一类二阶泛函微分方程的区间振动研究

Interval Oscillation for A Class of Second Order Functional Differential Equations

考虑一类二阶泛函微分方程:(r(t)ψ(x(t))φ(x′(t)))′+p(t)φ(x′(t))+f(t,x(t),x(τ(t)),x′(t),x′(τ(t)))=0。其中,r,τ∈C([t0,∞),(0,∞)),limt→∞τ(t)=∞,ψ∈C(R,R),φ(u)=|u|α-1u,α为大于零的常数,p∈C([t0,∞),R),f∈C([t0,∞)×R4,R),t0>0,通过构造适当的Riccati变换,并利用积分平均方法及完全平方技术,得到了方程的解振动的一些新的充分条件,推广了相关文献的结果。

The second order functional differential equation of the form(r(t)ψ(x(t))φ(x'(t)))'+p(t)φ(x'(t))+f(t,x(t),x(τ(t)),x'(t),x'(τ(t)))=0 is considered,where r,τ∈C([t0,∞),(0,∞)),limt→∞τ(t)=∞,ψ∈C(R,R)φ(u)=-u-α-1u,α is a conatant more than zero,p∈C([t0,∞),R),f∈C([t0,∞)×R4,R),t0>0.By using generalized Riccati transformation,some new sufficient conditions for solutions of the equation to oscillate are obtained through the method of integral average and perfect square.The results improve some of the known results in the literature.