Remarks on Stability Conditions for the Differential Equation x″ + a(t)ƒ(x) = 0

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Abstract
Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfies where a(t) > 0, t ≧ t0 and = max (−a′(t), 0), and then every solution of (1) satisfies (2).

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