| Title:
|
Asymptotic behavior of solutions of a $2n^{th}$ order nonlinear differential equation (English) |
| Author:
|
Lin, C. S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2002 |
| Pages:
|
665-672 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for \[ \left\rbrace \begin{array}{ll}(-1)^n u^{(2n)} + f(t,u) = 0,\hspace{5.0pt}\text{in} \hspace{5.0pt}(\alpha , \infty ), u^{(i)}(\xi ) = 0, \quad i = 0,1,\dots , n-1, \hspace{5.0pt} \text{and} \hspace{5.0pt}\xi \in (\alpha , \infty ), \end{array}\right.\] must be unbounded, provided $f(t,z)z\ge 0$, in $E \times \mathbb R$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E \times I$. (B) Every bounded solution for $(-1)^n u^{(2n)} + f(t,u) = 0$, in $\mathbb R$, must be constant, provided $f(t,z)z\ge 0$ in $\mathbb R \times \mathbb R$ and for every bounded subset $I$, $f(t,z)$ is bounded in $\mathbb R \times I$. (English) |
| Keyword:
|
asymptotic behavior |
| Keyword:
|
higher order differential equation |
| MSC:
|
34C10 |
| MSC:
|
34C11 |
| MSC:
|
34D05 |
| idZBL:
|
Zbl 1023.34032 |
| idMR:
|
MR1923270 |
| . |
| Date available:
|
2009-09-24T10:55:19Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127752 |
| . |
| Reference:
|
[1] M. Biernacki: Sur l’equation differentielle $y^{(4)} + A(x)y = 0$.Ann. Univ. Mariae Curie-Skłodowska 6 (1952), 65–78. MR 0064230 |
| Reference:
|
[2] S. P. Hastings and A. C. Lazer: On the asymptotic behavior of solutions of the differential equation $y^{(4)} = p(x)y$.Czechoslovak Math. J. 18(93) (1968), 224–229. MR 0226110 |
| Reference:
|
[3] G. D. Jones: Asymptotic behavior of solutions of a fourth order linear differential equation.Czechoslovak Math. J. 38(113) (1988), 578–584. Zbl 0672.34052, MR 0962901 |
| Reference:
|
[4] G. D. Jones: Oscillatory solutions of a fourth order linear differential equation.Lecture notes in pure and apllied Math. Vol 127, 1991, pp. 261–266. MR 1096762 |
| Reference:
|
[5] M. K. Kwong and A. Zettl: Norm Inequalities for Derivatives and Differences. Lecture notes in Mathematics, 1536.Springer-Verlag, Berlin, 1992. MR 1223546 |
| Reference:
|
[6] M. Švec: Sur le comportement asymtotique des intégrales de l’équation differentielle $y^{(4)} + Q(x)y = 0$.Czechoslovak Math. J. 8(83) (1958), 230–245. MR 0101355 |
| . |
