On the existence of periodic solutions for certain differential equations (Q1765439)

The paper is concerned with the existence of periodic solutions for two classes of second- and third-order nonlinear differential equations. By means of topological degree theory, it is established that if under certain conditions, the differential equation \[ x^{\prime\prime}+\lambda\phi(t)f(x)h(x^{\prime})=0 \] has a periodic solution, then there exists a periodic solution of \[ x^{\prime\prime}+\lambda_{\sigma}\phi(t)g(x,x^{\prime},\sigma)=0. \] A similar result is proved also for the third-order nonlinear differential equation \[ x^{\prime\prime\prime}+\lambda\phi(t)g(x,x^{\prime},x^{\prime\prime} ,\sigma)=0. \] The paper concludes with illustrative examples.