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

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

References:

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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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