Periodic Solutions Of Third-Order Neutral Differential Equations With Delays

This thesis applies the tools and methods of nonlinear analysis to discuss the existence and uniqueness of periodic solutions and the existence and multiplicity of positive periodic solutions of some third-order neutral delay differential equations.The main results of this thesis are as follows:Firstly,we discuss the 2π-periodic solutions the problem for third-order neutral delay differential equation where δ>0,|c|<1 are constants;f:R×R×R→R is a continuous function which is 2π periodic in t and τ is the positive constant.By the method of positive operator perturbation,we establish the maximum principle for periodic problems of the corresponding linear equation.Based on the maximum principle,the existence and uniqueness results of 2π-periodic solutions for the problem are obtained by adopting the monotone iterative method of upper and lower solutions.Secondly,we discuss the positive 2π-periodic solutions problem for the thirdorder neutral differential equations with multiple delays where a:R→（0,+∞）is a continuous function which is 2π periodic in t;f:R ×[0,+∞)ⁿ→[0,+∞)is a continuous function which is 2π periodic in t andτ₁,τ₂,…,τ_n are positive constants.We build the strong positive estimation of the corresponding linear equation and select the appropriate cone.Under the inequality conditions allowing the nonlinear term f（t,x₁,x₂,…,x_n）to grow superlinearly or sublinearly on x₁,x₂,…,x_n,the existence and multiplicity results of positive 2π-periodic solutions for the problem are obtained by using fixed point index theory in cone.Finally,we discuss the positive 2π-periodic solutions problem for the third-order neutral differential equations with delay in an ordered Banach space E,where f:R×K²→K is a continuous function which is 2π periodic in t,τ is the positive constant and K is the cone of positive element in E.Under the condition that the nonlinear term f satisfies the non-compactness measure,the existence results of positive 2π-periodic solutions for the problem are obtained by applying fixed point index theorem of condensing mapping.The result extends some results in finite dimensional space to infinite dimensional space.