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Acta Mathematicae Applicatae Sinica

, Volume 6, Issue 1, pp 67–73 | Cite as

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)
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,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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Science Press, Beijing, China and Allerton Press, Inc., New York, U.S.A. 1990

Authors and Affiliations

  • Yu Yuanhong 
    • 1
  1. 1.Institute of Applied MathematicsAcademia SinicaChina

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