# American Institute of Mathematical Sciences

2014, 13(6): 2465-2474. doi: 10.3934/cpaa.2014.13.2465

## Some eigenvalue problems with non-local boundary conditions and applications

 1 Faculdade de Matemática, Universidade Federal do Pará, Belém, Brazil 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tar a s/n, 41012-Seville 3 Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, C/. Tarfia s/n, 41012 - Sevilla

Received  December 2013 Revised  April 2014 Published  July 2014

In this paper we study an elliptic eigenvalue problem with non-local boundary condition. We prove the existence of the principal eigenvalue and its main properties. As consequence, we show the existence and uniqueness of positive solution of a nonlinear problem arising from population dynamics.
Citation: 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
##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, \emph{SIAM Review.}, 18 (1976), 620. [2] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, (2003). doi: 10.1002/0470871296. [3] W. A. Day, Extensions of a property of heat equation to linear thermoelasticity and other theories,, \emph{Quart. Appl. Math.}, 40 (1982), 319. [4] J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation,, \emph{J. Differential Equations}, 127 (1996), 295. doi: 10.1006/jdeq.1996.0071. [5] A. Gladkov and M. Guedda, Semilinear heat equation with absorption and a nonlocal boundary condition,, \emph{Appl. Anal.}, 91 (2012), 2267. doi: 10.1080/00036811.2011.601297. [6] A. Gladkov and M. Guedda, Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition,, \emph{Nonlinear Anal.}, 74 (2011), 4573. doi: 10.1016/j.na.2011.04.027. [7] C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions,, \emph{Quart. Appl. Math.}, 53 (1995), 173. [8] C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Positive solutions of nonlinear problems,, \emph{J. Comput. Appl. Math.}, 88 (1998), 225. doi: 10.1016/S0377-0427(97)00215-X. [9] Y. Wang, Solutions to nonlinear elliptic equations with a nonlocal boundary condition,, \emph{Electron. J. Differential Equations}, 05 (2002).

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##### References:
 [1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,, \emph{SIAM Review.}, 18 (1976), 620. [2] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, (2003). doi: 10.1002/0470871296. [3] W. A. Day, Extensions of a property of heat equation to linear thermoelasticity and other theories,, \emph{Quart. Appl. Math.}, 40 (1982), 319. [4] J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation,, \emph{J. Differential Equations}, 127 (1996), 295. doi: 10.1006/jdeq.1996.0071. [5] A. Gladkov and M. Guedda, Semilinear heat equation with absorption and a nonlocal boundary condition,, \emph{Appl. Anal.}, 91 (2012), 2267. doi: 10.1080/00036811.2011.601297. [6] A. Gladkov and M. Guedda, Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition,, \emph{Nonlinear Anal.}, 74 (2011), 4573. doi: 10.1016/j.na.2011.04.027. [7] C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions,, \emph{Quart. Appl. Math.}, 53 (1995), 173. [8] C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. Positive solutions of nonlinear problems,, \emph{J. Comput. Appl. Math.}, 88 (1998), 225. doi: 10.1016/S0377-0427(97)00215-X. [9] Y. Wang, Solutions to nonlinear elliptic equations with a nonlocal boundary condition,, \emph{Electron. J. Differential Equations}, 05 (2002).
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