时间模上一类二阶泛函动态方程振荡的充分条件

Sufficient conditions of oscillation for certain second-order functional dynamic equations on time scales

研究了一类时间模上二阶Emden-Fowler型变时滞的中立型泛函动态方程{a(t)φ([x(t)+p(t)g(x(τ(t)))]^Δ)}^Δ+q1(t)f1(φ1(x(^Δ1(t))))+q2(t)f2(φ2(x(^Δ2(t))))=0的振荡性,其中,φ(u)=|u|^α-1u(α>0),φ1(u)=|u|^β-1u(β>0),φ2(u)=|u|^γ-1u(γ>0)。利用时间模上的有关理论和广义黎卡提变换技术,并借助各种不等式,得到了该方程振荡的一些新的充分条件,推广并丰富了一些已有结果。最后,给出了一些有趣的实例以说明文中的结果。

This paper is concerned with oscillatory behavior of the following second-order Emden-Fowler variable delay neutral functional dynamic equation { a ( t )φ([ x ( t )+ p ( t ) g ( x (τ( t )))]^Δ)}^Δ+ q1 ( t ) f1 (φ1 ( x (^Δ1 ( t ))))+ q2 ( t ) f2 (φ2 ( x (^Δ2 ( t ))))= 0 on a time scale T, where φ( u )=|u|α- 1 u(α>0),φ1 ( u )=|u|β- 1 u(β>0) and φ2 ( u )=|u|γ- 1 u(γ>0). By using the time scales theory and the Riccati transformation as well as the inequality technique, we establish some new sufficient conditions of oscillation for the equation. Our results deal with some cases not covered by the existing results in the literature. Finally, some interesting examples are given to illustrate the versatility of our results.