Periodic Solutions Of Third-order Ordinary Differential Equations

In this paper,by using Leray-Schauder fixed point theorem and Schauder fixed point theorem of completely continuous operators,the Fourier analysis method and fixed point index theory in cones,we deal with the existence and uniqueness of 2?-periodic solutions of general third-order ordinary differential equation.(?)where Lu(?)is a third-order ordinary differential operator,f:R × R3?R is a continuous function.The mainly discusses from the following four aspects:1.The existence and uniqueness of the solution of the third-order linear differen-tial equation are discussed by Fourier analysis method,and the solution operator is proved to be bounded.the Leray-Schauder fixed point theorem is applied under the condition that the nonlinear term f satisfies a growth,the existence and uniqueness of the 2?-periodic solution of the general third-order nonlinear differential equation are obtained.2.Under the nonlinear term f satisfying the superlinear growth condition and Nagumo-type growth condition,Leray-Schauder fixed point theorem is used to ob-tain a general third-order nonlinear differential equation the existence of 2?-periodic solutions.3.The existence and uniqueness of the 2?-periodic solution of the general third-order nonlinear differential equation are obtained,by establishing a suitable convex closed set and under the nonlinear term f satisfies relatively weak conditions,using Schauder fixed point theorem of completely continuous operators.4.Using the fixed point index theory of cones,under the conditions that the nonlinear term f satisfies some easily verifiable inequalities,some existence results of the positive 2?-periodic solutions of the general third-order nonlinear differential equation was obtained that allow f satisfies the superlinear or sublinear growth on x,y,z.