By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

In the current booklet a scientific exposition of the consequences concerning virtually periodic ideas of impulsive differential equations is given and the opportunity of their program is illustrated.

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**Example text**

K→±∞ 24 1 Impulsive Diﬀerential Equations and Almost Periodicity We shall prove that the set of sequences {tjk } is uniformly almost periodic. ε Let ε > 0 and p be an -almost period of the sequence {αk }. 10 it follows that the set of sequences {tjk } is uniformly almost periodic. We shall use the following properties of the uniformly almost periodic sequences. 2 ([138]). Let the set of sequences {tjk }, tjk = tk+j − tk , k, j = ±1, ±2, . , be uniformly almost periodic. 27) where i(s, t) is the number of points tk in the interval (s, t).

K=±1,±2,... 3 Almost Periodic Functions In this part we shall consider the main deﬁnitions and properties of almost periodic piecewise continuous functions. 12. The function ϕ ∈ P C[R, Rn ] is said to be almost periodic, if the following holds: (a) {tk } ∈ U AP S. (b) For any ε > 0 there exists a real number δ = δ(ε) > 0 such that, if the points t and t belong to one and the same interval of continuity of ϕ(t) and satisfy the inequality |t − t | < δ, then ||ϕ(t ) − ϕ(t )|| < ε. (c) For any ε > 0 there exists a relatively dense set T such that, if τ ∈ T , then ||ϕ(t+τ )−ϕ(t)|| < ε for all t ∈ R satisfying the condition |t−tk | > ε, k = ±1, ±2, .

10 it follows that the set of sequences {tjk } is uniformly almost periodic. We shall use the following properties of the uniformly almost periodic sequences. 2 ([138]). Let the set of sequences {tjk }, tjk = tk+j − tk , k, j = ±1, ±2, . , be uniformly almost periodic. 27) where i(s, t) is the number of points tk in the interval (s, t). 3 ([138]). Let the set of sequences {tjk }, tjk = tk+j − tk , k, j = ±1, ±2, . , be uniformly almost periodic. Then for each ε > 0 there exists a positive number l = l( ) such that for each interval A of a length l, there exist a subinterval I ⊂ A of a length ε > 0, and an integer number q such that tqk − r < ε, k = ±1, ±2, .

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