| Title:
|
A note on the existence of positive solutions of one-dimensional $p$-Laplacian boundary value problems (English) |
| Author:
|
Liu, Yuji |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
55 |
| Issue:
|
3 |
| Year:
|
2010 |
| Pages:
|
241-264 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper is concerned with the existence of positive solutions of a multi-point boundary value problem for higher-order differential equation with one-dimensional $p$-Laplacian. Examples are presented to illustrate the main results. The result in this paper generalizes those in existing papers. (English) |
| Keyword:
|
one-dimension $p$-Laplacian differential equation |
| Keyword:
|
nonlocal boundary value problem |
| Keyword:
|
positive solution |
| Keyword:
|
fixed-point theorem |
| MSC:
|
34B10 |
| MSC:
|
34B15 |
| MSC:
|
34B18 |
| MSC:
|
35B10 |
| idZBL:
|
Zbl 1224.34070 |
| idMR:
|
MR2657836 |
| DOI:
|
10.1007/s10492-010-0010-z |
| . |
| Date available:
|
2010-07-20T13:46:54Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140397 |
| . |
| Reference:
|
[1] Agarwal, R. P., O'Regan, D., Wong, P. J. Y.: Positive Solutions of Differential, Difference and Integral Equations.Kluwer Academic Dordrecht (1999). Zbl 1157.34301, MR 1680024 |
| Reference:
|
[2] Bai, C., Fang, J.: Existence of multiple positive solutions for nonlinear $m$-point boundary value problems.Appl. Math. Comput. 140 (2003), 297-305. Zbl 1030.34026, MR 1953901, 10.1016/S0096-3003(02)00227-8 |
| Reference:
|
[3] Bai, C., Fang, J.: Existence of multiple positive solutions for nonlinear multi-point boundary value problems.J. Math. Anal. Appl. 281 (2003), 76-85. MR 1980075, 10.1016/S0022-247X(02)00509-7 |
| Reference:
|
[4] delPino, M. A., Elgueta, M., Manásevich, R. F.: A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0$, $u(0)=u(T)=0$, $p>1$.J. Differ. Equ. 80 (1989), 1-13. MR 1003248, 10.1016/0022-0396(89)90093-4 |
| Reference:
|
[5] delPino, M. A., Manásevich, R. F.: Multiple solutions for the $p$-Laplacian under global nonresonance.Proc. Am. Math. Soc. 112 (1991), 131-138. MR 1045589 |
| Reference:
|
[6] Gaines, R. E., Mawhin, J. L.: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math., Vol. 568.Springer Berlin (1977). MR 0637067, 10.1007/BFb0089537 |
| Reference:
|
[7] Guo, Y., Ge, W.: Three positive solutions for the one dimension $p$-Laplacian.J. Math. Anal. Appl. 286 (2003), 491-508. MR 2008845, 10.1016/S0022-247X(03)00476-1 |
| Reference:
|
[8] Ji, D., Feng, M., Ge, W.: Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity.Appl. Math. Comput. 196 (2008), 511-520. Zbl 1138.34014, MR 2388707, 10.1016/j.amc.2007.06.028 |
| Reference:
|
[9] Karakostas, G. L.: Triple positive solutions for the $\Phi$-Laplacian when $\Phi$ is a sup-multiplicative-like function.Electron. Diff. Equ. 69 (2004), 1-13. Zbl 1057.34010, MR 2057656 |
| Reference:
|
[10] Karakostas, G. L.: Positive solutions for the $\Phi$-Laplacian when $\Phi$ is a sup-multiplicative-like function.Electron. Diff. Equ. 68 (2004), 1-12. Zbl 1057.34010, MR 2057655 |
| Reference:
|
[11] Kosmatov, N.: Symmetric solutions of a multi-point boundary value problem.J. Math. Anal. Appl. 309 (2005), 25-36. Zbl 1085.34011, MR 2154024, 10.1016/j.jmaa.2004.11.008 |
| Reference:
|
[12] Lan, K. D.: Multiple positive solutions of semilinear differential equations with singularities.J. Lond. Math. Soc., Ser. II 63 (2001), 690-704. Zbl 1032.34019, MR 1825983, 10.1112/S002461070100206X |
| Reference:
|
[13] Lian, W.-C., Wong, F.: Existence of positive solutions for higher-order generalized $p$-Laplacian BVPs.Appl. Math. Lett. 13 (2000), 35-43. Zbl 0964.34018, MR 1772689, 10.1016/S0893-9659(00)00051-3 |
| Reference:
|
[14] Liang, R., Peng, J., Shen, J.: Positive solutions to a generalized second order three-point boundary value problem.Appl. Math. Comput. 196 (2008), 931-940. Zbl 1140.34313, MR 2388746, 10.1016/j.amc.2007.07.025 |
| Reference:
|
[15] Liu, B.: Positive solutions of a nonlinear four-point boundary value problems.Appl. Math. Comput. 155 (2004), 179-203. Zbl 1068.34011, MR 2078102, 10.1016/S0096-3003(03)00770-7 |
| Reference:
|
[16] Liu, Y., Ge, W.: Multiple positive solutions to a three-point boundary value problem with $p$-Laplacian.J. Math. Anal. Appl. 277 (2003), 293-302. Zbl 1026.34028, MR 1954477, 10.1016/S0022-247X(02)00557-7 |
| Reference:
|
[17] Lü, H., O'Regan, D., Zhong, C.: Multiple positive solutions for the one dimensional singular $p$-Laplacian.Appl. Math. Comput. 133 (2002), 407-422. MR 1924626, 10.1016/S0096-3003(01)00240-5 |
| Reference:
|
[18] Ma, R.: Existence of positive solutions for superlinear semipositone $m$-point boundary-value problems.Proc. Edinb. Math. Soc., II. Ser. 46 (2003), 279-292. Zbl 1069.34036, MR 1998561, 10.1017/S0013091502000391 |
| Reference:
|
[19] Ma, R.: Multiple nonnegative solutions of second-order systems of boundary value problems.Nonlinear Anal., Theory Methods Appl. 42 (2000), 1003-1010. Zbl 0973.34014, MR 1780450, 10.1016/S0362-546X(99)00152-2 |
| Reference:
|
[20] Ma, R.: Nodal solutions for a second-order $m$-point boundary value problem.Czech. Math. J. 56 (2006), 1243-1263. Zbl 1164.34329, MR 2280807, 10.1007/s10587-006-0092-7 |
| Reference:
|
[21] Ma, R., Castaneda, N.: Existence of solutions of nonlinear $m$-point boundary value problems.J. Math. Anal. Appl. 256 (2001), 556-567. Zbl 0988.34009, MR 1821757, 10.1006/jmaa.2000.7320 |
| Reference:
|
[22] Ma, R., Thompson, B.: Positive solutions for nonlinear $m$-point eigenvalue problems.J. Math. Anal. Appl. 297 (2004), 24-37. Zbl 1057.34011, MR 2079645, 10.1016/j.jmaa.2003.12.046 |
| Reference:
|
[23] Wang, Y., Zhao, W., Ge, W.: Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional $p$-Laplacian.J. Math. Anal. Appl. 326 (2007), 641-654. Zbl 1119.34050, MR 2277809, 10.1016/j.jmaa.2006.03.028 |
| Reference:
|
[24] Webb, J. R. L.: Positive solutions of some three point boundary value problems via fixed point index theory.Nonlinear Anal., Theory Methods Appl. 47 (2001), 4319-4332. Zbl 1042.34527, MR 1975828, 10.1016/S0362-546X(01)00547-8 |
| Reference:
|
[25] Zhang, G., Sun, J.: Positive solutions of $m$-point boundary value problems.J. Math. Anal. Appl. 291 (2004), 406-418. Zbl 1069.34037, MR 2038179, 10.1016/j.jmaa.2003.11.034 |
| Reference:
|
[26] Zhang, Z., Wang, J.: On existence and multiplicity of positive solutions to singular multi-point boundary value problems.J. Math. Anal. Appl. 295 (2004), 502-512. Zbl 1056.34018, MR 2072028, 10.1016/j.jmaa.2004.03.057 |
| . |
