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2014, 13(3): 1223-1236. doi: 10.3934/cpaa.2014.13.1223

Disconjugacy and extremal solutions of nonlinear third-order equations

1. 

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

Received  June 2013 Revised  November 2013 Published  December 2013

In this paper, we make an exhaustive study of the third order linear operators $u''' +Mu$, $u'''+Mu'$ and $u'''+Mu''$ coupled with $(k, 3-k)$-conjugate boundary conditions, where $k=1,2$. We obtain the optimal intervals on which the Green's functions are of one sign. The main tool is the disconjugacy theory. As an application of our results, we develop a monotone iteration method to obtain positive solutions of the nonlinear problem $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.
Citation: Ruyun Ma, Yanqiong Lu. Disconjugacy and extremal solutions of nonlinear third-order equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1223-1236. doi: 10.3934/cpaa.2014.13.1223
References:
[1]

F. Bernis and L. A. Peleter, Two problems from draining flows involving third-order ordinary differential equation ,, \emph{SIAM J. Math. Anal.}, 27 (1996), 515. doi: 10.1137/S0036141093260847.

[2]

A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems ,, \emph{J. Math. Anal. Appl.}, 185 (1994), 302. doi: 10.1006/jmaa.1994.1250.

[3]

A. Cabada and R. Enguica, Positive solutions of fourth order problems with clamped beam boundary conditions ,, \emph{Nonlinear Analysis}, 74 (2011), 3112. doi: 10.1016/j.na.2011.01.027.

[4]

W. A. Coppel, Disconjugacy. Lecture Notes in Mathematics, 220,, Springer-Verlag, (1971).

[5]

U. Elias, Eigenvalue problems for the equations $Ly + \lambda p(x)y=0$ ,, \emph{Journal of Differential Equations}, 29 (1978), 28. doi: 10.1016/0022-0396(78)90039-6.

[6]

U. Elias, Oscillation Theory of Two-Term Differential Equations (Mathematics and Its Applications), 396,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-017-2517-0.

[7]

M. Gregus, Third order linear differential equations (Mathematics and its Applications),, Reidel, (1987). doi: 10.1007/978-94-009-3715-4.

[8]

L. K. Jackson, Existence and uniqueness of solutions of boundary value problems for third order differential equations ,, \emph{J. Differential Equations}, 13 (1973), 432. doi: 10.1016/0022-0396(73)90002-8.

[9]

G. Klaasen, Differential inequalities and existence theorems for second and third order boundary value problems ,, \emph{J. Differential Equations}, 10 (1971), 529. doi: 10.1016/0022-0396(71)90010-6.

[10]

S. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem ,, \emph{J. Math. Anal. Appl.}, 323 (2006), 413. doi: 10.1016/j.jmaa.2005.10.037.

[11]

R. Ma, Multiplicity results for a third order boundary value problem at resonance ,, \emph{Nonlinear Anal.}, 32 (1998), 493. doi: 10.1016/S0362-546X(97)00494-X.

[12]

D. J. O'Regan, Topological transversality: Application to third-order boundary value problem ,, \emph{SIAM J. Math. Anal.}, 19 (1987), 630. doi: 10.1137/0518048.

[13]

B. P. Rynne, Global bifurcation for 2$m$th-order boundary value problems and infinitely many solutions of superlinear problems ,, \emph{J. Differential Equations}, 188 (2003), 461. doi: 10.1016/S0022-0396(02)00146-8.

[14]

W. C. Troy, Solution of third order differential equations relevant to draining and coating flows ,, \emph{SIAM J. Math. Anal.}, 24 (1993), 155. doi: 10.1137/0524010.

[15]

Q. Yao and Y. Feng, The existence of solutions for a third order two-point boundary value problem ,, \emph{Appl. Math. Lett.}, 15 (2002), 227. doi: 10.1016/S0893-9659(01)00122-7.

show all references

References:
[1]

F. Bernis and L. A. Peleter, Two problems from draining flows involving third-order ordinary differential equation ,, \emph{SIAM J. Math. Anal.}, 27 (1996), 515. doi: 10.1137/S0036141093260847.

[2]

A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems ,, \emph{J. Math. Anal. Appl.}, 185 (1994), 302. doi: 10.1006/jmaa.1994.1250.

[3]

A. Cabada and R. Enguica, Positive solutions of fourth order problems with clamped beam boundary conditions ,, \emph{Nonlinear Analysis}, 74 (2011), 3112. doi: 10.1016/j.na.2011.01.027.

[4]

W. A. Coppel, Disconjugacy. Lecture Notes in Mathematics, 220,, Springer-Verlag, (1971).

[5]

U. Elias, Eigenvalue problems for the equations $Ly + \lambda p(x)y=0$ ,, \emph{Journal of Differential Equations}, 29 (1978), 28. doi: 10.1016/0022-0396(78)90039-6.

[6]

U. Elias, Oscillation Theory of Two-Term Differential Equations (Mathematics and Its Applications), 396,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-017-2517-0.

[7]

M. Gregus, Third order linear differential equations (Mathematics and its Applications),, Reidel, (1987). doi: 10.1007/978-94-009-3715-4.

[8]

L. K. Jackson, Existence and uniqueness of solutions of boundary value problems for third order differential equations ,, \emph{J. Differential Equations}, 13 (1973), 432. doi: 10.1016/0022-0396(73)90002-8.

[9]

G. Klaasen, Differential inequalities and existence theorems for second and third order boundary value problems ,, \emph{J. Differential Equations}, 10 (1971), 529. doi: 10.1016/0022-0396(71)90010-6.

[10]

S. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem ,, \emph{J. Math. Anal. Appl.}, 323 (2006), 413. doi: 10.1016/j.jmaa.2005.10.037.

[11]

R. Ma, Multiplicity results for a third order boundary value problem at resonance ,, \emph{Nonlinear Anal.}, 32 (1998), 493. doi: 10.1016/S0362-546X(97)00494-X.

[12]

D. J. O'Regan, Topological transversality: Application to third-order boundary value problem ,, \emph{SIAM J. Math. Anal.}, 19 (1987), 630. doi: 10.1137/0518048.

[13]

B. P. Rynne, Global bifurcation for 2$m$th-order boundary value problems and infinitely many solutions of superlinear problems ,, \emph{J. Differential Equations}, 188 (2003), 461. doi: 10.1016/S0022-0396(02)00146-8.

[14]

W. C. Troy, Solution of third order differential equations relevant to draining and coating flows ,, \emph{SIAM J. Math. Anal.}, 24 (1993), 155. doi: 10.1137/0524010.

[15]

Q. Yao and Y. Feng, The existence of solutions for a third order two-point boundary value problem ,, \emph{Appl. Math. Lett.}, 15 (2002), 227. doi: 10.1016/S0893-9659(01)00122-7.

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