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A spectral countability condition for almost automorphy of solutions of differential equations. (English) Zbl 1120.34044

From the authors’ introduction: Let us consider equations of the form
\[ {dx\over dt}= A(t)x+ f(t), \]
where \(A(t)\) is a (generally unbounded) linear operator on a Banach space \(X\) which is periodic, and \(f\) is an \(X\)-valued almost automorphic function on \(\mathbb{R}\). We are interested in conditions for which every bounded mild solution of this equation is almost automorphic.

MSC:

34G10 Linear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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[1] Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. · Zbl 0978.34001
[2] Boles R. Basit, A generalization of two theorems of M. I. Kadec on the indefinite integral of abstract almost periodic functions, Mat. Zametki 9 (1971), 311 – 321 (Russian). · Zbl 0208.37904
[3] Bolis Basit and A. J. Pryde, Ergodicity and stability of orbits of unbounded semigroup representations, J. Aust. Math. Soc. 77 (2004), no. 2, 209 – 232. · Zbl 1076.46044
[4] Arno Berger, Stefan Siegmund, and Yingfei Yi, On almost automorphic dynamics in symbolic lattices, Ergodic Theory Dynam. Systems 24 (2004), no. 3, 677 – 696. · Zbl 1058.37007
[5] Ralph Chill and Jan Prüss, Asymptotic behaviour of linear evolutionary integral equations, Integral Equations Operator Theory 39 (2001), no. 2, 193 – 213. · Zbl 1011.45004
[6] Toka Diagana, Gaston Nguerekata, and Nguyen Van Minh, Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3289 – 3298. · Zbl 1053.34050
[7] Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. · Zbl 0456.35001
[8] Yoshiyuki Hino and Satoru Murakami, Almost automorphic solutions for abstract functional differential equations, J. Math. Anal. Appl. 286 (2003), no. 2, 741 – 752. · Zbl 1046.34088
[9] Yoshiyuki Hino, Toshiki Naito, Nguyen Van Minh, and Jong Son Shin, Almost periodic solutions of differential equations in Banach spaces, Stability and Control: Theory, Methods and Applications, vol. 15, Taylor & Francis, London, 2002. · Zbl 1026.34001
[10] B. M. Levitan and V. V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. · Zbl 0499.43005
[11] James Liu, Gaston N’Guérékata, and Nguyen van Minh, A Massera type theorem for almost automorphic solutions of differential equations, J. Math. Anal. Appl. 299 (2004), no. 2, 587 – 599. · Zbl 1081.34054
[12] T. Naito, N. V. Minh, R. Miyazaki, and Y. Hamaya, Boundedness and almost periodicity in dynamical systems, J. Differ. Equations Appl. 7 (2001), no. 4, 507 – 527. · Zbl 1009.39001
[13] Gaston M. N’Guerekata, Almost automorphic functions and applications to abstract evolution equations, African Americans in mathematics, II (Houston, TX, 1998) Contemp. Math., vol. 252, Amer. Math. Soc., Providence, RI, 1999, pp. 71 – 76. · Zbl 0943.34048
[14] Gaston M. N’Guerekata, Almost automorphic and almost periodic functions in abstract spaces, Kluwer Academic/Plenum Publishers, New York, 2001. · Zbl 0974.34058
[15] Gaston M. N’Guérékata, Topics in almost automorphy, Springer-Verlag, New York, 2005. · Zbl 1073.43004
[16] Wolfgang M. Ruess and Vũ Quá»’c Phóng, Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. Differential Equations 122 (1995), no. 2, 282 – 301. · Zbl 0837.34067
[17] Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. · Zbl 0913.58051
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