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May  2013, 33(5): 1857-1882. doi: 10.3934/dcds.2013.33.1857

## Almost periodic and almost automorphic solutions of linear differential 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  December 2011 Revised  May 2012 Published  December 2012

We analyze the existence of almost periodic (respectively, almost automorphic, recurrent) solutions of a linear non-homogeneous differential (or difference) equation in a Banach space, with almost periodic (respectively, almost automorphic, recurrent) coefficients. Under some conditions we prove that one of the following alternatives is fulfilled:
(i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm;
(ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable.
If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.
Citation: Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
##### References:
 [1] B. R. Basit, A connection between the almost periodic functions of Levitan and almost automorphic functions, Vestnik Moskov. Univ. Ser. I Mat. Meh., 26 (1971), 11-15.  Google Scholar [2] B. R. Basit, Les fonctions abstraites presques automorphiques et presque périodiques au sens de levitan, et leurs différence, Bull. Sci. Math. (2), 101 (1977), 131-148.  Google Scholar [3] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  Google Scholar [4] N. Bourbaki, "Espaces Vectoriels Topologiques," Hermann, Paris, 1955.  Google Scholar [5] I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979.  Google Scholar [6] I. U. Bronsteyn, "Nonautonomous Dynamical Systems," Kishinev, "Shtiintsa", 1984 (in Russian).  Google Scholar [7] T. Caraballo and D. N. Cheban, On the structure of the global attractor for non-autonomous difference equations with weak convergence, Comm. Pure Applied Analysis, 11 (2012), 809-828. doi: 10.3934/cpaa.2012.11.809.  Google Scholar [8] D. N. Cheban, Global attractors of infinite-dimensional dynamical systems, I, Bulletin of Academy of Sciences of Republic of Moldova, Mathematics, 2 (1994), 2-21.  Google Scholar [9] D. N. Cheban, Uniform exponential stability of linear almost periodic systems in a Banach spaces, Electronic Journal of Differential Equations, 2000 (2000), 1-18.  Google Scholar [10] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528 pp. doi: 10.1142/9789812563088.  Google Scholar [11] D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations, Journal of Dynamics and Differential Equations, 20 (2008), 669-697. doi: 10.1007/s10884-008-9101-x.  Google Scholar [12] P. Cieutat and A. Haraux, Exponential decay and existence of almost periodic solutions for some linear forced differential equations, Portugaliae Mathematica, 59 (2002), Fasc. 2, Nova Série, 141-158.  Google Scholar [13] J. Egawa, A characterization of almost automorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 203-206.  Google Scholar [14] H. Falun, Existence of almost periodic solutions for dissipative, Ann. of Diff. Eqs., 6 (1990), 271-279.  Google Scholar [15] J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.  Google Scholar [16] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.  Google Scholar [17] P. Milnes, Almost automorphic functions and totally bounded groups, Rocky Mountain J. Math., 7 (1977), 231-250.  Google Scholar [18] K. Petersen, "Ergodic Theory," Cambridge University Press. Cambridge - New York - Port Chester - Melbourn - Sydney, 1989, 342 pp.  Google Scholar [19] R. J. Sacker and G. R. Sell, Existence of Dichotomies and Invariant Splittings for Linear Differential Systems, I, Journal of Differential Equations, 15 (1974), 429-458.  Google Scholar [20] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in banach spaces, Journal of Differential Equations, 113 (1994), 17-67 doi: 10.1006/jdeq.1994.1113.  Google Scholar [21] G. R. Sell, "Topological Dynamics and Differential Equations," Van Nostrand-Reinbold, London, 1971.  Google Scholar [22] L. Schwartz, "Analyse Mathématique," volume I. Hermann, 1967. Google Scholar [23] B. A. Shcherbakov, "Topologic Dynamics and Poisson Stability of Solutions of Differential Equations," Ştiinţa, Chişinău, 1972. (In Russian)  Google Scholar [24] 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 [25] B. A. Shcherbakov, The nature of the recurrence of the solutions of linear differential systems, An. Şti. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.), 21 (1975), 57-59. (in Russian).  Google Scholar [26] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian)  Google Scholar [27] W. Shen W. and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp.  Google Scholar [28] K. S. Sibirsky, "Introduction to Topological Dynamics," Noordhoff, Leyden, 1975.  Google Scholar [29] Y. V. Trubnikov and A. I. Perov, "The Differential Equations with Monotone Nonlinearity," Nauka i Tehnika. Minsk, 1986 (in Russian).  Google Scholar [30] P. Walters, "Ergodic Theory - Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin, 1975, 198 pp.  Google Scholar

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##### References:
 [1] B. R. Basit, A connection between the almost periodic functions of Levitan and almost automorphic functions, Vestnik Moskov. Univ. Ser. I Mat. Meh., 26 (1971), 11-15.  Google Scholar [2] B. R. Basit, Les fonctions abstraites presques automorphiques et presque périodiques au sens de levitan, et leurs différence, Bull. Sci. Math. (2), 101 (1977), 131-148.  Google Scholar [3] S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  Google Scholar [4] N. Bourbaki, "Espaces Vectoriels Topologiques," Hermann, Paris, 1955.  Google Scholar [5] I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979.  Google Scholar [6] I. U. Bronsteyn, "Nonautonomous Dynamical Systems," Kishinev, "Shtiintsa", 1984 (in Russian).  Google Scholar [7] T. Caraballo and D. N. Cheban, On the structure of the global attractor for non-autonomous difference equations with weak convergence, Comm. Pure Applied Analysis, 11 (2012), 809-828. doi: 10.3934/cpaa.2012.11.809.  Google Scholar [8] D. N. Cheban, Global attractors of infinite-dimensional dynamical systems, I, Bulletin of Academy of Sciences of Republic of Moldova, Mathematics, 2 (1994), 2-21.  Google Scholar [9] D. N. Cheban, Uniform exponential stability of linear almost periodic systems in a Banach spaces, Electronic Journal of Differential Equations, 2000 (2000), 1-18.  Google Scholar [10] D. N. Cheban, "Global Attractors of Non-Autonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528 pp. doi: 10.1142/9789812563088.  Google Scholar [11] D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$-monotone differential equations, Journal of Dynamics and Differential Equations, 20 (2008), 669-697. doi: 10.1007/s10884-008-9101-x.  Google Scholar [12] P. Cieutat and A. Haraux, Exponential decay and existence of almost periodic solutions for some linear forced differential equations, Portugaliae Mathematica, 59 (2002), Fasc. 2, Nova Série, 141-158.  Google Scholar [13] J. Egawa, A characterization of almost automorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 203-206.  Google Scholar [14] H. Falun, Existence of almost periodic solutions for dissipative, Ann. of Diff. Eqs., 6 (1990), 271-279.  Google Scholar [15] J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.  Google Scholar [16] B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.  Google Scholar [17] P. Milnes, Almost automorphic functions and totally bounded groups, Rocky Mountain J. Math., 7 (1977), 231-250.  Google Scholar [18] K. Petersen, "Ergodic Theory," Cambridge University Press. Cambridge - New York - Port Chester - Melbourn - Sydney, 1989, 342 pp.  Google Scholar [19] R. J. Sacker and G. R. Sell, Existence of Dichotomies and Invariant Splittings for Linear Differential Systems, I, Journal of Differential Equations, 15 (1974), 429-458.  Google Scholar [20] R. J. Sacker and G. R. Sell, Dichotomies for linear evolutionary equations in banach spaces, Journal of Differential Equations, 113 (1994), 17-67 doi: 10.1006/jdeq.1994.1113.  Google Scholar [21] G. R. Sell, "Topological Dynamics and Differential Equations," Van Nostrand-Reinbold, London, 1971.  Google Scholar [22] L. Schwartz, "Analyse Mathématique," volume I. Hermann, 1967. Google Scholar [23] B. A. Shcherbakov, "Topologic Dynamics and Poisson Stability of Solutions of Differential Equations," Ştiinţa, Chişinău, 1972. (In Russian)  Google Scholar [24] 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 [25] B. A. Shcherbakov, The nature of the recurrence of the solutions of linear differential systems, An. Şti. Univ. "Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.), 21 (1975), 57-59. (in Russian).  Google Scholar [26] B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian)  Google Scholar [27] W. Shen W. and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp.  Google Scholar [28] K. S. Sibirsky, "Introduction to Topological Dynamics," Noordhoff, Leyden, 1975.  Google Scholar [29] Y. V. Trubnikov and A. I. Perov, "The Differential Equations with Monotone Nonlinearity," Nauka i Tehnika. Minsk, 1986 (in Russian).  Google Scholar [30] P. Walters, "Ergodic Theory - Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin, 1975, 198 pp.  Google Scholar
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