    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

 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:
  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.  Google Scholar  R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2 (1989), 91-110. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  S. N. Bernstein, Sur les équations du calcul des variations, Ann. Sci. Ecole Norm. Sup., 29 (1912), 431-485. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  J. Ehme and J. Henderson, Existence and local uniqueness for nonlinear Lidstone boundary value problems, J. Inequal. Pure Appl. Math., 1 (2000), 1-9. Google Scholar  P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems, J. Qual. Theory Differ. Equ., 2 (2000), 1-8. Google Scholar  D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, Boston, 1988. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  M. Nagumo, $\ddotU$ber die Differentialgleichung $y''=f(t,y,y')$, Proc. Phys. Math. Soc. Japan, 19 (1937), 861-866. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Q. Yao, Existence of $n$ positive solutions to general Lidstone boundary value problems, Acta Math. Sinica (Chin. Ser.), 48 (2005), 365-376. Google Scholar  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.  Google Scholar

show all references

##### References:
  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.  Google Scholar  R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2 (1989), 91-110. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  S. N. Bernstein, Sur les équations du calcul des variations, Ann. Sci. Ecole Norm. Sup., 29 (1912), 431-485. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  J. Ehme and J. Henderson, Existence and local uniqueness for nonlinear Lidstone boundary value problems, J. Inequal. Pure Appl. Math., 1 (2000), 1-9. Google Scholar  P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems, J. Qual. Theory Differ. Equ., 2 (2000), 1-8. Google Scholar  D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones," Academic Press, Boston, 1988. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  M. Nagumo, $\ddotU$ber die Differentialgleichung $y''=f(t,y,y')$, Proc. Phys. Math. Soc. Japan, 19 (1937), 861-866. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  Q. Yao, Existence of $n$ positive solutions to general Lidstone boundary value problems, Acta Math. Sinica (Chin. Ser.), 48 (2005), 365-376. Google Scholar  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.  Google Scholar
  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  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  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  Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227  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  Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205  Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032  Kun-Peng Jin, Jin Liang, Ti-Jun Xiao. Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077  Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885  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  Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015  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  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  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  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  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  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  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  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  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

2020 Impact Factor: 1.916