Asymptotic behavior of a class of nonlinear differential equations of 𝑛th order

Kong, Qingkai

doi:10.1090/S0002-9939-1988-0947667-1

Asymptotic behavior of a class of nonlinear differential equations of $n$th order
HTML articles powered by AMS MathViewer

by Qingkai Kong PDF

Proc. Amer. Math. Soc. 103 (1988), 831-838 Request permission

Abstract:

In this paper we obtain a result of the asymptotic behavior of the $n$th order equation ${u^{(n)}} + f(t,u,u’, \ldots ,{u^{(n - 1)}}) = 0$ under some assumptions. For $n = 2$ and $f(t,u,u’) \equiv f(t,u)$, it revises the result given by Jingcheng Tong, which is not true in general.

References

Jing Cheng Tong, The asymptotic behavior of a class of nonlinear differential equations of second order, Proc. Amer. Math. Soc. 84 (1982), no. 2, 235–236. MR 637175, DOI 10.1090/S0002-9939-1982-0637175-4
Donald S. Cohen, The asymptotic behavior of a class of nonlinear differential equations, Proc. Amer. Math. Soc. 18 (1967), 607–609. MR 212289, DOI 10.1090/S0002-9939-1967-0212289-3
Paul R. Beesack, On Lakshmikantham’s comparison method for Gronwall inequalities, Ann. Polon. Math. 35 (1977/78), no. 2, 187–222. MR 460764, DOI 10.4064/ap-35-2-187-222
U. D. Dhongade and S. G. Deo, A nonlinear generalization of Bihari’s inequality, Proc. Amer. Math. Soc. 54 (1976), 211–216. MR 430694, DOI 10.1090/S0002-9939-1976-0430694-2
Kurt Kreith, Extremal solutions for a class of nonlinear differential equations, Proc. Amer. Math. Soc. 79 (1980), no. 3, 415–421. MR 567983, DOI 10.1090/S0002-9939-1980-0567983-8