Publicado

2010-01-01

A Variational Characterization of the Fucik Spectrum and Applications

Palabras clave:


Fucik spectrum, Saddle point principle, Asymptotic behavior (es)

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Autores/as

  • Alfonso Castro Harvey Mudd College
  • Chen Chang University of Texas at San Antonio
We characterize the {\it Fucik spectrum} (see \cite{fucik}) of a class selfadjoint operators. Our characterization relies on Lyapunov-Schmidt reduction arguments. We use this characterization to establish the existence of solutions for a semilinear wave equation. This work has been motivated by the authors' results in \cite{chcastro1} where one dimensional second order ordinary differential equations are studied.
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A Variational Characterization of the Fucik Spectrum and Applications

Una caracterización variacional del espectro de Fucik y aplicaciones
ALFONSO CASTRO1, CHEN CHANG2

1Harvey Mudd College, Claremont, USA. Email: castro@math.hmc.edu 
2UTSA, San Antonio, USA. Email: chen.chang@utsa.edu 


Abstract

We characterize the \it Fucik spectrum (see [7]) of a class selfadjoint operators. Our characterization relies on Lyapunov-Schmidt reduction arguments. We use this characterization to establish the existence of solutions for a semilinear wave equation. This work has been motivated by the authors results in [4] where one dimensional second order ordinary differential equations are studied.

Key words: Fucik spectrum, Saddle point principle, Asymptotic behavior.


2000 Mathematics Subject Classification: 35J20, 35J25, 35J60.

Resumen

Se caracteriza el espectro de Fucik (véase [7]) de una clase de operadores autoadjuntos. Basamos esta caracterización en el método de reducción de Lyapunov-Schmidt. Usamos esta caracterización para demostrar la existencia de soluciones a una ecuación de onda semilineal. Este trabajo ha sido motivado por los resultados de los autores en [4] donde se estudian ecuaciones diferenciales ordinarias de segundo orden.

Palabras clave: Espectro de Fucik, principio de puntos de silla, comportamiento asintótico.


Texto completo disponible en PDF


References

[1] A. K. Ben-Naoum, C. Fabry, and D. Smets, `Resonance with respect to the Fucik Spectrum´, Electron. J. Differential Equations, 37 (2000), 1-21.

[2] H. Brezis and L. Nirenberg, `Forced Vibrations for a Nonlinear Wave Equation´, Comm. on Pure and Applied Mathematics 31, (1978), 1-30.

[3] A. Castro, `Hammerstein Integral Equations with Indefinite Kernel´, Math. and Math. Sci. 1, (1978), 187-201.

[4] A. Castro and C. Chang, `Asymptotic Behavior of the Potential and Existence of a Periodic Solution for a Second Order Differential Equation´, Applicable Analysis 82, 11 (2003), 1029-1038.

[5] M. Cuesta and J. P. Gossez, `A Variational Approach to Nonresonance with Respect to the Fucik Spectrum´,Nonlinear Analysis T.M.A. 19, 5 (1992), 487-500.

[6] M. Cuesta, D. G. de Figueiredo, and J. P. Gossez, `The Beginning of the Fucik Spectrum for the p-Laplacian´,J. Differential Equations 159, 1 (1999), 212-238.

[7] S. Fucik, `Boundary Value Problems with Jumping Nonlinearities´, Casopis Pest. Mat. 101, (1976), 69-87.

[8] E. Massa, `On a Variational Characterization of a Part of the Fucik Spectrum and a Superlinear Equation for the Neumann p-Laplacian in Dimension One´, Adv. Differential Equations 9, 5-6 (2004a), 699-720.

[9] E. Massa, `On a Variational Characterization of the Fucik Spectrum of the Laplacian and a Superlinear Sturm-Liouville Equation´, Proc. Roy. Soc. Edinburgh Sect. A 134, 3 (2004b), 557-577.

[10] E. Massa and B. Ruf, `On the Fucik Spectrum for Elliptic Systems´, Topol. Methods Nonlinear Analysis 27, 2 (2006), 195-228.

[11] D. G. de Figueiredo and J. P. Gossez, `On the First Curve of the Fucik Spectrum of an Elliptic Operator´,Differential and Integral Equations 7, 5-6 (1994), 1285-1302.

[12] D. G. de Figueiredo and B. Ruf, `On the Periodic Fucik Spectrum and a Superlinear Sturm-Liouville Equation´,Proc. Roy. Soc. Edinburgh Sect. A. 123, 1 (1993), 95-107.

(Recibido en enero de 2009. Aceptado en abril de 2010)

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv44n1a03, 
    AUTHOR  = {Castro, Alfonso and Chang, Chen}, 
    TITLE   = {{A Variational Characterization of the Fucik Spectrum and Applications}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2010}, 
    volume  = {44}, 
    number  = {1}, 
    pages   = {23-40} 
}

Cómo citar

APA

Castro, A. y Chang, C. (2010). A Variational Characterization of the Fucik Spectrum and Applications. Revista Colombiana de Matemáticas, 44(1), 23–40. https://revistas.unal.edu.co/index.php/recolma/article/view/28591

ACM

[1]
Castro, A. y Chang, C. 2010. A Variational Characterization of the Fucik Spectrum and Applications. Revista Colombiana de Matemáticas. 44, 1 (ene. 2010), 23–40.

ACS

(1)
Castro, A.; Chang, C. A Variational Characterization of the Fucik Spectrum and Applications. rev.colomb.mat 2010, 44, 23-40.

ABNT

CASTRO, A.; CHANG, C. A Variational Characterization of the Fucik Spectrum and Applications. Revista Colombiana de Matemáticas, [S. l.], v. 44, n. 1, p. 23–40, 2010. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/28591. Acesso em: 29 mar. 2024.

Chicago

Castro, Alfonso, y Chen Chang. 2010. «A Variational Characterization of the Fucik Spectrum and Applications». Revista Colombiana De Matemáticas 44 (1):23-40. https://revistas.unal.edu.co/index.php/recolma/article/view/28591.

Harvard

Castro, A. y Chang, C. (2010) «A Variational Characterization of the Fucik Spectrum and Applications», Revista Colombiana de Matemáticas, 44(1), pp. 23–40. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/28591 (Accedido: 29 marzo 2024).

IEEE

[1]
A. Castro y C. Chang, «A Variational Characterization of the Fucik Spectrum and Applications», rev.colomb.mat, vol. 44, n.º 1, pp. 23–40, ene. 2010.

MLA

Castro, A., y C. Chang. «A Variational Characterization of the Fucik Spectrum and Applications». Revista Colombiana de Matemáticas, vol. 44, n.º 1, enero de 2010, pp. 23-40, https://revistas.unal.edu.co/index.php/recolma/article/view/28591.

Turabian

Castro, Alfonso, y Chen Chang. «A Variational Characterization of the Fucik Spectrum and Applications». Revista Colombiana de Matemáticas 44, no. 1 (enero 1, 2010): 23–40. Accedido marzo 29, 2024. https://revistas.unal.edu.co/index.php/recolma/article/view/28591.

Vancouver

1.
Castro A, Chang C. A Variational Characterization of the Fucik Spectrum and Applications. rev.colomb.mat [Internet]. 1 de enero de 2010 [citado 29 de marzo de 2024];44(1):23-40. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/28591

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