Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

Abstract

This paper studies the existence of a positive solution to the second-order periodic boundary value problem $$u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0<t<2\pi,\ ~u(0)=u(2\pi),\ u^{\prime}(0)=u^{\prime}(2\pi),$$ where the nonlinear term f(t,u) may be singular at t=0, t=2π and u=0. When there exist the limit functions lim u→+0 f(t,u)/u and lim u→+∞ f(t,u)/u, we prove that the problem has a positive solution provided that the integrations of the limit functions are appropriate.