Several Problems Of Lienard Equation

This thesis has been devoted to studying several problems of Lienard equation and consists of five sections.In section one, the origin and research methods on the problem of Lienard equation are introduced and the main contents in the thesis are summarized.In section two, the existence of periodic solutions for the Lienard equation is investigated. The sufficient conditions for the existence of nontrivial periodic solutions and for which there exists at least a periodic solution are obtained by studying the differential system which is equivalent to the Lienard equation.In section three, the nonexistence of periodic solutions for the generalized Lienard equation x"+f(x)ψ(x’)x’+g(x)ψ(x’)=0is investigated. Two sufficient conditions for the nonexistence of periodic solutions for the differential system which is equivalent to the generalized Lienard equation are obtained by applying Poincare tangent curve method.In section four, the existence of limit cycles for the Lienard system is investigated. Hypothesis "G(±∞)=+∞" is canceled in the thesis and the sufficient condition for the existence of limit cycles for the Lienard system is obtained by applying ring domain theorem.In the section five, the oscillation of solutions of the Lienard equation is investigated. Two sufficient conditions of oscillation of the nontrivial solutions of the differential system are obtained by studying the differential system above which is equivalent to the Lienard equation.