广义Emden-Fowler方程的振动性

Oscillation Criteria of Generalized Emden-Fowler Equations

本文考虑广义Emden-Fowler方程(r(t)|z′(t)|^(α-1)z′(t))′+q(t)|x(σ(t))|^(β-1)x(σ(t))=0,t≥t_0,其中z(t)=x(t)+p(t)x(τ(t)),α≥β>0.利用Riccati变换,积分平均技巧和Hardy-Littlewood-Polya的一个不等式在两种情况∫_t_0~∞r^(-1/α)(t)dt=∞和∫_t_0~∞r^(-1/α)(t)dt<∞下建立了所考虑方程的若干新的振动准则,所得结果推广和改进了最近文献中的一些熟知定理.文中也给出了例子说明我们结果的应用.

In this paper, we are concerned with the generalized Emden-Fowler equation of the form （r（t）｜z′（t）｜^α-1z′（t））′＋q（t）｜x（σ（t））｜^β-1x（σ（t））=0,t≥t0,where z（t）=x（t）＋p（t）x（τ（t））,α≥β〉0.By use of the Riccati transformation, integral averaging technique and an inequality due to Hardy-Littlewood-Polya, new oscillation criteria are established both for the case ∫t0^∞r^-1/α（t）dt=∞,and the case ∫t0^∞r^-1/α（t）dt〈∞.These criteria here derived extend and improve some known results in the recent literature. Some examples are given to illustrate our main results.