The Periodic Solutions For A Class Of Higher-Order Neutral Functional Differential Equations

In this paper, by using monotone iterative method of upper and lower solutions and fixed point theorem of completely continuous operators, we deal with the ex-istence and uniqueness of periodic solutions to the higher-order neutral functional differential equation where ?> 0,0<c< 1, are constants,f:R x [0,?)m+1? [0,?) is a continuous function which is 27?-periodic in t and ?1,?2,…,?m? 0 are constants. Existence results of periodic solutions are obtained to the equation. Moreover, by using the fixed point index theory in cones and the Krasnoselskii's fixed point theorem of the sum of a completely continuous operator and a contractive operator, we deal with the existence and multiplicity of positive periodic solutions to the higher-order neutral functional differential equation where ?>0, M> 0 are constants,f:R ×[0, ?)m ?[0,?) is a continuous function which is 2?-periodic in t and ?1, T2,…, ?m? 0 are constants.The main results of this paper are as follows:1. With the aid of the existence and uniqueness of solutions for corresponding higher-order linear differential equation, we build a new maximum principle by the method of positive operator perturbation, and obtain the existence and uniqueness of periodic solutions under some weak conditions by the method of monotone itera-tive of upper and lower solutions for the higher-order neutral functional differential equation.2. By the demonstration of spectral radius for the linear operator of the corre-sponding higher-order linear differential equation, we obtain the existence of periodic-solutions for the higher-order neutral functional differential equation by using the fixed point theorem of completely continuous operators under the linear growth conditions.3. Under the conditions concerning the first eigenvalue of the corresponding linear differential equation, the results of the existence of positive solutions for the higher-order neutral functional differential equation are obtained in the case of su-perlinear and sublinear by constructing a suitable cone and applying the fixed-point index theory in cone.4. By the demonstration of spectral radius for the linear operator of the corre-sponding higher-order linear differential equation, we use the Krasnoselskii's fixed point theorem of the sum of a completely continuous operator and a contractive op-erator, and we obtain the existence of positive solutions for the higher-order neutral functional differential equation.