Limit boundary value problem for nonlinear differential equation of the second order. (English) Zbl 0616.34009
Consider the limit boundary value problem (LBVP)
\[
(*)\quad \ddot x=X(t,x,\dot x)a\dot x(0)+bx(0)=cx(+\infty)=const.
\]
Under certain conditions, the author establishes necessary and sufficient conditions for the existence of the solution of this LBVP. The asymptotic behavior of bounded solutions of (*) is discussed.
Reviewer: K.S.Miller
MSC:
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34E99 | Asymptotic theory for ordinary differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
References:
| [1] | Сансоне Дж., О{\(\beta\)}ыкновенные дифференпиальные еравнения, T. II, гл. XII, § 7, Москова, 1954. · Zbl 0342.02023 |
| [2] | Мышкнс А. Д., Щербина Г. В., Об одной предельные краевой задаче, не удовлеворяющей условию условию С. Н. Бернштейна и имеющей приложение в теории капиллюрных ювлений,Дuфф. Урае. 12 (1976), 991–998. · JFM 29.0445.02 |
| [3] | Щерббина Г. Б., Достаточные условия разрешнмости одной нелиейвой краевой задачи на полуоси,Дuфф. Урае., 11 (1975), 2189–2195. · Zbl 0324.20051 |
| [4] | Hartman P., Wintner A., On the non-increasing solutions ofy”=f(x,y,y’), Amer. J. Math., 73 (1951), 390–404. · Zbl 0042.32601 · doi:10.2307/2372184 |
| [5] | Клоков Ю. А., Одна предельнаая краевая задача уравнения \(\ddot x + \dot xf(x,\dot x) + \varphi (x) = 0\) ,Изе. ВУЗ ое, Mam., 6 (1959), 72–79. |
| [6] | Клоков ю. А., Метод решения предельной краевой для обыкновенного дифференциального уравнения вторго порядка,Mam. сб., 53 (1961), 219–232. · JFM 27.0465.06 |
| [7] | Клоков ю. А., Об одной краевой задаче с условием на бесконечности для обыкновенного дифференциального уравнения второго порядка,УМН, 17:6 (1962), 145–149. · Zbl 0342.02023 |
| [8] | Taliaferro S. D., Asymptotic behavior of solutions ofy”=ф(t)y {\(\lambda\)},J. Math. Anal. Appl., 66 (1978), 95–134. · Zbl 0399.34048 · doi:10.1016/0022-247X(78)90272-X |
| [9] | Liang Zhongchao, Asymptotic charactor of the solutions of a class of second order nonlinear differential equation. (in Chinese)Shuxue Jinzhan, 9 (1966), 251–264. |
| [10] | Liang Zhongchao, Asymptotic behavior of the solutions of a class of second order nonlinear differential equations,Acta Math. Sinica, 24 (1981), 45–54. · Zbl 0494.34032 |
| [11] | Liang Zhongchao, The boundary value problem on an infinite interval for nonlinear differential equation of second order,Acta Appl. Math. Sinica, 4 (1981), 272–279. · Zbl 0476.34018 |
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