Abstract
We study the oscillation and asymptotic behavior of third-order nonlinear delay differential equation with piecewise constant argument of the form(r2(t)(r1(t)x'(t))')'+p(t)x'(t)+f(t,x([t]))=0. We establish several sufficient conditions which insure that any solution of this equation oscillates or converges to zero. Some examples are given to illustrate the importance of our results.
Highlights
We study the oscillation and asymptotic behavior of third-order nonlinear delay differential equation with piecewise constant argument of the form r2 t r1 txtptxtft, x t 0
Throughout this paper, we assume that xf t, x ≥ 0 and that there exist functions q t and φ x such that i q t is continuous on 0, ∞ with q t > 0, ii φ x is continuously differentiable and nondecreasing on −∞, ∞, φ x /x ≥ K > 0 for x / 0, Journal of Applied Mathematics iii |f t, x | ≥ q t |φ x |, x / 0, t ≥ 0, iv Kq t − p t ≥ 0 and is not identically zero in any subinterval of 0, ∞
The delay functional differential equations provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon its past history
Summary
In the last few decades, there has been increasing interest in obtaining sufficient conditions for the oscillatory of solutions of different classes of the first-order differential equations with piecewise constant arguments, see 6–9 and the references therein. As far as we know, there are not works studying the oscillation and asymptotic behavior of third-order delay differential equations with piecewise constant argument. Motivated by this fact, in the present paper, we will investigate the oscillatory and asymptotic behavior of a certain class of third-order equation 1.1 with damping.
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