Acta Mathematicae Applicatae Sinica

, Volume 6, Issue 1, pp 67–73

# Oscillations caused by several retarded and advanced arguments

• Yu Yuanhong
Article

## Abstract

Consider a retarded differential equation
$$x^{a - 1} (t)x'(t) + P_0 (t)x^a (t) + \sum\limits_{i = 1}^N {P_i (t)x^a (g_i (t)) = 0,} g_i (t)< t,$$
(1)
$$x^{a - 1} (t)x'(t) - P_0 (t)x^a (t) - \sum\limits_{i = 1}^N {P_i (t)x^a (g_i (t)) = 0,} g_i (t) > t,$$
(2)
wherea=m/n, m andn are odd natural numbers,P0(t),Pi(t) andgi(t) are continuous functions, andPi(t) are positive-valued on [t0, ∞), limgi(t)=∞,i=1, 2, ...,N. We prove the following
Theorem. Suppose that there is a constantT such that
$$\mathop {\inf }\limits_{\mu > 0,t \geqslant T} \frac{\alpha }{\mu }\sum\limits_{i = 1}^N {P_i (t)\exp [\alpha B_i + \mu T_i (t)]} > 1.$$
(3)

Then all solutions of (1) and (2) are oscillatory.

Here$$B_i = \mathop {\inf }\limits_{t \geqslant T} \int_{D_i } {P_0 (s)ds > - \infty }$$Di=[gi(t),t],Ti(t)=tgi(t), for (1), andDi=[t,gi(t)].Ti(t)=gi(t)−t for (2),i=1, 2, ...,N.

### Keywords

Differential Equation Continuous Function Natural Number Math Application Advanced Argument

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