Existence Of Periodic Solutions For Two Classes Of Second-order Nonlinear Ordinary Differential Equations

In this paper,the existence of solutions for two classes of nonlinear ordinary differential equations is studiod by using time map estimation,implicit function theorem and Mawhin continuation theory.The main work is as follows:1.The equation in the form of x"=a(t)/x3 for the stability of the mass wave respiration in the Bose-Einstein condensed state,the propagation of the guided wave in the optical fiber,electromagnetic capture of a neutral atom near the charging line has important applications.In view of the related physical applications,we study the existence of T-periodic solutions of the singular second order ordinary differential equations with sign changing weight.The necessary condition for the equation to have a positive periodic solution is that?0T a(t)<0.And Bravo and Torres conjecture that it is also a sufficient condition.The conjecture is proved by them under the premise of where a+,a_>0.In a more general case where a+1 ao,a_>0,We prove that the conjecture is still correct by using the time map estimation and implicit function theorem,and generalize the main results of Bravo and Torres,2.We study the diffrential inclusion x"+?(x)sgnx'+f(t,x,x')(?)?(t)with T-periodic right-hand side,where,?:R?R+ is continuous,f:R3?R are continuous,?:R?R is Lebesgue measurable,bounded and T-periodic.The models for example a mechanical system with one degree of freedom sub-jected to dry friction and periodic external force.Moreover,when the solution is a constant,the spring oscillator is in a static state and when the solution is not a constant,the spring oscillator is in a moving state.Therefore,we get the exis-tence of the T-periodic solution by using the Mawhin continuation theorem.And the sufficient and necessary condition for the existence of the non-constant solution is given.