈R under the case of superlinear and sublinear conditions.where a∈ C(R,(0,+∞))is a ω-periodic functions;f:R x[0,+∞)xRn →[0,+∞)is a continuous function which is ω-periodic in t and T1,T2,…Tn>0 are constants.