Study On Periodic Solutions Of Several Nonlinear Differential Equations

In this thesis,we mainly use the coincidence degree theorem to study the existence and uniqueness of periodic solutions of several types of nonlinear ordinary differential equations,which mainly consists of three parts.In the first part,we study the existence of positive periodic solutions of Liebau-type differential equations x"+cx'=r(t)x(t)?-s(t)x(t)?and x"+cx'=r(t)|x(t)|?-s(t)|x(t)|?,and at the same time,we study existence of positive periodic solutions of generalized Liebau-type differential equations with p-Laplacian(?p(x'(t)))'+f(t,x(t))x'(t)+r(t)x(t)?-s(t)x(t)?=0,where r(t),s(t)are continuous periodic functions,f:R × R?R is a continuous function,c?R ?>0,?>0 and ???.In the second part,we study the existence of periodic solutions of the forced pendulum equations x"+kx'+a(t)sin x=e(t),where a(t),e(t)are continuous periodic functions,k?R.In the third part,we study the existence of periodic solutions of fourth-order differential equations u(4)+pu"-a(t)u?+b(t)u?=0 and u(4)+pu"-a(t)|t|?+b(t)|u|?=0,and at the same time,we study uniqueness of positive periodic solutions for fourth-order differential equations with a small parameter ?u(4)+pu"-?a(t)u?+b(t)u?=0,where a(t),b(t)are continuous periodic functions,p?R,?>0,?>0 and ???.