Abstract
New oscillation results to fourth order delay differential equations with damping
Highlights
We consider the fourth-order trinomial differential equation with delay argument+ p(t)y (t) + q(t)y(τ(t)) = 0, for t ≥ t0. (E)Throughout the paper, the following hypotheses will be made:(H1) p, q, τ ∈ C([t0, ∞), R) such that p(t) ≥ 0, q(t) > 0, τ(t) ≤ t for all t ≥ t0 and lim τ(t) = ∞. t→∞(H2) ri(t) ∈ C([t0, ∞), R), ri(t) > 0,∞ ds = ∞, t0 ri(s) i = 1, 2, 3.J
We offer a comparison theorem that relates properties of solutions of (E) with those of second-order differential equations
There has been an open problem regarding the study of sufficient conditions ensuring oscillation of all solutions of fourth-order differential equation with damping
Summary
We consider the fourth-order trinomial differential equation with delay argument. + p(t)y (t) + q(t)y(τ(t)) = 0, for t ≥ t0. Prototypes of higher-order trinomial differential equations with delay, which have been primarily studied in the literature are such that a difference in the derivative order between the first and the middle term differs either by one or two [4, 9]. By means of the Riccati technique, they presented some sufficient conditions under which any solution of (E0) oscillates or tends to zero as t → ∞. Their crucial “preliminary” theorem ensures a constant sign of the first-derivative y(t) provided an auxiliary third-order differential equation z (t) + p(t)z(t) = 0. As an application of that principle, we will use the Riccati transformation technique to establish a new sufficient condition ensuring oscillation of all solutions of the studied trinomial equation (E). The criterion derived directly involves a coefficient p(t) pertaining to a damped term and does not depend on solutions of the auxiliary differential equation
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