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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) \[ x'(t)=A(t)x([t])+f(t), \]
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.

MSC:

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
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References:

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[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
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