Original article
Comportamiento cualitativo de las soluciones de ecuaciones diferenciales sin la condición signum
Published 1999-04-15
Keywords
- Nonnegative damping,
- continuability in the future,
- boundedness,
- nonlinear differential equations
How to Cite
Martínez Sánchez, F. R., & Ruiz Chaveco, A. I. (1999). Comportamiento cualitativo de las soluciones de ecuaciones diferenciales sin la condición signum. Revista Integración, Temas De matemáticas, 17(1), 11–25. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/876
Abstract
In this paper we study the behaviour of solutions of second order damped nonlinear differential equation, x''+\phi(t,x,x') + a(t)g(x)k(x')=0, without the signum condition: x g(x) > 0 for all x different 0. We establish sufficient conditions for the continuability in the future and boundedness of solutions of this equation. Our results generalize a number of existing results.
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References
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[2] Burton, T.A. (1977): “A continuation result for differential equations”,Proc.Amer. Math. Soc.67(2): 272–276.
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[4] Burton, T.A.; Grimmer, R. (1971): “On the asymptotic behaviour of solutionsofx′′+a(t)f(x) = 0”,Proc. Cambridge Philos. Soc.70: 77–88.
[5] Heidel, J.W. (1972): “A Lyapunov function for a generalized Li ́enard equation”,J. Math. Analysis Applic.39: 192–197.
[6] Huang, L. (1994): “On the necessary and sufficient conditions for the bounded-ness of the solutions of the nonlinear oscillating equation”,Nonlinear Analysis,Theory, Methods& Applications23(11): 1467–1475.
[7] Hurewicz, W. (1958):Lectures on Ordinary Differential Equations, The M.I.T.Press, Cambridge, Massachussetts and London.
[8] Martínez–Sánchez, F.R.; Ruiz–Chaveco, A.I. (2001): “Prolongabilidad al futuroy acotamiento de las soluciones de ecuaciones y sistemas de ecuaciones diferenciales de segundo orden”,Revista Ciencias Matemáticas U.H.(por aparecer).
[9] Martínez–Sánchez, F.R.; Ruiz–Chaveco, A.I. (2001): “On continuability of solu-tions of nonlinear differential equation without the signum condition”.ElectronicJournal of Qualitative Theory of Differential Equations(submitted for publication).
[10] Repilado–Ramírez, J.A.; Ruiz–Chaveco, A.I. (1985): “Sobre el comportamiento de las soluciones de la ecuación x′′+g(x)x′+a(t)f(x) = 0 (I)”,Revista Ciencias Matemáticas U.H.VI(1): 65–71.
[2] Burton, T.A. (1977): “A continuation result for differential equations”,Proc.Amer. Math. Soc.67(2): 272–276.
[3] Burton, T.A.; Grimmer, G. (1971): “On continuability of solutions of secondorder differential equations”,Proc. Amer. Math. Soc.29: 277–283.
[4] Burton, T.A.; Grimmer, R. (1971): “On the asymptotic behaviour of solutionsofx′′+a(t)f(x) = 0”,Proc. Cambridge Philos. Soc.70: 77–88.
[5] Heidel, J.W. (1972): “A Lyapunov function for a generalized Li ́enard equation”,J. Math. Analysis Applic.39: 192–197.
[6] Huang, L. (1994): “On the necessary and sufficient conditions for the bounded-ness of the solutions of the nonlinear oscillating equation”,Nonlinear Analysis,Theory, Methods& Applications23(11): 1467–1475.
[7] Hurewicz, W. (1958):Lectures on Ordinary Differential Equations, The M.I.T.Press, Cambridge, Massachussetts and London.
[8] Martínez–Sánchez, F.R.; Ruiz–Chaveco, A.I. (2001): “Prolongabilidad al futuroy acotamiento de las soluciones de ecuaciones y sistemas de ecuaciones diferenciales de segundo orden”,Revista Ciencias Matemáticas U.H.(por aparecer).
[9] Martínez–Sánchez, F.R.; Ruiz–Chaveco, A.I. (2001): “On continuability of solu-tions of nonlinear differential equation without the signum condition”.ElectronicJournal of Qualitative Theory of Differential Equations(submitted for publication).
[10] Repilado–Ramírez, J.A.; Ruiz–Chaveco, A.I. (1985): “Sobre el comportamiento de las soluciones de la ecuación x′′+g(x)x′+a(t)f(x) = 0 (I)”,Revista Ciencias Matemáticas U.H.VI(1): 65–71.