一类二阶非线性微分方程的解的渐近性

Asymptotic Behaviour of the Solutions to a Class of Second Order Nonlinear Differential Equation

关键词:  微分方程  渐近性  非线性  Schauder不动点定理

Key words:  differential equation  asymptotic behavior  nonlinear  Schauder's fixed point theorem

摘要:

利用Schauder不动点定理,研究二阶非线性微分方程u″=f(t,u,u'),t ≥ 1解(f∈C[[1,∞)×R×R,R])的渐近性,给出方程解渐近于直线at+b(a,b∈R)的一个充分条件.从而推广文献[2]定理1的结果,简化文献[3]中定理4成立的条件.

Abstract:

This paper concerned about the asymptotic behavior of solutions of the nonlinear differential equation:u″=f(t,u,u'),t ≥ 1 where f∈C[[1,∞)×R×R,R].In our approach a key role is played by Schauder's fixed point theorem.A suffcient condition is given for this equation to have solutions behaving asymptotically like linear functions at+b where a,b∈R.Our results improve theorem 1 of Reference[2].And the conditions of theorem 4 of Reference[3] are simplified.