ISSN:
1618-3932
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Consider a retarded differential equation (1) $$x^{a - 1} (t)x'(t) + P_0 (t)x^a (t) + \sum\limits_{i = 1}^N {P_i (t)x^a (g_i (t)) = 0,} g_i (t)〈 t,$$ and an advanced differential equation (2) $$x^{a - 1} (t)x'(t) - P_0 (t)x^a (t) - \sum\limits_{i = 1}^N {P_i (t)x^a (g_i (t)) = 0,} g_i (t) 〉 t,$$ 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 (3) $$\mathop {\inf }\limits_{\mu 〉 0,t \geqslant T} \frac{\alpha }{\mu }\sum\limits_{i = 1}^N {P_i (t)\exp [\alpha B_i + \mu T_i (t)]} 〉 1.$$ 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02014717
