具有次线性中立项的二阶Emden-Fowler时滞微分方程的振动性

Oscillation criteria of second order Emden-Fowler time-delay differential equations with a sub-linear neutral term

研究含有次线性中立项的二阶Emden-Fowler时滞微分方程(r(t)([x(t)+p(t)xθ(τ(t))]′)α)′+q(t)xβ(σ(t))=0解的振动性,其中α,β,θ均为正奇数之商,0<θ≤1,β≥α。利用Riccati变换,积分平均和不等式技巧,建立了方程的三个新的振动准则。所得结果将经典的Leighton[1]和Kneser[2]振动准则推广到含有次线性中立项的超线性Emden-Fowler时滞微分方程。而且,新的结果不仅推广和改进了最近文献中出现的关于该方程当0<θ<1时的振动准则,同时也改进,推广和简化了方程当θ=1或者p(t)=0时的振动准则,所得准则的有效性通过若干例子给出了说明。

Here,the oscillation criteria for solutions of the second order Emden-Fowler time-delay differential equations with sublinear neutral term(r(t)([x(t)+p(t)xθ(τ(t))]′)α)′+q(t)xβ(σ(t))=0 were studied.α,β,θwere quotients of positive odd numbers and 0<θ≤1,β≥α.By using Riccati transformation,integral averaging and inequality techniques,3 new oscillation criteria for solutions of the equations were established to extend the classical Leighton and Kneser oscillation criteria to solutions of the equations.Furthermore,the new obtained results could not only generalize and improve the established oscillation criteria for solutions of the equations when 0<θ<1 published in recent literature,and but also improve,generalize and simplify the oscillation criteria for solutions of the equations whenθ=1 or p(t)=0.The effectiveness of the obtained criteria was illustrated with several examples.