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We study necessary and sufficient conditions for the abstract functional differential equation ẋ =Ax+Fx t +f(t) to have almost periodic, quasi periodic solutions with the same structure of spectrum as f. The main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f. The obtained results extend recent results to abstract functional differential equations.

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Abstract. We obtain a condition on the spectrum of the characteristic operator for the zero solution of Volterra difference equations on a Banach space to be exponen tially asymptotically stable. Moreover, we apply some results obtained here to derive stability properties for some differential equations with piecewise continuous delays.

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