| Title:
|
Nodal solutions for a second-order $m$-point boundary value problem (English) |
| Author:
|
Ma, Ruyun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
1243-1263 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the existence of nodal solutions of the $m$-point boundary value problem \[ u^{\prime \prime }+ f(u)=0, \quad 0<t<1, u^{\prime }(0)=0, \quad u(1)=\sum ^{m-2}_{i=1} \alpha _i u(\eta _i) \] where $\eta _i\in \mathbb{Q}$ $(i=1, 2, \cdots , m-2)$ with $0<\eta _1<\eta _2<\cdots <\eta _{m-2}<1$, and $\alpha _i\in \mathbb{R}$ $(i=1, 2, \cdots , m-2)$ with $\alpha _i>0$ and $0<\sum \nolimits ^{m-2}_{i=1} \alpha _i < 1$. We give conditions on the ratio $f(s)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques. (English) |
| Keyword:
|
multiplicity results |
| Keyword:
|
eigenvalues |
| Keyword:
|
bifurcation methods |
| Keyword:
|
nodal zeros |
| Keyword:
|
multi-point boundary value problems |
| MSC:
|
34B10 |
| MSC:
|
34C23 |
| MSC:
|
34G20 |
| MSC:
|
34L20 |
| MSC:
|
47J15 |
| MSC:
|
47N20 |
| idZBL:
|
Zbl 1164.34329 |
| idMR:
|
MR2280807 |
| . |
| Date available:
|
2009-09-24T11:42:37Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128143 |
| . |
| Reference:
|
[1] A. Ambrosetti, P. Hess: Positive solutions of asymptotically linear elliptic eigenvalue problems.J. Math. Anal. Appl. 73 (1980), 411–422. MR 0563992, 10.1016/0022-247X(80)90287-5 |
| Reference:
|
[2] A. Castro, P. Drábek, J. M. Neuberger: A sign-changing solution for a superlinear Dirichlet problem. II.Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), pp. 101–107. MR 1976635 |
| Reference:
|
[3] L. H. Erbe, H. Wang: On the existence of positive solutions of ordinary differential equations.Proc. Amer. Math. Soc. 120 (1994), 743–748. MR 1204373, 10.1090/S0002-9939-1994-1204373-9 |
| Reference:
|
[4] R. Ma: Existence of positive solutions for superlinear semipositone $m$-point boundary-value problems.Proc. Edin. Math. Soc. 46 (2003), 279–292. Zbl 1069.34036, MR 1998561, 10.1017/S0013091502000391 |
| Reference:
|
[5] R. Ma, B. Thompson: Nodal solutions for nonlinear eigenvalue problems.Nonlinear Analysis, Theory Methods Appl. 59 (2004), 717–718. MR 2096325 |
| Reference:
|
[6] Y. Naito, S. Tanaka: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations.Nonlinear Analysis TMA 56 (2004), 919–935. MR 2036055, 10.1016/j.na.2003.10.020 |
| Reference:
|
[7] P. H. Rabinowitz: Some global results for nonlinear eigenvalue problems.J. Funct. Anal. 7 (1971), 487–513. Zbl 0212.16504, MR 0301587, 10.1016/0022-1236(71)90030-9 |
| Reference:
|
[8] I. Rachůnková: On four-point boundary value problem without growth conditions.Czechoslovak Math. J. 49 (1999), 241–248. MR 1692485, 10.1023/A:1022491900369 |
| Reference:
|
[9] B. Ruf, P. N. Srikanth: Multiplicity results for ODEs with nonlinearities crossing all but a finite number of eigenvalues.Nonlinear Analysis TMA 10 (1986), 157–163. MR 0825214, 10.1016/0362-546X(86)90043-X |
| Reference:
|
[10] B. P. Rynne: Global bifurcation for $2m$th-order boundary value problems and infinity many solutions of superlinear problems.J. Differential Equations 188 (2003), 461–472. MR 1954290, 10.1016/S0022-0396(02)00146-8 |
| Reference:
|
[11] J. R. Webb: Positive solutions of some three-point boundary value problems via fixed point index theory.Nonlinear Analysis 47 (2001), 4319–4332. Zbl 1042.34527, MR 1975828, 10.1016/S0362-546X(01)00547-8 |
| Reference:
|
[12] X. Xu: Multiple sign-changing solutions for some $m$-point boundary value problems.Electronic Journal of Differential Equations 89 (2004), 1–14. Zbl 1058.34013, MR 2075428 |
| . |
