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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

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 & 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.  Google Scholar

[2]

R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis,, Differential Integral Equations, 2 (1989), 91.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. N. Bernstein, Sur les équations du calcul des variations,, Ann. Sci. Ecole Norm. Sup., 29 (1912), 431.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[11]

J. Ehme and J. Henderson, Existence and local uniqueness for nonlinear Lidstone boundary value problems,, J. Inequal. Pure Appl. Math., 1 (2000), 1.   Google Scholar

[12]

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

[13]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones,", Academic Press, (1988).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Nagumo, $\ddotU$ber die Differentialgleichung $y''=f(t,y,y')$,, Proc. Phys. Math. Soc. Japan, 19 (1937), 861.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Q. Yao, Existence of $n$ positive solutions to general Lidstone boundary value problems,, Acta Math. Sinica (Chin. Ser.), 48 (2005), 365.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. P. Agarwal, On fourth-order boundary value problems arising in beam analysis,, Differential Integral Equations, 2 (1989), 91.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. N. Bernstein, Sur les équations du calcul des variations,, Ann. Sci. Ecole Norm. Sup., 29 (1912), 431.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Ehme and J. Henderson, Existence and local uniqueness for nonlinear Lidstone boundary value problems,, J. Inequal. Pure Appl. Math., 1 (2000), 1.   Google Scholar

[12]

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

[13]

D. Guo and V. Lakshmikantham, "Nonlinear Problems in Abstract Cones,", Academic Press, (1988).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. Nagumo, $\ddotU$ber die Differentialgleichung $y''=f(t,y,y')$,, Proc. Phys. Math. Soc. Japan, 19 (1937), 861.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Q. Yao, Existence of $n$ positive solutions to general Lidstone boundary value problems,, Acta Math. Sinica (Chin. Ser.), 48 (2005), 365.   Google Scholar

[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.  Google Scholar

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