摘要
考虑二阶非线性泛函微分方程y″(t) +a(t) f (y(t) ) +b(t) y(t-τ) +c(t) y′(t) =0 (* )y″(t) +a(t) f(y(t) ) +b(t) g(y(t- τ) ) +c(t) y′(t) =0 ,(* * )其中 a∈ C1([0 ,∞ ,(0 ,∞ ) ) ,b∈ C([0 ,∞ ) ,R) ,c∈ C([0 ,∞ ) ,(0 ,∞ ) ) ,f ,g∈ C(R,R)且存在常数λ>0 ,μ>0 ,使当 u≠ 0时有 f(u)u ]≥ λ,g2 (u)≤ μu2 .文章得到方程 (* * )所有解有界的一个充分条件为 ,存在函数 h∈ C1([0 ,∞ ) ,(0 ,∞ ) ) ,使得h(t)≥ b2 (t)a′(t) +2 a(t) c(t) , h′(t)≤ 0 ,∫∞ h(s) ds<∞ .
Consider the second-order nonlinear functi onal differential equationy″(t)+a(t)f(y(t))+b(t)y(t-τ)+c(t)y′(t)=0(*)y″(t)+a(t)f(y(t))+b(t)g(y(t-τ))+c(t)y′(t)=0,(**)where a∈C 1([0,∞,(0,∞)),b∈C([0,∞),R),c∈C([0, ∞),(0,∞)),f,g∈C(R,R) and there exists a constant λ>0,μ>0,such thatf(u)u]≥λ and g 2(u)≤μu 2 for u≠0.A sufficient condit ion for the boundedness of all solutions of (**)is obtained,that is,there exists a function h∈C 1(≥λ and g 2(u)≤μu 2 for u≠0.A sufficient condit ion for the boundedness of all solutions of (**)is obtained,that is,there exists a function h∈C 1([0,∞),(0,∞)),such thath(t)≥b 2(t)a′(t)+2a(t)c(t),h′(t)≤0, ∫ ∞h(s)ds<∞.
出处
《太原师范学院学报（自然科学版）》
2003年第3期5-7,共3页
Journal of Taiyuan Normal University:Natural Science Edition
