American Institute of Mathematical Sciences

Advanced
September  2012, 11(5): 1825-1838. doi: 10.3934/cpaa.2012.11.1825

Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case

Zhiming Guo 1, , Zhi-Chun Yang 2, and  Xingfu Zou 3,

1. 

School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006
2. 

College of Mathematics Science, Chongqing Normal University, Chongqing 400047
3. 

Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7

Received  March 2011 Revised  September 2011 Published  March 2012

This paper deals with a class of non-local second order differential equations subject to the homogeneous Dirichlet boundary condition. The main concern is positive steady state of the boundary value problem, especially when the equation does not enjoy the monotonicity. Nonexistence, existence and uniqueness of positive steady state for the problem are addressed. In particular, developed is a technique that combines the method of super-sub solutions and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the existence and uniqueness of a positive steady state. Two examples are given to illustrate the obtained results.
Keywords: Dirichlet boundary value problems, uniqueness., Non-local second order differential equation, positive solutions, existence.
Mathematics Subject Classification: Primary: 34B1.
Citation: Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space,, SIAM Review, 18 (1976), 620. doi: 10.1137/1018114. Google Scholar

[2]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19,, American Mathematical Society, (1998). Google Scholar

[3]

P. Freitas and G. Sweers, Positivity results for a nonlocal elliptic equation,, Proceedings of the Royal Society of Edinburgh, 128A (1998), 697. doi: 10.1017/S0308210500021727. Google Scholar

[4]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0. Google Scholar

[5]

P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems,, Math. Z., 154 (1977), 17. doi: 10.1007/BF01215108. Google Scholar

[6]

D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations,, Diff. Eqns. Dynam. Syst., 11 (2003), 117. Google Scholar

[7]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326. Google Scholar

[8]

C. V. Pao, "Nonlinear Parablic and Elliptic Equations,", Plenum, (1992). doi: 10.1007/978-1-4615-3034-3. Google Scholar

[9]

M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations,", Springer-Verlag, (1984). doi: 10.1007/978-1-4612-5282-5. Google Scholar

[10]

J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains,, Proc. Royal Soc. London. A, 457 (2001), 1841. Google Scholar

[11]

D. Xu and X.-Q. Zhao, A nonlocal reaction diffusion population model with stage structure,, Canadian Applied Mathematics Quarterly, 11 (2003), 303. Google Scholar

[12]

X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay,, Canad. Appl. Math. Quart., 17 (2009), 271. Google Scholar

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space,, SIAM Review, 18 (1976), 620. doi: 10.1137/1018114. Google Scholar

[2]

L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19,, American Mathematical Society, (1998). Google Scholar

[3]

P. Freitas and G. Sweers, Positivity results for a nonlocal elliptic equation,, Proceedings of the Royal Society of Edinburgh, 128A (1998), 697. doi: 10.1017/S0308210500021727. Google Scholar

[4]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0. Google Scholar

[5]

P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems,, Math. Z., 154 (1977), 17. doi: 10.1007/BF01215108. Google Scholar

[6]

D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations,, Diff. Eqns. Dynam. Syst., 11 (2003), 117. Google Scholar

[7]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326. Google Scholar

[8]

C. V. Pao, "Nonlinear Parablic and Elliptic Equations,", Plenum, (1992). doi: 10.1007/978-1-4615-3034-3. Google Scholar

[9]

M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations,", Springer-Verlag, (1984). doi: 10.1007/978-1-4612-5282-5. Google Scholar

[10]

J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure-I. Traveling wave fronts on unbounded domains,, Proc. Royal Soc. London. A, 457 (2001), 1841. Google Scholar

[11]

D. Xu and X.-Q. Zhao, A nonlocal reaction diffusion population model with stage structure,, Canadian Applied Mathematics Quarterly, 11 (2003), 303. Google Scholar

[12]

X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay,, Canad. Appl. Math. Quart., 17 (2009), 271. Google Scholar

[1]

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
[2]

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
[3]

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
[4]

Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155
[5]

Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[6]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273
[7]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[8]

Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[9]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[10]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[11]

Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465
[12]

Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475
[13]

Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373
[14]

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
[15]

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
[16]

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
[17]

Juan C. Pozo, Vicente Vergara. Fundamental solutions and decay of fully non-local problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 639-666. doi: 10.3934/dcds.2019026
[18]

Jinggang Tan. Positive solutions for non local elliptic problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 837-859. doi: 10.3934/dcds.2013.33.837
[19]

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 & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[20]

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

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences