On the qualitative behaviors of solutions of third order nonlinear differential equations

On some classes of nonoscillatory solutions of third-order nonlinear differential equations

“…Let z ∈ M 0 . Without loss of generality, we suppose that there exists t 0 ≥ a such that z(t) > 0, z [1] (t) < 0, z [2] (t) > 0 for t ≥ t 0 . Because (z [2] (t)) = q(t)f (z(t)) > 0 for t ≥ t 0 , z [2] is a positive increasing function on the interval [t 0 , ∞), and so we have that (z [1] (t)) ≥ z [2] (t 0 )r(t) for t ≥ t 0 .…”

“…Without loss of generality, we suppose that there exists t 0 ≥ a such that z(t) > 0, z [1] (t) < 0, z [2] (t) > 0 for t ≥ t 0 . Because (z [2] (t)) = q(t)f (z(t)) > 0 for t ≥ t 0 , z [2] is a positive increasing function on the interval [t 0 , ∞), and so we have that (z [1] (t)) ≥ z [2] (t 0 )r(t) for t ≥ t 0 . Integrating this On some classes of nonoscillatory solutions of third-order nonlinear differential equations inequality twice on the interval [t, ∞), t ≥ t 0 and taking into account the facts that −∞ < z [1] (∞) ≤ 0 and 0 ≤ z(∞) < ∞, we obtain…”

“…Assume that z ∈ M B 0 . Without loss of generality, we suppose that there exists t 0 ≥ a such that z(t) > 0, z [1] (t) < 0, z [2] (t) > 0 for t ≥ t 0 . Let z(∞) = l z > 0.…”

“…The object of this paper is to continue in study of the nonlinear case and to provide some other results for the equations of the form (N A ). For the sake of brevity, we introduce the following notation z [0] = z, z [1] = 1 p z , z [2] = 1 r 1 p z = 1 r z [1] , z [3] = 1 r 1 p z = z [2] .…”

“…The set N (N A ) can be divided into the following four classes in the same way as in [7,8]: [1] (t) < 0, z(t)z [2] (t) > 0 for t ≥ T z }, [1] (t) > 0, z(t)z [2] (t) < 0 for t ≥ T z }, [1] (t) > 0, z(t)z [2] (t) > 0 for t ≥ T z }.…”