The Study Of Nondegeneracy And Periodic Solution Of Three Kinds Of Nonlinear Differential Equations

With the development of modern society,more and more scholars are interested in the study of the existence and uniqueness of periodic solutions of ordinary differential equations.An important method that studying the existence and uniqueness of periodic solutions is to treat the differential equation as a linearized disturbance.Thus,the problem is decomposed into the study of the nondegeneracy of the trivial solution of the linear system and how the solution of the system appears under disturbance.An important purpose of studying the nondegeneracy of linear differential equations is to study the existence and uniqueness of periodic solutions of nonlinear differential equations.In this thesis,we firstly discuss the nondegeneracy of periodic solution for a thirdorder linear differential equationBy the Writinger inequality,we give a nondegeneracy condition of the above equation （?） By the nondegeneracy of third-order linear differential equation,we consider the existence and uniqueness of periodic solution of the third-order nonlinear differential equation with semilinear and superlinear terms.Secondly,by the properties of neutral operator and the Writinger inequality,we investigate the nondegeneracy of a kind of neutral differential equation as follows,（?）Later,from the nondegeneracy and Manásevish-Mawhin continuation theorem,we obtain the existence and uniqueness of the following neutral differential equation （?）Afterwards,we consider the nondegeneracy of a kind of Liénard differential equation （?） Then by applications of the nondegeneracy and the Manasevish-Mawhin continuation theorem,we obtain the existence and uniqueness of the following Liénard differential equation （?） where s∈R,h∈L¹（S_T）is a T-periodic function and ∫₀^T h（t）dt=0,g∈C（R,R）is a strictly monotone function.