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Positive solutions of a fourth-order boundary value problem involving derivatives of all orders
1. | Department of Mathematics, Qingdao Technological University, No 11 Fushun Road, Qingdao, Shandong Province, China |
2. | Department of Mathematics, Xuzhou Normal University, Xuzhou 221116 |
References:
[1] |
A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415-426.
doi: 10.1016/S0022-247X(86)80006-3. |
[2] |
R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2 (1989), 91-110. |
[3] |
R. P. Agarwal and G. Akrivis, Boundary value problems occurring in plate deflection theory, J. Comput. Appl. Math., 8 (1982), 145-154.
doi: 10.1016/0771-050X(82)90035-3. |
[4] |
R. P. Agarwal and P. J. Y. Wong, Lidstone polynomials and boundary value problems, Comput. Math. Appl., 17 (1989), 1397-1421.
doi: 10.1016/0898-1221(89)90023-0. |
[5] |
R. P. Agarwal, D. O'Regan and S. Staněk, Singular Lidstone boundary value problem with given maximal values for solutions, Nonlinear Anal., 55 (2003), 859-881.
doi: 10.1016/j.na.2003.06.001. |
[6] |
Z. Bai and W. Ge, Solutions of $2n$th Lidstone boundary value problems and dependence on higher order derivatives, J. Math. Anal. Appl., 279 (2003), 442-450.
doi: 10.1016/S0022-247X(03)00011-8. |
[7] |
S. N. Bernstein, Sur les équations du calcul des variations, Ann. Sci. Ecole Norm. Sup., 29 (1912), 431-485. |
[8] |
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.
doi: 10.1006/jmaa.1999.6500. |
[9] |
C. De Coster, C. Fabry and F. Munyamarere, Nonresonance conditions for fourth-order nonlinear boundary value problems, Int. J. Math. Math. Sci., 17 (1994), 725-740.
doi: 10.1155/S0161171294001031. |
[10] |
M. A. Del Pino and R. F. Manasevich, Existence for a fourth-order nonlinear boundary problem under a twoparameter nonresonance condition, Proc. Amer. Math. Soc., 112 (1991), 81-86.
doi: 10.1090/S0002-9939-1991-1043407-9. |
[11] |
J. Ehme and J. Henderson, Existence and local uniqueness for nonlinear Lidstone boundary value problems, J. Inequal. Pure Appl. Math., 1 (2000), 1-9. |
[12] |
P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems, J. Qual. Theory Differ. Equ., 2 (2000), 1-8. |
[13] |
D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, Boston, 1988. |
[14] |
H. Feng, D. Ji and W. Ge, Existence and uniqueness of solutions for a fourth-order boundary value problem, Nonlinear Anal., 70 (2009), 3561-3566.
doi: 10.1016/j.na.2008.07.013. |
[15] |
F. Li and Z. Liu, Multiple positive solutions of some nonlinear operator equations and their applications, Acta Math. Sinica (Chin. Ser.), 41 (1998), 97-102. |
[16] |
Z. Liu and F. Li, Multiple positive solutions of nonlinear two-point value problems, J. Math. Anal. Appl., 203 (1996), 610-625.
doi: 10.1006/jmaa.1996.0400. |
[17] |
Y. Ma, Existence of positive solutions of Lidstone boundary value problems, J. Math. Anal. Appl., 314 (2006), 97-108.
doi: 10.1016/j.jmaa.2005.03.059. |
[18] |
M. Nagumo, $\ddotU$ber die Differentialgleichung $y''=f(t,y,y')$, Proc. Phys. Math. Soc. Japan, 19 (1937), 861-866. |
[19] |
Y. Wang, On $2n$th-order Lidstone boundary value problems, J. Math. Anal. Appl., 312 (2005), 383-400.
doi: 10.1016/j.jmaa.2005.03.039. |
[20] |
Y. Wang, On fourth-order elliptic boundary value problems with nonmonotone nonlinear function, J. Math. Anal. Appl., 307 (2005), 1-11.
doi: 10.1016/j.jmaa.2004.09.063. |
[21] |
Z. Wei, Existence of positive solutions for $2n$th-order singular sublinear boundary value problems, J. Math. Anal. Appl., 306 (2005), 619-636.
doi: 10.1016/j.jmaa.2004.10.037. |
[22] |
Z. Wei, Positive solutions for $2n$th-order singular sub-linear $m$-point boundary value problems, Appl. Math. Comput., 182 (2006), 1280-1295.
doi: 10.1016/j.amc.2006.05.014. |
[23] |
Z. Wei, A necessary and sufficient condition for $2n$th-order singular superlinear $m$-point boundary value problems, J. Math. Anal. Appl., 327 (2007), 930-947.
doi: 10.1016/j.jmaa.2006.04.056. |
[24] |
Z. Yang, Existence and uniqueness of psoitive solutions for a higher order boundary value problem, Comput. Math. Appl., 54 (2007), 220-228.
doi: 10.1016/j.camwa.2007.01.018. |
[25] |
Z. Yang, D. O'Regan and R. P. Agarwal, Positive solutions of a second-order boundary value problem via integro-differential quation arguments, Appl. Anal., 88 (2009), 1197-1211.
doi: 10.1080/00036810903157212. |
[26] |
Q. Yao, Existence of $n$ positive solutions to general Lidstone boundary value problems, Acta Math. Sinica (Chin. Ser.), 48 (2005), 365-376. |
[27] |
B. Zhang and X. Liu, Existence of multiple symmetric positive solutions of higher order Lidstone problems, J. Math. Anal. Appl., 284 (2003), 672-689.
doi: 10.1016/S0022-247X(03)00386-X. |
show all references
References:
[1] |
A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415-426.
doi: 10.1016/S0022-247X(86)80006-3. |
[2] |
R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2 (1989), 91-110. |
[3] |
R. P. Agarwal and G. Akrivis, Boundary value problems occurring in plate deflection theory, J. Comput. Appl. Math., 8 (1982), 145-154.
doi: 10.1016/0771-050X(82)90035-3. |
[4] |
R. P. Agarwal and P. J. Y. Wong, Lidstone polynomials and boundary value problems, Comput. Math. Appl., 17 (1989), 1397-1421.
doi: 10.1016/0898-1221(89)90023-0. |
[5] |
R. P. Agarwal, D. O'Regan and S. Staněk, Singular Lidstone boundary value problem with given maximal values for solutions, Nonlinear Anal., 55 (2003), 859-881.
doi: 10.1016/j.na.2003.06.001. |
[6] |
Z. Bai and W. Ge, Solutions of $2n$th Lidstone boundary value problems and dependence on higher order derivatives, J. Math. Anal. Appl., 279 (2003), 442-450.
doi: 10.1016/S0022-247X(03)00011-8. |
[7] |
S. N. Bernstein, Sur les équations du calcul des variations, Ann. Sci. Ecole Norm. Sup., 29 (1912), 431-485. |
[8] |
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.
doi: 10.1006/jmaa.1999.6500. |
[9] |
C. De Coster, C. Fabry and F. Munyamarere, Nonresonance conditions for fourth-order nonlinear boundary value problems, Int. J. Math. Math. Sci., 17 (1994), 725-740.
doi: 10.1155/S0161171294001031. |
[10] |
M. A. Del Pino and R. F. Manasevich, Existence for a fourth-order nonlinear boundary problem under a twoparameter nonresonance condition, Proc. Amer. Math. Soc., 112 (1991), 81-86.
doi: 10.1090/S0002-9939-1991-1043407-9. |
[11] |
J. Ehme and J. Henderson, Existence and local uniqueness for nonlinear Lidstone boundary value problems, J. Inequal. Pure Appl. Math., 1 (2000), 1-9. |
[12] |
P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems, J. Qual. Theory Differ. Equ., 2 (2000), 1-8. |
[13] |
D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, Boston, 1988. |
[14] |
H. Feng, D. Ji and W. Ge, Existence and uniqueness of solutions for a fourth-order boundary value problem, Nonlinear Anal., 70 (2009), 3561-3566.
doi: 10.1016/j.na.2008.07.013. |
[15] |
F. Li and Z. Liu, Multiple positive solutions of some nonlinear operator equations and their applications, Acta Math. Sinica (Chin. Ser.), 41 (1998), 97-102. |
[16] |
Z. Liu and F. Li, Multiple positive solutions of nonlinear two-point value problems, J. Math. Anal. Appl., 203 (1996), 610-625.
doi: 10.1006/jmaa.1996.0400. |
[17] |
Y. Ma, Existence of positive solutions of Lidstone boundary value problems, J. Math. Anal. Appl., 314 (2006), 97-108.
doi: 10.1016/j.jmaa.2005.03.059. |
[18] |
M. Nagumo, $\ddotU$ber die Differentialgleichung $y''=f(t,y,y')$, Proc. Phys. Math. Soc. Japan, 19 (1937), 861-866. |
[19] |
Y. Wang, On $2n$th-order Lidstone boundary value problems, J. Math. Anal. Appl., 312 (2005), 383-400.
doi: 10.1016/j.jmaa.2005.03.039. |
[20] |
Y. Wang, On fourth-order elliptic boundary value problems with nonmonotone nonlinear function, J. Math. Anal. Appl., 307 (2005), 1-11.
doi: 10.1016/j.jmaa.2004.09.063. |
[21] |
Z. Wei, Existence of positive solutions for $2n$th-order singular sublinear boundary value problems, J. Math. Anal. Appl., 306 (2005), 619-636.
doi: 10.1016/j.jmaa.2004.10.037. |
[22] |
Z. Wei, Positive solutions for $2n$th-order singular sub-linear $m$-point boundary value problems, Appl. Math. Comput., 182 (2006), 1280-1295.
doi: 10.1016/j.amc.2006.05.014. |
[23] |
Z. Wei, A necessary and sufficient condition for $2n$th-order singular superlinear $m$-point boundary value problems, J. Math. Anal. Appl., 327 (2007), 930-947.
doi: 10.1016/j.jmaa.2006.04.056. |
[24] |
Z. Yang, Existence and uniqueness of psoitive solutions for a higher order boundary value problem, Comput. Math. Appl., 54 (2007), 220-228.
doi: 10.1016/j.camwa.2007.01.018. |
[25] |
Z. Yang, D. O'Regan and R. P. Agarwal, Positive solutions of a second-order boundary value problem via integro-differential quation arguments, Appl. Anal., 88 (2009), 1197-1211.
doi: 10.1080/00036810903157212. |
[26] |
Q. Yao, Existence of $n$ positive solutions to general Lidstone boundary value problems, Acta Math. Sinica (Chin. Ser.), 48 (2005), 365-376. |
[27] |
B. Zhang and X. Liu, Existence of multiple symmetric positive solutions of higher order Lidstone problems, J. Math. Anal. Appl., 284 (2003), 672-689.
doi: 10.1016/S0022-247X(03)00386-X. |
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