| Title:
|
The method of upper and lower solutions for a Lidstone boundary value problem (English) |
| Author:
|
Guo, Yanping |
| Author:
|
Gao, Ying |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
639-652 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we develop the monotone method in the presence of upper and lower solutions for the $2$nd order Lidstone boundary value problem \[ u^{(2n)}(t)=f(t,u(t),u^{\prime \prime }(t),\dots ,u^{(2(n-1))}(t)),\quad 0<t<1, u^{(2i)}(0)=u^{(2i)}(1)=0,\quad 0\le i\le n-1, \] where $f\:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ is continuous. We obtain sufficient conditions on $f$ to guarantee the existence of solutions between a lower solution and an upper solution for the higher order boundary value problem. (English) |
| Keyword:
|
$n$-parameter eigenvalue problem |
| Keyword:
|
Lidstone boundary value problem |
| Keyword:
|
lower solution |
| Keyword:
|
upper solution |
| MSC:
|
34B15 |
| idZBL:
|
Zbl 1081.34019 |
| idMR:
|
MR2153088 |
| . |
| Date available:
|
2009-09-24T11:26:11Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128008 |
| . |
| Reference:
|
[1] A. R. Aftabizadeh: Existence and uniqueness theorems for fourth-order boundary value problems.J. Math. Anal. Appl. 116 (1986), 415–426. Zbl 0634.34009, MR 0842808, 10.1016/S0022-247X(86)80006-3 |
| Reference:
|
[2] C. De Coster, C. Fabry and F. Munyamarere: Nonresonance conditions for fourth-order nonlinear boundary problems.Internat. J. Math. Sci. 17 (1994), 725–740. MR 1298797, 10.1155/S0161171294001031 |
| Reference:
|
[3] M. A. Del Pino and R. F. Manasevich: Existence for a fourth-order boundary value problem under a two parameter nonresonance condition.Proc. Amer. Math. Soc. 112 (1991), 81–86. MR 1043407, 10.2307/2048482 |
| Reference:
|
[4] C. P. Gupta: Existence and uniqueness theorem for a bending of an elastic beam equation.Appl. Anal. 26 (1988), 289–304. MR 0922976, 10.1080/00036818808839715 |
| Reference:
|
[5] R. A. Usmani: A uniqueness theorem for a boundary value problem.Proc. Amer. Math. Soc. 77 (1979), 327–335. Zbl 0424.34019, MR 0545591, 10.1090/S0002-9939-1979-0545591-4 |
| Reference:
|
[6] R. Agarwal: On fourth-order boundary value problems arising in beam analysis.Differential Integral Equations 2 (1989), 91–110. Zbl 0715.34032, MR 0960017 |
| Reference:
|
[7] A. Cabada: The method of lower and upper solutions for second, third, fourth and higher order boundary value problems.J. Math. Anal. Appl. 185 (1994), 302–320. Zbl 0807.34023, MR 1283059, 10.1006/jmaa.1994.1250 |
| Reference:
|
[8] C. De Coster and L. Sanchez: Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth-order O. D. E.Riv. Mat. Pura. Appl. 14 (1994), 1129–1138. MR 1275354 |
| Reference:
|
[9] P. Korman: A maximum principle for fourth-order ordinary differential equations.Appl. Anal. 33 (1989), 267–273. Zbl 0681.34016, MR 1030113, 10.1080/00036818908839878 |
| Reference:
|
[10] J. Schröder: Fourth-order two-point boundary value problems.Nonlinear Anal. 8 (1984), 107–114. 10.1016/0362-546X(84)90063-4 |
| Reference:
|
[11] Zhanbing Bai: The method of lower and upper solutions for a bending of an elastic beam equation.J. Math. Anal. Appl. 248 (2000), 195–402. MR 1772591, 10.1006/jmaa.2000.6887 |
| Reference:
|
[12] R. Y. Ma, J. H. Zhang and S. M. Fu: The method of lower and upper solutions for fourth-order two-point boundary value problems.J. Math. Anal. Appl. 215 (1997), 415–422. MR 1490759, 10.1006/jmaa.1997.5639 |
| Reference:
|
[13] J. M. Davis and J. Henderson: Triple positive symmetric solutions for a Lidstone boundary value problem.Differential Equations Dynam. Systems 7 (1999), 321–330. MR 1861076 |
| Reference:
|
[14] J. M. Davis, P. W. Eloe and J. Henderson: Triple positive solutions and dependence on higher order derivatives.J. Math. Anal. Appl. 237 (1999), 710–720. MR 1710319, 10.1006/jmaa.1999.6500 |
| Reference:
|
[15] J. M. Davis, J. Henderson and P. J. Y. Wong: General Lidstone problems: multiplicity and symmetry of solutions.J. Math. Anal. Appl. 251 (2000), 527–548. MR 1794756, 10.1006/jmaa.2000.7028 |
| Reference:
|
[16] D. Gilbarg and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order.Springer-Verlag, New York, 1977. MR 0473443 |
| . |
