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

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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
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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.
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[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.

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