Global structure of positive solutions for superlinear th-boundary value problems
Czechoslovak Mathematical Journal (2010)
- Volume: 60, Issue: 1, page 161-172
- ISSN: 0011-4642
Access Full Article
topAbstract
topHow to cite
topMa, Ruyun, and An, Yulian. "Global structure of positive solutions for superlinear $2m$th-boundary value problems." Czechoslovak Mathematical Journal 60.1 (2010): 161-172. <http://eudml.org/doc/37998>.
@article{Ma2010,
abstract = {We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem \[ \begin\{aligned\} (-1)^mu^\{(2m)\}(t)&=\lambda a(t)f(u(t)),\ \ \ \ \ 0<t<1, \\ u^\{(2i)\}(0)&=u^\{(2i)\}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end\{aligned\} \]
where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _\{s\rightarrow 0^+\}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.},
author = {Ma, Ruyun, An, Yulian},
journal = {Czechoslovak Mathematical Journal},
keywords = {multiplicity results; Lidstone boundary value problem; eigenvalues; bifurcation methods; positive solutions; multiplicity; Lidstone boundary value problem; eigenvalue; bifurcation method; positive solution},
language = {eng},
number = {1},
pages = {161-172},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global structure of positive solutions for superlinear $2m$th-boundary value problems},
url = {http://eudml.org/doc/37998},
volume = {60},
year = {2010},
}
TY - JOUR
AU - Ma, Ruyun
AU - An, Yulian
TI - Global structure of positive solutions for superlinear $2m$th-boundary value problems
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 161
EP - 172
AB - We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem \[ \begin{aligned} (-1)^mu^{(2m)}(t)&=\lambda a(t)f(u(t)),\ \ \ \ \ 0<t<1, \\ u^{(2i)}(0)&=u^{(2i)}(1)=0,\ \ \ \ i=0,1,2,\cdots ,m-1 . \end{aligned} \]
where $a\in C([0,1], [0,\infty ))$ and $a(t_0)>0$ for some $t_0\in [0,1]$, $f\in C([0,\infty ),[0,\infty ))$ and $f(s)>0$ for $s>0$, and $f_0=\infty $, where $f_0=\lim _{s\rightarrow 0^+}f(s)/s$. We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.
LA - eng
KW - multiplicity results; Lidstone boundary value problem; eigenvalues; bifurcation methods; positive solutions; multiplicity; Lidstone boundary value problem; eigenvalue; bifurcation method; positive solution
UR - http://eudml.org/doc/37998
ER -
References
top- Gupta, C. P., 10.1080/00036818808839715, Appl. Anal. 26 (1988), 289-304. (1988) Zbl0611.34015MR0922976DOI10.1080/00036818808839715
- Agarwal, R. P., Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore (1986). (1986) Zbl0619.34019MR1021979
- Agarwal, R. P., Wong, P. J. Y., 10.1016/0898-1221(89)90023-0, Comput. Math. Appl. 17 (1989), 1397-1421. (1989) Zbl0682.65049MR0992124DOI10.1016/0898-1221(89)90023-0
- Yang, Y. S., 10.1090/S0002-9939-1988-0958062-3, Proc. Amer. Math. Soc. 104 (1988), 175-180. (1988) Zbl0671.34016MR0958062DOI10.1090/S0002-9939-1988-0958062-3
- Pino, M. A. Del, Manásevich, R. F., Multiple solutions for the -Laplacian under global nonresonance, Proc. Amer. Math. Soc. 112 (1991), 131-138. (1991) MR1045589
- Ma, R., Wang, H., 10.1080/00036819508840401, Appl. Anal. 59 (1995), 225-231. (1995) Zbl0841.34019MR1378037DOI10.1080/00036819508840401
- Ma, R., Zhang, J., Fu, S., 10.1006/jmaa.1997.5639, J. Math. Anal. Appl. 215 (1997), 415-422. (1997) Zbl0892.34009MR1490759DOI10.1006/jmaa.1997.5639
- Bai, Z., Wang, H., 10.1016/S0022-247X(02)00071-9, J. Math. Anal. Appl. 270 (2002), 357-368. (2002) Zbl1006.34023MR1915704DOI10.1016/S0022-247X(02)00071-9
- Bai, Z., Ge, W., 10.1016/S0022-247X(03)00011-8, J. Math. Anal. Appl. 279 (2003), 442-450. (2003) MR1974036DOI10.1016/S0022-247X(03)00011-8
- Yao, Q., 10.1016/S0096-3003(02)00152-2, Applied Mathematics and Computation 137 (2003), 477-485. (2003) Zbl1093.34515MR1950111DOI10.1016/S0096-3003(02)00152-2
- Li, Y., Abstract existence theorems of positive solutions for nonlinear boundary value problems, Nonlinear Anal. TMA 57 (2004), 211-227. (2004) Zbl1064.47058MR2056428
- Li, F., Li, Y., Liang, Z., 10.1016/j.jmaa.2006.09.025, J. Math. Anal. Appl. 331 (2007), 958-977. (2007) Zbl1119.34014MR2313694DOI10.1016/j.jmaa.2006.09.025
- Rynne, B. P., 10.1016/S0022-0396(02)00146-8, J. Differential Equations 188 (2003), 461-472. (2003) MR1954290DOI10.1016/S0022-0396(02)00146-8
- Rynne, B. P., Solution curves of th order boundary value problems, Electron. J. Differential Equations 32 (2004), 1-16. (2004) Zbl1060.34011MR2036216
- Bari, R., Rynne, B. P., 10.1016/j.jmaa.2003.08.043, J. Math. Anal. Appl. 292 (2004), 17-22. (2004) MR2050212DOI10.1016/j.jmaa.2003.08.043
- Ma, R., 10.1016/j.amc.2004.10.014, Appl. Math. Comput. 168 (2005), 1219-1231. (2005) Zbl1082.34023MR2171774DOI10.1016/j.amc.2004.10.014
- Ma, R., 10.1016/j.jmaa.2005.06.045, J. Math. Anal. Appl. 319 (2006), 424-434. (2006) Zbl1098.34012MR2227914DOI10.1016/j.jmaa.2005.06.045
- Ma, R., 10.1016/j.jmaa.2005.03.078, J. Math. Anal. Appl. 314 (2006), 254-265. (2006) Zbl1085.34015MR2183550DOI10.1016/j.jmaa.2005.03.078
- Elias, U., 10.1016/0022-0396(78)90039-6, J. Diff. Equations, 29 (1978), 28-57. (1978) Zbl0369.34008MR0486759DOI10.1016/0022-0396(78)90039-6
- Whyburn, G. T., Topological Analysis, Princeton University Press, Princeton (1958). (1958) Zbl0080.15903MR0099642
- Xu, J., Han, X., 10.1016/j.na.2006.10.017, Nonlinear Analysis TMA 67 (2007), 3350-3356. (2007) Zbl1136.34016MR2350891DOI10.1016/j.na.2006.10.017
- Rabinowitz, P., 10.1016/0022-1236(71)90030-9, J. Funct. Anal. 7 (1971), 487-513. (1971) Zbl0212.16504MR0301587DOI10.1016/0022-1236(71)90030-9
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.