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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)

${x}^{\text{'}}\left(t\right)=A\left(t\right)x\left(\left[t\right]\right)+f\left(t\right),$

where $A\left(t\right)$ 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.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K12 Growth, boundedness, comparison of solutions of functional-differential equations 34K14 Almost and pseudo-periodic solutions of functional differential equations 43A60 Almost periodic functions on groups, etc.; almost automorphic functions
##### References:
 [1] Shah, S. M.; Wiener, Joseph: Advanced differential equations with piecewise constant argument deviations, Int. J. Math. math. Sci. 6, No. 4, 671-703 (1983) · Zbl 0534.34067 · doi:10.1155/S0161171283000599 [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 · doi:10.1016/0022-247X(84)90248-8 [3] Van Minh, Nguyen; 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 · doi:10.1016/j.jmaa.2006.02.079 [4] N’guérékata, Gaston: Almost automorphy and almost periodic function in abstract spaces, (2001) [5] Van Minh, Nguyen; 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 · doi:10.1090/S0002-9939-06-08528-5 [6] Levitan, B. M.; Zhikov, V. V.: Almost periodic functions and differential equations, (1982) · Zbl 0499.43005 [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) · Zbl 0228.43011 · doi:10.1007/BF01410036