Disconjugacy and extremal solutions of nonlinear third-order equations

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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

Ruyun Ma ^1, and Yanqiong Lu ^1,

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

Received June 2013 Revised November 2013 Published December 2013

Abstract
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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 $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.

Keywords: lower and upper solutions., disconjugacy, Third-order equations, positive solutions.

Mathematics Subject Classification: Primary: 34B10; Secondary: 34B1.

Citation: Ruyun Ma, Yanqiong Lu. Disconjugacy and extremal solutions of nonlinear third-order equations. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1223-1236. doi: 10.3934/cpaa.2014.13.1223

References:

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[8]	L. K. Jackson, Existence and uniqueness of solutions of boundary value problems for third order differential equations , J. Differential Equations, 13 (1973), 432-437. doi: 10.1016/0022-0396(73)90002-8.
[9]	G. Klaasen, Differential inequalities and existence theorems for second and third order boundary value problems , J. Differential Equations, 10 (1971), 529-537. doi: 10.1016/0022-0396(71)90010-6.
[10]	S. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem , J. Math. Anal. Appl., 323 (2006), 413-425. doi: 10.1016/j.jmaa.2005.10.037.
[11]	R. Ma, Multiplicity results for a third order boundary value problem at resonance , Nonlinear Anal., 32 (1998), 493-500. doi: 10.1016/S0362-546X(97)00494-X.
[12]	D. J. O'Regan, Topological transversality: Application to third-order boundary value problem , SIAM J. Math. Anal., 19 (1987), 630-641. 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 , J. Differential Equations, 188 (2003), 461-472. doi: 10.1016/S0022-0396(02)00146-8.
[14]	W. C. Troy, Solution of third order differential equations relevant to draining and coating flows , SIAM J. Math. Anal., 24 (1993), 155-171. doi: 10.1137/0524010.
[15]	Q. Yao and Y. Feng, The existence of solutions for a third order two-point boundary value problem , Appl. Math. Lett., 15 (2002), 227-232. 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 , SIAM J. Math. Anal., 27 (1996), 515-527. doi: 10.1137/S0036141093260847.
[2]	A. Cabada, The method of lower and upper solutions for second, third, fourth and higher order boundary value problems , J. Math. Anal. Appl., 185 (1994), 302-320. doi: 10.1006/jmaa.1994.1250.
[3]	A. Cabada and R. Enguica, Positive solutions of fourth order problems with clamped beam boundary conditions , Nonlinear Analysis, 74 (2011), 3112-3122. doi: 10.1016/j.na.2011.01.027.
[4]	W. A. Coppel, Disconjugacy. Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin-New York, 1971.
[5]	U. Elias, Eigenvalue problems for the equations $Ly + \lambda p(x)y=0$ , Journal of Differential Equations, 29 (1978), 28-57. 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, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-94-017-2517-0.
[7]	M. Gregus, Third order linear differential equations (Mathematics and its Applications), Reidel, Dordrecht, 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 , J. Differential Equations, 13 (1973), 432-437. doi: 10.1016/0022-0396(73)90002-8.
[9]	G. Klaasen, Differential inequalities and existence theorems for second and third order boundary value problems , J. Differential Equations, 10 (1971), 529-537. doi: 10.1016/0022-0396(71)90010-6.
[10]	S. Li, Positive solutions of nonlinear singular third-order two-point boundary value problem , J. Math. Anal. Appl., 323 (2006), 413-425. doi: 10.1016/j.jmaa.2005.10.037.
[11]	R. Ma, Multiplicity results for a third order boundary value problem at resonance , Nonlinear Anal., 32 (1998), 493-500. doi: 10.1016/S0362-546X(97)00494-X.
[12]	D. J. O'Regan, Topological transversality: Application to third-order boundary value problem , SIAM J. Math. Anal., 19 (1987), 630-641. 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 , J. Differential Equations, 188 (2003), 461-472. doi: 10.1016/S0022-0396(02)00146-8.
[14]	W. C. Troy, Solution of third order differential equations relevant to draining and coating flows , SIAM J. Math. Anal., 24 (1993), 155-171. doi: 10.1137/0524010.
[15]	Q. Yao and Y. Feng, The existence of solutions for a third order two-point boundary value problem , Appl. Math. Lett., 15 (2002), 227-232. doi: 10.1016/S0893-9659(01)00122-7.

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