Almost periodic and almost automorphic solutions of linear differential equations

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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

Tomás Caraballo ^1, and David Cheban ^2,

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

Abstract
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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.

Keywords: Almost periodic solution, almost automorphic solutions, cocycle, uniform asymptotic stability, linear nonautonomous contractive dynamical systems., nonautonomous dynamical systems.

Mathematics Subject Classification: 34G10, 34K06, 34K14, 37B20, 37B25, 37B55, 37C75, 37L15, 37L30, 39A05, 39A24, 93D2.

Citation: Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete and 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.
[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.
[3]	S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.
[4]	N. Bourbaki, "Espaces Vectoriels Topologiques," Hermann, Paris, 1955.
[5]	I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979.
[6]	I. U. Bronsteyn, "Nonautonomous Dynamical Systems," Kishinev, "Shtiintsa", 1984 (in Russian).
[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.
[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.
[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.
[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.
[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.
[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.
[13]	J. Egawa, A characterization of almost automorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 203-206.
[14]	H. Falun, Existence of almost periodic solutions for dissipative, Ann. of Diff. Eqs., 6 (1990), 271-279.
[15]	J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.
[16]	B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.
[17]	P. Milnes, Almost automorphic functions and totally bounded groups, Rocky Mountain J. Math., 7 (1977), 231-250.
[18]	K. Petersen, "Ergodic Theory," Cambridge University Press. Cambridge - New York - Port Chester - Melbourn - Sydney, 1989, 342 pp.
[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.
[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.
[21]	G. R. Sell, "Topological Dynamics and Differential Equations," Van Nostrand-Reinbold, London, 1971.
[22]	L. Schwartz, "Analyse Mathématique," volume I. Hermann, 1967.
[23]	B. A. Shcherbakov, "Topologic Dynamics and Poisson Stability of Solutions of Differential Equations," Ştiinţa, Chişinău, 1972. (In Russian)
[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.
[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).
[26]	B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian)
[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.
[28]	K. S. Sibirsky, "Introduction to Topological Dynamics," Noordhoff, Leyden, 1975.
[29]	Y. V. Trubnikov and A. I. Perov, "The Differential Equations with Monotone Nonlinearity," Nauka i Tehnika. Minsk, 1986 (in Russian).
[30]	P. Walters, "Ergodic Theory - Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin, 1975, 198 pp.

show all references

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.
[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.
[3]	S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.
[4]	N. Bourbaki, "Espaces Vectoriels Topologiques," Hermann, Paris, 1955.
[5]	I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979.
[6]	I. U. Bronsteyn, "Nonautonomous Dynamical Systems," Kishinev, "Shtiintsa", 1984 (in Russian).
[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.
[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.
[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.
[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.
[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.
[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.
[13]	J. Egawa, A characterization of almost automorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 203-206.
[14]	H. Falun, Existence of almost periodic solutions for dissipative, Ann. of Diff. Eqs., 6 (1990), 271-279.
[15]	J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988.
[16]	B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982.
[17]	P. Milnes, Almost automorphic functions and totally bounded groups, Rocky Mountain J. Math., 7 (1977), 231-250.
[18]	K. Petersen, "Ergodic Theory," Cambridge University Press. Cambridge - New York - Port Chester - Melbourn - Sydney, 1989, 342 pp.
[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.
[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.
[21]	G. R. Sell, "Topological Dynamics and Differential Equations," Van Nostrand-Reinbold, London, 1971.
[22]	L. Schwartz, "Analyse Mathématique," volume I. Hermann, 1967.
[23]	B. A. Shcherbakov, "Topologic Dynamics and Poisson Stability of Solutions of Differential Equations," Ştiinţa, Chişinău, 1972. (In Russian)
[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.
[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).
[26]	B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian)
[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.
[28]	K. S. Sibirsky, "Introduction to Topological Dynamics," Noordhoff, Leyden, 1975.
[29]	Y. V. Trubnikov and A. I. Perov, "The Differential Equations with Monotone Nonlinearity," Nauka i Tehnika. Minsk, 1986 (in Russian).
[30]	P. Walters, "Ergodic Theory - Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin, 1975, 198 pp.

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