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May  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. Google Scholar

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

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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