Oscillations caused by several retarded and advanced arguments

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))

and an advanced differential equation

$$x^{a - 1} (t)x'(t) - P_0 (t)x^a (t) - \sum\limits_{i = 1}^N {P_i (t)x^a (g_i (t)) = 0,} g_i (t) > t,$$

((2))

wherea=m/n, m andn are odd natural numbers,P ₀(t),P _i(t) andg _i(t) are continuous functions, andP _i(t) are positive-valued on [t ₀, ∞), limg _i(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$B_i = \mathop {\inf }\limits_{t \geqslant T} \int_{D_i } {P_0 (s)ds > - \infty }$ D _i=[g _i(t),t],T _i(t)=t−g _i(t), for (1), andD _i=[t,g _i(t)].T _i(t)=g _i(t)−t for (2),i=1, 2, ...,N.