具非线性中立项时滞微分方程解的振动性

在α≠1且β∈(0,1)情形下研究了一阶具非线性中立项时滞微分方程x(t)-pxα(t-τ)]′+q(t)xβ(t-σ),t≥t0解的振动性和非振动性,在β∈(0,1)情形下获得了上述方程所有有界解振动的充要条件,同时在α∈(1,∞)且β∈(0,1)情形下获得了上述方程存在无界正解的充分条件.这些新的结果填补了已有文献中空白.

Oscillation of solutions of delay differential equations with a nonlinear neutral term

The oscillatory behavior of solutions of the delay differential equations with nonlinear neutral termx(t)-px~α(t-τ)]′+q(t)x~β(t-σ),t≥t_0is studied in the case when α≠1 and β∈(0,1).The necessary and sufficient conditions are obtained which guarantee every bounded solutions of the above equation oscillates in the case β∈(0,1),and the sufficient conditions are also obtained for the existence of an unbounded positive solution of the above equation in the case α∈(1,∞) and β∈(0,1).These results fill a gap in the literature.