American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On improvement of summability properties in nonautonomous Kolmogorov equations
  • CPAA Home
  • This Issue
  • Next Article
    Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation
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

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:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. A. Coppel, Disconjugacy. Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin-New York, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Gregus, Third order linear differential equations (Mathematics and its Applications), Reidel, Dordrecht, 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 , J. Differential Equations, 13 (1973), 432-437. 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 , J. Differential Equations, 10 (1971), 529-537. 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 , J. Math. Anal. Appl., 323 (2006), 413-425. doi: 10.1016/j.jmaa.2005.10.037.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

W. A. Coppel, Disconjugacy. Lecture Notes in Mathematics, 220, Springer-Verlag, Berlin-New York, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Gregus, Third order linear differential equations (Mathematics and its Applications), Reidel, Dordrecht, 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 , J. Differential Equations, 13 (1973), 432-437. 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 , J. Differential Equations, 10 (1971), 529-537. 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 , J. Math. Anal. Appl., 323 (2006), 413-425. doi: 10.1016/j.jmaa.2005.10.037.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89
[2]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040
[3]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337
[4]

John R. Graef, Bo Yang. Positive solutions of a third order nonlocal boundary value problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 89-97. doi: 10.3934/dcdss.2008.1.89
[5]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
[6]

Júlio Cesar Santos Sampaio, Igor Leite Freire. Symmetries and solutions of a third order equation. Conference Publications, 2015, 2015 (special) : 981-989. doi: 10.3934/proc.2015.0981
[7]

Chunyan Ji, Yang Xue, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4737-4754. doi: 10.3934/dcdsb.2020122
[8]

Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381
[9]

João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
[10]

Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[11]

Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
[12]

Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257
[13]

Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209-218. doi: 10.3934/proc.2011.2011.209
[14]

John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515
[15]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6069-6090. doi: 10.3934/dcdsb.2021003
[16]

Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
[17]

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103
[18]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012
[19]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[20]

Gang Li, Fen Gu, Feida Jiang. Positive viscosity solutions of a third degree homogeneous parabolic infinity Laplace equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1449-1462. doi: 10.3934/cpaa.2020071

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy