Abstract
In this paper, we prove that the space \((\mathit{PAA}(\mathbb{R},\mathbb{X}),\|\cdot\|_{0})\) is complete. This not only gives an affirmative answer to a basic problem in this field, but also enables us to obtain an existence and uniqueness theorem of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. An example is given to illustrate our theorem.
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References
Bochner, S.: Uniform convergence of monotone sequences of functions. Proc. Natl. Acad. Sci. USA 47, 582–585 (1961)
Diagana, T.: Existence of p-almost automorphic mild solution to some abstract differential equations. Int. J. Evol. Equ. 1(1), 57–67 (2005)
Goldstein, J.A., N’Guérékata, G.M.: Almost automorphic solutions of semilinear evolution equations. Proc. Am. Math. Soc. 133, 2401–2408 (2005)
Liang, J., Zhang, J., Xiao, T.J.: Composition of pseudo almost automorphic functions, J. Math. Anal. Appl. (to appear)
N’Guérékata, G.M.: Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations. Semigroup Forum 69, 80–86 (2004)
N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005)
Veech, W.A.: Almost automorphic functions on groups. Am. J. Math. 87, 719–751 (1965)
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Communicated by Jerome A. Goldstein
The work was supported partly by the National Natural Science Foundation of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.
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Xiao, TJ., Liang, J. & Zhang, J. Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76, 518–524 (2008). https://doi.org/10.1007/s00233-007-9011-y
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DOI: https://doi.org/10.1007/s00233-007-9011-y