1Harvey Mudd College, Claremont, USA. Email: castro@math.hmc.edu
2UTSA, San Antonio, USA. Email: chen.chang@utsa.edu
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.
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.
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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. [ Links ]
[2] H. Brezis and L. Nirenberg, `Forced Vibrations for a Nonlinear Wave Equation´, Comm. on Pure and Applied Mathematics 31, (1978), 1-30. [ Links ]
[3] A. Castro, `Hammerstein Integral Equations with Indefinite Kernel´, Math. and Math. Sci. 1, (1978), 187-201. [ Links ]
[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. [ Links ]
[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. [ Links ]
[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. [ Links ]
[7] S. Fucik, `Boundary Value Problems with Jumping Nonlinearities´, Casopis Pest. Mat. 101, (1976), 69-87. [ Links ]
[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. [ Links ]
[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. [ Links ]
[10] E. Massa and B. Ruf, `On the Fucik Spectrum for Elliptic Systems´, Topol. Methods Nonlinear Analysis 27, 2 (2006), 195-228. [ Links ]
[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. [ Links ]
[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. [ Links ]
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}
}