Abstract
In this paper, we study the existence and uniqueness of almost automorphic solutions for non-autonomous linear and nonlinear differential equations in a Banach space, when the linear equation admits an exponential dichotomy.
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References
Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. 48, 2039–2043 (1962)
Campos, J., Tarallo, M.: Almost automorphic linear dynamics by Favard theory. J. Differ. Equ. 256, 1350–1367 (2014)
Caraballo, T., Cheban, D.: Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition I. J. Differ. Equ. 246, 108–128 (2009)
Caraballo, T., Cheban, D.: Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard’s separation condition II. J. Differ. Equ. 246, 1164–1186 (2009)
Chicone, C., Latushkin, Yu.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence (1999)
Chen, A., Chen, F., Deng, S.: On almost automorphic mild solutions for fractional semilinear initial value problems. Comput. Math. Appl. 59, 1318–1325 (2010)
Chen, Z., Lin, W.: Square-mean pseudo almost automorphic process and its application to stochastic evolution equations. J. Funct. Anal. 261, 69–89 (2011)
Cieutat, P., Ezzinbi, K.: Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities. J. Funct. Anal. 260, 2598–2634 (2011)
Coppel, W.A.: Dichotomy in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, New York (1978)
Diagana, T.: Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces. Springer, New York (2013)
Diagana, T., Hernández, E.M., dos Santos, J.P.C.: Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations. Nonlinear Anal. 71, 248–257 (2009)
Ding, H., Long, W., N’Guérékata, G.M.: Almost automorphic solutions of non-autonomous evolution equations. Nonlinear Anal. 70, 4158–4164 (2009)
Ding, H., Liang, J., Xiao, T.: Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces. Nonlinear Anal. 73, 1426–1438 (2010)
Dollard, J.D., Friedman, C.N.: Product Integration with Applications to Differential Equations. Addison-Wesley, Reading (1979)
Gill, R.D., Johansen, S.: A survey of product integration with a view toward application in survival analysis. Ann. Stat. 18, 1501–1555 (1990)
Hale, J.K.: Ordinary Differential Equations. Wiley-Interscience, New York (1969)
Johnson, R.A.: A linear, almost periodic equation with an almost automorphic solution. Proc. Am. Math. Soc. 82, 199–205 (1981)
Li, K.: Weighted pseudo almost automorphic solutions for nonautonomous SPDEs driven by Lévy noise. J. Math. Anal. Appl. 427, 686–721 (2015)
Lizama, C., Mesquita, J.G.: Almost automorphic solutions of dynamic equations on time scales. J. Funct. Anal. 265, 2267–2311 (2013)
Masani, P.R.: Multiplicative Riemann integration in normed rings. Trans. Am. Math. Soc. 61, 147–192 (1947)
N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic/Plenum Publishers, New York (2001)
N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)
N’Guérékata, G.M., Pankov, A.: Stepanov-like almost automorphic functions and monotone evolution equations. Nonlinear Anal. 68, 2658–2667 (2008)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Sacker, R., Sell, G.: Existence of dichotomies and invariant splitting for linear differential systems I [II, III]. J. Differ. Equ. 15 (1974), 429–458 [22, 478–496, 497–522 (1976)]
Slavík, A.: Product Integration, its History and Applications. Matfyzpress, Prague (2007)
Shen, W., Yi, Y.: Almost automorphic and almost periodic dynamics in skew-product semiflows. In: Memoirs of the American Mathematical Society, vol 647. American Mathematical Society, p. 93 (1998)
Volterra, V.: Sulle equazioni differenziali lineari. Rendiconti Accademia dei Lincei 4, 393–396 (1887)
Xiao, T., Zhu, X., Liang, J.: Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. Nonlinear Anal. 70, 4079–4085 (2009)
Acknowledgements
The authors are grateful to the anonymous referees for their careful reading of the manuscript and very helpful suggestions. The first author is supported by the National NSF of China (No. 11301001, No. 11671118), China Postdoctoral Science Foundation funded project (No. 2016M591697), NSF of Anhui Province of China (No. KJ2017A432, No. 1708085MA17). The second author is supported by the National Natural Science Foundation of China (Grant No. 11701375), the Natural Science Foundation of Shanghai Normal University (Grant No. SK201709).
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Communicated by Henrik Shahgholian.
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Zhu, H., Liao, FF. Almost Automorphic Solutions of Non-autonomous Differential Equations. Bull. Iran. Math. Soc. 44, 205–223 (2018). https://doi.org/10.1007/s41980-018-0015-z
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DOI: https://doi.org/10.1007/s41980-018-0015-z