Positive solutions of a fourth-order boundary value problem involving derivatives of all orders

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September 2012, 11(5): 1615-1628. doi: 10.3934/cpaa.2012.11.1615

Positive solutions of a fourth-order boundary value problem involving derivatives of all orders

Zhilin Yang ^1, and Jingxian Sun ^2,

1.	Department of Mathematics, Qingdao Technological University, No 11 Fushun Road, Qingdao, Shandong Province, China
2.	Department of Mathematics, Xuzhou Normal University, Xuzhou 221116

Received May 2010 Revised December 2011 Published March 2012

Abstract
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This paper is mainly concerned with the existence, multiplicity and uniqueness of positive solutions for the fourth-order boundary value problem \begin{eqnarray*} u^{(4)}=f(t,u,u^\prime,-u^{\prime\prime},-u^{\prime\prime\prime}),\\ u(0)=u^\prime(1)=u^{\prime\prime}(0)=u^{\prime\prime\prime}(1)=0, \end{eqnarray*} where $f\in C([0,1]\times\mathbb R_+^4,\mathbb R_+)(\mathbb R_+:=[0,\infty))$. Based on a priori estimates achieved by utilizing some integral identities and inequalities, we use fixed point index theory to prove the existence, multiplicity and uniqueness of positive solutions for the above problem. Finally, as a byproduct, our main results are applied to establish the existence, multiplicity and uniqueness of symmetric positive solutions for the fourth order Lidstone problem.

Keywords: integro-differential equation, positive solution, a priori estimate, symmetric positive solution., fixed point index, Fourth-order boundary value problem.

Mathematics Subject Classification: Primary: 34B18, 47H11; Secondary: 47N20, 45J05, 45M2.

Citation: Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1615-1628. doi: 10.3934/cpaa.2012.11.1615

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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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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