| Title:
|
Existence of one-signed solutions of nonlinear four-point boundary value problems (English) |
| Author:
|
Ma, Ruyun |
| Author:
|
Chen, Ruipeng |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
593-612 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$ -u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ) $$ and $$ u''+Mu=rg(t)f(u), \quad u(0)=u(\varepsilon ),\quad u(1)=u(1-\varepsilon ), $$ where $\varepsilon \in (0,{1}/{2})$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C([0,1],(0,+\infty ))$, $f\in C(\mathbb {R},\mathbb {R})$ with $sf(s)>0$ for $s\neq 0$. The proof of the main results is based upon bifurcation techniques. (English) |
| Keyword:
|
four-point boundary value problem |
| Keyword:
|
one-signed solution |
| Keyword:
|
bifurcation method |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| MSC:
|
34B18 |
| MSC:
|
34C23 |
| idZBL:
|
Zbl 1265.34053 |
| idMR:
|
MR2984621 |
| DOI:
|
10.1007/s10587-012-0052-3 |
| . |
| Date available:
|
2012-11-10T20:58:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143012 |
| . |
| Reference:
|
[1] Chu, J., Sun, Y., Chen, H.: Positive solutions of Neumann problems with singularities.J. Math. Anal. Appl. 337 (2008), 1267-1272 \MR 2386375. Zbl 1142.34315, MR 2386375, 10.1016/j.jmaa.2007.04.070 |
| Reference:
|
[2] Deimling, K.: Nonlinear Functional Analysis.Springer, Berlin (1985) \MR 0787404. Zbl 0559.47040, MR 0787404 |
| Reference:
|
[3] Jiang, D., Liu, H.: Existence of positive solutions to second order Neumann boundary value problems.J. Math. Res. Expo. 20 (2000), 360-364. Zbl 0963.34019, MR 1787796 |
| Reference:
|
[4] Li, X., Jiang, D.: Optimal existence theory for single and multiple positive solutions to second order Neumann boundary value problems.Indian J. Pure Appl. Math. 35 (2004), 573-586. Zbl 1070.34038, MR 2071406 |
| Reference:
|
[5] Li, Z.: Positive solutions of singular second-order Neumann boundary value problem.Ann. Differ. Equations 21 (2005), 321-326. Zbl 1090.34524, MR 2175705 |
| Reference:
|
[6] Ma, R., Thompson, B.: Nodal solutions for nonlinear eigenvalue problems.Nonlinear Anal., Theory Methods Appl. 59 (2004), 707-718. Zbl 1059.34013, MR 2096325, 10.1016/j.na.2004.07.030 |
| Reference:
|
[7] Miciano, A. R., Shivaji, R.: Multiple positive solutions for a class of semipositone Neumann two-point boundary value problems.J. Math. Anal. Appl. 178 (1993), 102-115. Zbl 0783.34016, MR 1231730, 10.1006/jmaa.1993.1294 |
| Reference:
|
[8] Rabinowitz, P. H.: 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:
|
[9] Rachůnková, I., Staněk, S., Tvrdý, M.: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations.Hindawi Publishing Corporation, New York (2008). MR 2572243 |
| Reference:
|
[10] Sun, J., Li, W.: Multiple positive solutions to second-order Neumann boundary value problems.Appl. Math. Comput. 146 (2003), 187-194. Zbl 1041.34013, MR 2007778, 10.1016/S0096-3003(02)00535-0 |
| Reference:
|
[11] Sun, J., Li, W., Cheng, S.: Three positive solutions for second-order Neumann boundary value problems.Appl. Math. Lett. 17 (2004), 1079-1084. Zbl 1061.34014, MR 2087758, 10.1016/j.aml.2004.07.012 |
| Reference:
|
[12] Sun, Y., Cho, Y. J., O'Regan, D.: Positive solution for singular second order Neumann boundary value problems via a cone fixed point theorem.Appl. Math. Comput. 210 (2009), 80-86 \MR 2504122. MR 2504122, 10.1016/j.amc.2008.11.025 |
| . |
