# American Institute of Mathematical Sciences

October  2011, 16(3): 703-717. doi: 10.3934/dcdsb.2011.16.703

## Almost periodic and asymptotically almost periodic solutions of Liénard equations

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 2 State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău

Received  July 2010 Revised  November 2010 Published  June 2011

The aim of this paper is to study the almost periodic and asymptotically almost periodic solutions on $(0,+\infty)$ of the Liénard equation

$x''+f(x)x'+g(x)=F(t),$

where $F: T\to R$ ($T= R_+$ or $R$) is an almost periodic or asymptotically almost periodic function and $g:(a,b)\to R$ is a strictly decreasing function. We study also this problem for the vectorial Liénard equation.
We analyze this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers [3, 7] to prove the existence of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of Liénard equations (both scalar and vectorial).

Citation: Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
##### References:
 [1] E. H. Ait Dads, P. Cieutat and L. Lachimi, Structure of the set of bounded solutions and existence of pseudo almost periodic solutions of a Lienard equation, Differential and Integral Equations, 20 (2007), 793-813.  Google Scholar [2] J. Campos and P. J. Torres P., On the structure of the set of bounded solutions on a periodic Liénard equation, Proc. Amer. Math. Soc., 127 (1999), 1453-1462. doi: 10.1090/S0002-9939-99-05046-7.  Google Scholar [3] T. Caraballo and D. N. Cheban, Levitan Almost Periodic and Almost Automorphic Solutions of Second-Order Monotone Differential Equations, J. Differential Equations, 251 (2011), 708-727. doi: 10.1016/j.jde.2011.04.021.  Google Scholar [4] T. Caraballo and D. N. Cheban, On the Structure of the Global Attractor for Non-autonomous Dynamical Systems with Weak Convergence, Communications on Pure and Applied Analysis, (2011), to appear. Google Scholar [5] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipstive Dynamical Systems," Interdisciplinary Mathematical Sciences 1, River Edge, NJ: World Scientific, 2004, 528pp.  Google Scholar [6] D. N. Cheban, Levitan Almost Periodic and Almost Automorphic Solutions of $V$-monotone Differential Equations, J. Dynamics and Differential Equations, 20 (2008), 669-697. doi: 10.1007/s10884-008-9101-x.  Google Scholar [7] D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations," Hindawi Publishing Corporation, New York, 2009, 203pp.  Google Scholar [8] P. Cieutat, On the structure of the set of bounded solutions on an almost periodic Liénard equation, Nonlinear Analysis, 58 (2004), 885-898. doi: 10.1016/j.na.2003.12.005.  Google Scholar [9] P. Cieutat, S. Fatajou and G. M. N'Guerekata, Bounded and almost automorphic solutions of Lineard equation with a singular nonlinearity, EJQTDE, 21 (2008), 1-15.  Google Scholar [10] M. Frechet, Les Fonctions Asymptotiquement Presque-Periodiques Continues, C. R. Acad. Sci. Paris, 213 (1941), 520-522.  Google Scholar [11] M. Frechet, Les Fonctions Asymptotiquement Presque-Periodiques, Rev. Sci., 79 (1941), 341-354.  Google Scholar [12] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.  Google Scholar [13] P. Martínez-Amores and P. J. Torres, Dynamics of periodic differential equation with a singular nonlinearity of attractive type, J. Math. Anal. Appl., 202 (1996), 1027-1039. doi: 10.1006/jmaa.1996.0358.  Google Scholar [14] B. A. Shcherbakov, "Topologic Dynamics and Poisson Stability of Solutions of Differential Equations," Ştiinţa, Chişinău, 1972. (In Russian)  Google Scholar [15] B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differential Equations, 11 (1975), 1246-1255.  Google Scholar [16] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian)  Google Scholar [17] V. V. Zhikov, Monotonicity in the Theory of Almost Periodic Solutions of Non-Linear operator Equations, Mat. Sbornik, 90 (1973), 214-228; English transl., Math. USSR-Sb., 19 (1974), 209-223.  Google Scholar

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##### References:
 [1] E. H. Ait Dads, P. Cieutat and L. Lachimi, Structure of the set of bounded solutions and existence of pseudo almost periodic solutions of a Lienard equation, Differential and Integral Equations, 20 (2007), 793-813.  Google Scholar [2] J. Campos and P. J. Torres P., On the structure of the set of bounded solutions on a periodic Liénard equation, Proc. Amer. Math. Soc., 127 (1999), 1453-1462. doi: 10.1090/S0002-9939-99-05046-7.  Google Scholar [3] T. Caraballo and D. N. Cheban, Levitan Almost Periodic and Almost Automorphic Solutions of Second-Order Monotone Differential Equations, J. Differential Equations, 251 (2011), 708-727. doi: 10.1016/j.jde.2011.04.021.  Google Scholar [4] T. Caraballo and D. N. Cheban, On the Structure of the Global Attractor for Non-autonomous Dynamical Systems with Weak Convergence, Communications on Pure and Applied Analysis, (2011), to appear. Google Scholar [5] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipstive Dynamical Systems," Interdisciplinary Mathematical Sciences 1, River Edge, NJ: World Scientific, 2004, 528pp.  Google Scholar [6] D. N. Cheban, Levitan Almost Periodic and Almost Automorphic Solutions of $V$-monotone Differential Equations, J. Dynamics and Differential Equations, 20 (2008), 669-697. doi: 10.1007/s10884-008-9101-x.  Google Scholar [7] D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations," Hindawi Publishing Corporation, New York, 2009, 203pp.  Google Scholar [8] P. Cieutat, On the structure of the set of bounded solutions on an almost periodic Liénard equation, Nonlinear Analysis, 58 (2004), 885-898. doi: 10.1016/j.na.2003.12.005.  Google Scholar [9] P. Cieutat, S. Fatajou and G. M. N'Guerekata, Bounded and almost automorphic solutions of Lineard equation with a singular nonlinearity, EJQTDE, 21 (2008), 1-15.  Google Scholar [10] M. Frechet, Les Fonctions Asymptotiquement Presque-Periodiques Continues, C. R. Acad. Sci. Paris, 213 (1941), 520-522.  Google Scholar [11] M. Frechet, Les Fonctions Asymptotiquement Presque-Periodiques, Rev. Sci., 79 (1941), 341-354.  Google Scholar [12] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.  Google Scholar [13] P. Martínez-Amores and P. J. Torres, Dynamics of periodic differential equation with a singular nonlinearity of attractive type, J. Math. Anal. Appl., 202 (1996), 1027-1039. doi: 10.1006/jmaa.1996.0358.  Google Scholar [14] B. A. Shcherbakov, "Topologic Dynamics and Poisson Stability of Solutions of Differential Equations," Ştiinţa, Chişinău, 1972. (In Russian)  Google Scholar [15] B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differential Equations, 11 (1975), 1246-1255.  Google Scholar [16] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian)  Google Scholar [17] V. V. Zhikov, Monotonicity in the Theory of Almost Periodic Solutions of Non-Linear operator Equations, Mat. Sbornik, 90 (1973), 214-228; English transl., Math. USSR-Sb., 19 (1974), 209-223.  Google Scholar
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