Basic Properties of Almost Periodic Functions

Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order

“…For referring and completeness, we state next some properties that relate these almost periodicity notions with phase spaces, which will be essential in the sequel. We refer the reader to [6,27] for most of the basic aspects of the theory of a.p. vector functions.…”

“…This result was extended by Amerio to include functions with values in a uniformly convex Banach space ( [2]). However, the result is no longer true in arbitrary Banach spaces, though a theorem due to Bochner establishes that if f ∈ AP (X) and the range of g is relatively compact then g ∈ AP (X) (See [6]). This property is related with the abstract Cauchy problem of second order since in many cases the operator S(t) is compact.…”

On the Representation of Shift-Varying Linear System Maps

“…It is unsatisfactory because according to the usual theory of integration, 4 It is also well known, at least for d = 1, that an alternative approach (see, for example, [8]) involves envisioning a sequence of progressively taller and narrower unit-integral functions centered at α = 0. 5 In this spirit, but with no attempt to adhere strictly to the concept of a generalized function, we introduce the following definition.…”

“…4 More specifically, with δ(0) = ∞ allowed, the integral is zero as a Lebesgue integral or an improper Riemann integral. 5 There is also a related theory of distributions [22] developed by L. Schwartz and S. Sobelov around 1948, in which the delta function is viewed as a linear functional on a certain type of space of infinitely differentiable functions.…”

“…For example, and for d = 1, continuous almost periodic functions [4] are uniformly continuous. So are bounded continuous functions x for which both lim t→∞ x(t) and lim t→−∞ x(t) exist.…”