Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space. (English) Zbl 1216.34075

The author considers the differential equation with piecewise constant argument (EPCA) \[ x'(t)=A(t)x([t])+f(t), \]
where \(A(t)\) is an \(X\)-valued 1-periodic operator and the forcing term is almost automorphic; \(X\) being a Banach space which does not contain any subspace isomorphic to \(c_0\). Using the concept of uniform spectrum due to Diagana-Minh-N’Guérékata combined with properties of almost automorphic sequences, the author proves that every bounded solution to (EPCA) is almost automorphic. The result generalizes a previous one by Nguyen Van Minh and Tran Dat in 2007.


34K30 Functional-differential equations in abstract spaces
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
Full Text: DOI


[1] Shah, S. M.; Wiener, Joseph, Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Math. Sci., 6, 4, 671-703 (1983) · Zbl 0534.34067
[2] Cooke, K. L.; Wiener, J., Retarded differential equations with piecewise constant argument delays, J. Math. Anal. Appl., 99, 265-297 (1984) · Zbl 0557.34059
[3] Minh, Nguyen Van; Dat, Tran, On the almost automorphy of bounded solutions of differential equations with piecewise constant argument, J. Math. Anal. Appl., 236, 165-178 (2007) · Zbl 1115.34068
[4] N’Guérékata, Gaston, Almost Automorphy and Almost Periodic Function in Abstract Spaces (2001), Kluwer Academic, Plenum Publisher: Kluwer Academic, Plenum Publisher New York, Berlin, Moscow · Zbl 1001.43001
[5] Minh, Nguyen Van; Naito, Toshiki; N’Guérékata, Gaston, A spectral countability condition for almost automorphy of solutions of differential equations, Proc. Amer. Math. Soc., 134, 3257-3266 (2006) · Zbl 1120.34044
[6] Levitan, B. M.; Zhikov, V. V., (Almost Periodic Functions and Differential Equations. Almost Periodic Functions and Differential Equations, Moscow Univ. Publ. House, 1978 (1982), Cambridge University Press), English translation · Zbl 0414.43008
[7] Basic, B., Generalisation of two theorems of M.I. Kadets concerning the indefinite integral of abstract almost periodic functions, Mat. Zametki, 9, 311-321 (1971), (in Russian)
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