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Compact boundedness in periodic functional differential equations with infinite delay

Synopsis It is the aim of this article to consider some problems arising from the non local-compactness of the phase space for functional differential equations. The compact boundedness, that is, the boundedness depending on each compact set involving the initial values, is proved to be implied from the ultimate boundedness for periodic systems of functional differential equations on Cγ: = {φ ∊ C((–∞,0]) Note that it is known that the compactness cannot be dropped in the above. An example is also given to show that the asymptotic stability is not necessarily uniform even for periodic functional differential equations on Co.