Existence Of Periodic Solutions For Liénard Equation With A Singularity

With the expanding application of nonlinear differential equations,the study of the periodic solutions for differential equations with singularities has more important science meaning and application value,moreover,the study of periodic solutions always has great significance in the study of the qualitative theory of differential equations.Therefore,many mathematician began concern the research of the periodic solutions.In this paper,the existence of periodic solutions of some Liénard equations with singularities is studied.The full text is divided into four parts,the main contents are as follow:Firstly,the first chapter introduces the research background and the development of periodic solutions of nonlinear differential equations with singularities,simply introduces some mathematical tools to study periodic solution,and summarizes the main work.And then,the existence of periodic solutions of a Liénard equation with a singularity of repulsive type is studied in the second chapter.By using the Manasevich and Mawhin's continuation theorem,we prove that the equation has at least one T-periodic solution.Furthermore,we also obtain that a sufficient and necessary condition for the existence of positive T-periodic solutions of the equation.Then,in the third chapter,based on the second chapter,the existence of positive periodic solutions of a Liénard equation with a singularity of repulsive type is studied.We prove that the equation has at least one positive T-periodic solution by using the Manasevich and Mawhin's continuation theorem.At last,based on the second chapter,the fourth chapter studies the existence of periodic solutions of a Liénard equation with a singularity.However,the difference is that the singularity of the restoring force in the equation that we study in this chapter is classified as indefinite type.We prove that this equation has at least one T-periodic solution by using the Manasevich and Mawhin's continuation theorem.The interesting point is that the weak singularity of restoring force in the equation that we study at x = 0 is allowed and the coefficient function f(x)in the friction term may have a singularity at x = 0,in addition,the singularity of the restoring force may be classified as indefinite type,that is the a(t)is a sign-changing function.In this case,the methods in the past work to estimate a priori bounds of periodic solutions are inapplicable to the study in this paper.In order to overcome the difficulty in estimating a priori bounds,the relation between the coefficient function in the friction term and the singularity of the restoring force at x = 0 is analyzed systematically,and reveals that this relation has influence on the upper and lower bounds for positive periodic solutions of the equations.On this basis,the new distinctive methods to estimate a priori bounds for periodic solutions of the differential equations with singularities are formed.