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

Previous Article
Collocation methods for differential equations with piecewise linear delays
CPAA Home
This Issue
Next Article
On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion

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

Abstract
Related Papers

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 and 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), 620-709. doi: 10.1137/1018114.
[2]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998.
[3]	P. Freitas and G. Sweers, Positivity results for a nonlocal elliptic equation, Proceedings of the Royal Society of Edinburgh, 128A (1998), 697-715. doi: 10.1017/S0308210500021727.
[4]	W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.
[5]	P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 17-18. doi: 10.1007/BF01215108.
[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), 117-139.
[7]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[8]	C. V. Pao, "Nonlinear Parablic and Elliptic Equations," Plenum, New York, 1992. doi: 10.1007/978-1-4615-3034-3.
[9]	M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations," Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.
[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), 1841-1853.
[11]	D. Xu and X.-Q. Zhao, A nonlocal reaction diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11 (2003), 303-319.
[12]	X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay, Canad. Appl. Math. Quart., 17 (2009), 271-281.

show all references

References:

[1]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach space, SIAM Review, 18 (1976), 620-709. doi: 10.1137/1018114.
[2]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998.
[3]	P. Freitas and G. Sweers, Positivity results for a nonlocal elliptic equation, Proceedings of the Royal Society of Edinburgh, 128A (1998), 697-715. doi: 10.1017/S0308210500021727.
[4]	W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.
[5]	P. Hess, On uniqueness of positive solutions to nonlinear elliptic boundary value problems, Math. Z., 154 (1977), 17-18. doi: 10.1007/BF01215108.
[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), 117-139.
[7]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[8]	C. V. Pao, "Nonlinear Parablic and Elliptic Equations," Plenum, New York, 1992. doi: 10.1007/978-1-4615-3034-3.
[9]	M. H. Protter and H. F. Weinberger, "Maximum Principle in Differential Equations," Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-5282-5.
[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), 1841-1853.
[11]	D. Xu and X.-Q. Zhao, A nonlocal reaction diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11 (2003), 303-319.
[12]	X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction diffusion equations with delay, Canad. Appl. Math. Quart., 17 (2009), 271-281.

[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]	Yinuo Wang, Chuandong Li, Hongjuan Wu, Hao Deng. Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1767-1776. doi: 10.3934/dcdss.2022005
[3]	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
[4]	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
[5]	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
[6]	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
[7]	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
[8]	Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[9]	Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[10]	Antonio Greco, Vincenzino Mascia. Non-local sublinear problems: Existence, comparison, and radial symmetry. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 503-519. doi: 10.3934/dcds.2019021
[11]	Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[12]	Rafael Abreu, Cristian Morales-Rodrigo, Antonio Suárez. Some eigenvalue problems with non-local boundary conditions and applications. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2465-2474. doi: 10.3934/cpaa.2014.13.2465
[13]	Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475
[14]	Zhaoquan Xu, Chufen Wu. Spreading speeds for a class of non-local convolution differential equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4479-4492. doi: 10.3934/dcdsb.2020108
[15]	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
[16]	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
[17]	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
[18]	Yu Guo, Xiao-Bao Shu, Qianbao Yin. Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021236
[19]	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
[20]	Juan C. Pozo, Vicente Vergara. Fundamental solutions and decay of fully non-local problems. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 639-666. doi: 10.3934/dcds.2019026

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Zhiming Guo Zhi-Chun Yang Xingfu Zou
on Google Scholar
Zhiming Guo Zhi-Chun Yang Xingfu Zou

[Back to Top]