一类高阶非线性微分方程解的渐近性

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A HIGH ORDER NONLINEAR DIFFERENTIAL EQUATION

研究了 2n阶微分方程的渐近性 ,得到了如下两个结果 .在E×R上有f(t,z)z≥ 0 ,且对于每一有界子集I,f(t,z)在E×I上有界 ,则 (A)方程(- 1) nu( 2n) +f(t,u) =0 ,E =(α ,∞ ) ,u(i) (ξ) =0 ,i=0 ,1,2 ,… ,n- 1,ξ∈ (α ,∞ ) ,的每一非平凡解都是无界 .(B)假设在R×R上f(t,z)z≥ 0 ,且对于每一有界子集I,f(t,z)在R×I上有界 ,则方程 (- 1) nu( 2n) +f(t,u) =0在R内的每一有界解都是常数 .这些结论推广了JonesGD (1991)

The authors prove two results. (A) Every nontrivial solution for(-1)~nu^((2n))+f(t, u)=0, E=(α, ∞), u^((i))(ξ)= 0, i= 0, 1, 2, …, n-1, ξ∈(α, ∞), must be unbounded, provided f(t, z)z≥0, in E×R and for every bounded subset I, f(t, z) is bounded in E×I. (B) Every bounded solution for (-1)~nu^((2n))+f(t, u)=0, in R, must be constant, provided f(t, z)z ≥0 in R×R and for every bounded subset I, is bounded in R×I.