Existence of a Non-Oscillating solution for a Second Order Nonlinear ODE

In this paper we have considered the following nonlinear ordinary differential equation. $$y''(x) + F(x, y(x)) = 0\tag{0.1}$$ where \(F(t,x(t))\) is real valued function on \([0,\infty) \times R\), \(x\geq 0\). We have given sufficient conditions for the existence of a non oscillating solution for equation (0.1). These conditions are generalized with respect to the nonlinear function \(F\) and are in the spirit of the classical result by Atkinson [1].