Limits of solutions to a nonlinear second-order ODE

In this paper, the existence of solutions to the equation ẍ+2f(t)ẋ+β(t)x+g(t,x)=0, t≥0, is discussed. Our approach allows us achieve extension to the case of the whole real line, for which the existence of homoclinic solutions having zero limit at ±∞±∞ is deduced. The result is obtained through the method of the Lyapunov function and differential inequalities.