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

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

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.
Citation: Zhilin Yang, Jingxian Sun. Positive solutions of a fourth-order boundary value problem involving derivatives of all orders. Communications on Pure & 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. 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.

[3]

R. P. Agarwal and G. Akrivis, Boundary value problems occurring in plate deflection theory,, J. Comput. Appl. Math., 8 (1982), 145. 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. 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. 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. 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.

[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. 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. 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. 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.

[12]

P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems,, J. Qual. Theory Differ. Equ., 2 (2000), 1.

[13]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones,", Academic Press, (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. 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.

[16]

Z. Liu and F. Li, Multiple positive solutions of nonlinear two-point value problems,, J. Math. Anal. Appl., 203 (1996), 610. 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. 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.

[19]

Y. Wang, On $2n$th-order Lidstone boundary value problems,, J. Math. Anal. Appl., 312 (2005), 383. 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. 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. 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. 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. 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. 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. 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.

[27]

B. Zhang and X. Liu, Existence of multiple symmetric positive solutions of higher order Lidstone problems,, J. Math. Anal. Appl., 284 (2003), 672. 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. 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.

[3]

R. P. Agarwal and G. Akrivis, Boundary value problems occurring in plate deflection theory,, J. Comput. Appl. Math., 8 (1982), 145. 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. 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. 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. 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.

[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. 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. 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. 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.

[12]

P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems,, J. Qual. Theory Differ. Equ., 2 (2000), 1.

[13]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones,", Academic Press, (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. 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.

[16]

Z. Liu and F. Li, Multiple positive solutions of nonlinear two-point value problems,, J. Math. Anal. Appl., 203 (1996), 610. 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. 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.

[19]

Y. Wang, On $2n$th-order Lidstone boundary value problems,, J. Math. Anal. Appl., 312 (2005), 383. 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. 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. 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. 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. 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. 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. 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.

[27]

B. Zhang and X. Liu, Existence of multiple symmetric positive solutions of higher order Lidstone problems,, J. Math. Anal. Appl., 284 (2003), 672. doi: 10.1016/S0022-247X(03)00386-X.

[1]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269

[2]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225

[3]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[4]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1

[5]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337

[6]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885

[7]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[8]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015

[9]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[10]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[11]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276

[12]

John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89

[13]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775

[14]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319

[15]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[16]

Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261

[17]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084

[18]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018153

[19]

J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905

[20]

Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]