Ecuación de Lazer-Solimini con retraso dependiente del estado: una demostración alternativa

Lazer-Solimini Equation with State-Dependent Delay: An Alternative Demonstration

En este trabajo se hace una demostración alternativa de los resultados [6], donde se estudió la existencia de soluciones T-periódicas para una familia de ecuaciones del tipo Lazer-Solimini con retraso dependiente del estado. Las herramientas utilizadas en la demostración son una combinación de cotas a priori y grado de coincidencia.

In this paper, an alternative proof of results in [6] is given; there, the existence of T-periodic solutions of a family of Lazer-Solimini equations with state-dependent delay is studied. The tools used in the proof are a combination of a priori bounds and coincidence degree.

Ecuación de Lazer-Solimini con retraso dependiente del estado: una demostración alternativa Lazer-Solimini Equation with State-Dependent Delay: An Alternative Demonstration ALEXANDER GUTIÉRREZ¹

¹Universidad Tecnológica de Pereira, Pereira, Colombia. Email: alexguti@utp.edu.co

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Resumen

Palabras clave: Ecuación de Lazer-Solimini, retraso dependientedel estado, grado de coincidencia, soluciones periódicas.

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2000 Mathematics Subject Classification: 34K13, 34C25.

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Abstract

In this paper, an alternative proof of results in [6] is given; there, the existence of T-periodic solutions of a family of Lazer-Solimini equations with state-dependent delay is studied. The tools used in the proof are a combination of a priori bounds and coincidence degree.

Key words: Lazer-Solimini equation, State-dependent delay, Coincidence degree, Periodic solution.

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Texto completo disponible en PDF

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Referencias

[1] Hans-Otto Walther , F. H. , T. Krisztin, and J. Wu, Functional Differential Equations with State-Dependent Delay: Theory and Applications, 'Handbook of Differential Equations: Ordinary Differential Equations, A. Canada, P. Drbek, A. Fonda (Eds.)', (2006), Vol. 3, Elsevier, North-Holand, p. 435-545.

[2] A. Capietto, J. Mawhin, and F. Zanolin, 'Continuation Theorems for Periodic Perturbations of Autonomous Systems', Trans. Amer. Math. Soc. 329, (1992), 41-72.

[3] R. D. Driver, 'A Two-Body Problem of Classical Electrodynamics: The One-Dimensional Case', Ann. Phys. 21, (1963), 122-142.

[4] B. Du, C. Bai, and X. Zhao, 'Problems of Periodic for a Type of Duffing Equation with State-Dependent Delay', J. Comput. Appl. Math. 233, (2010), 2807-2813.

[5] W. B. Gordon, 'Conservative Dynamical Systems Involving Strong Forces', Trans. Amer. Math. Soc. 204, (1975), 113-135.

[6] A. Gutiérrez and P. J. Torres, 'The Lazer-Solimini Equation with State-Dependent Delay', Appl. Math. Lett. 25, (2012), 643-647.

[7] D. Hao and S. Ma, 'Semilinear Duffing Equations Crossing Resonance Points', J. Differential Equations 133, (1997), 98-116.

[8] A. C. Lazer and S. Solimini, 'On Periodic Solutions of Nonlinear Differential Equations with Singularities', Proc. Amer. Math. Soc. 99, 1 (1987), 109-114.

[9] S. Ma, Z. Wang, and J. Yu, 'Coincidence Degree and Periodic Solutions of Duffing Equations', Nonlinear Anal. 34, (1998), 443-460.

[10] J. Mawhin, 'Equivalence Theorems for Nonlinear Operator Equations and Coincidence Degree Theory for Some Mappings in Locally Convex Topological Vector Spaces', J. Difference Equations Appl. 12, (1972), 610-636.

[11] S. P. Travis, 'A One-Dimensional Two-Body Problem of Classical Electrodynamics', SIAM J. Appl. Math. 28, 3 (1975), 611-632.

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(Recibido en julio de 2012. Aceptado en enero de 2013)

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

@ARTICLE{RCMv47n1a03,
    AUTHOR  = {Gutiérrez, Alexander},
    TITLE   = {{Ecuación de Lazer-Solimini con retraso dependiente del estado: una demostración alternativa}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2013},
    volume  = {47},
    number  = {1},
    pages   = {29--38}
}

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