Periodic solutions of the second-order differential equations with singularity Original Research Article

In this paper, we deal with the existence of periodic solutions of the second-order differential equations x″+g(x)=p(t) with singularity near origin. By using the phase-plane analysis methods, we prove that the given equation has at least one periodic solution when g(x) exhibits semilinear condition for positive large x and singularity condition at origin and the primitive function View the MathML source tends to positive infinity as x→0+. Moreover, we construct a counter-example to show that our result is no longer valid when G(x) tends to some finite constant as x→0+.