American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays
  • CPAA Home
  • This Issue
  • Next Article
    Defect correction for spectral computations for a singular integral operator
June  2006, 5(2): 251-259. doi: 10.3934/cpaa.2006.5.251

Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems

Grey Ballard 1, , John Baxley 1, and  Nisrine Libbus 1,

1. 

Department of Mathematics, Wake Forest University, P.O. Box 7388, Winston-Salem, NC 27109, United States, United States

Received  March 2005 Revised  June 2005 Published  March 2006

We consider nonlinear boundary value problems with multiple solutions. A method is proposed for the computation of such solutions which depends crucially on known a priori qualitative information about the behavior of the solutions. The method is a two-stage method where the second stage is a shooting method and initial values of the shooting parameters are found in a first stage which approximates the boundary value problem with a discrete approximation. Both nonsingular and singular problems are considered.
Keywords: shooting, Multiple solutions, nonlinear boundary value problems., iterative methods, qualitative behavior.
Mathematics Subject Classification: Primary: 34B15, 65L10; Secondary: 65L12, 34B16, 34B18, 34A4.
Citation: Grey Ballard, John Baxley, Nisrine Libbus. Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems. Communications on Pure and Applied Analysis, 2006, 5 (2) : 251-259. doi: 10.3934/cpaa.2006.5.251
[1]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[2]

Felix Sadyrbaev, Inara Yermachenko. Multiple solutions of nonlinear boundary value problems for two-dimensional differential systems. Conference Publications, 2009, 2009 (Special) : 659-668. doi: 10.3934/proc.2009.2009.659
[3]

John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84
[4]

Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489
[5]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061
[6]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045
[7]

Alfonso Castro, Jorge Cossio, Carlos Vélez. Existence and qualitative properties of solutions for nonlinear Dirichlet problems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 123-140. doi: 10.3934/dcds.2013.33.123
[8]

Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure and Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785
[9]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[10]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[11]

K. Q. Lan. Multiple positive eigenvalues of conjugate boundary value problems with singularities. Conference Publications, 2003, 2003 (Special) : 501-506. doi: 10.3934/proc.2003.2003.501
[12]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377
[13]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283
[14]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[15]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596
[16]

Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627
[17]

Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103
[18]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[19]

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
[20]

John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291-299. doi: 10.3934/proc.2013.2013.291

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy