By Gani T. Stamov

In the current e-book a scientific exposition of the consequences with regards to virtually periodic ideas of impulsive differential equations is given and the potential of their program is illustrated.

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**Additional info for Almost periodic solutions of impulsive differential equations**

**Example text**

Let the set of sequences {tjk }, tjk = tk+j − tk , k, j = ±1, ±2, . , be uniformly almost periodic, and let the function Φ(t) be almost periodic in sense of Bohr. Then the sequence {Φ(tk )} is almost periodic. 11 ([139]). 3 Almost Periodic Functions 25 such that T − sn = {tk − sn } is uniformly convergent for n → ∞ to the set T1 ∈ B. 6. The set of sequences {tjk }, tjk = tk+j − tk , k, j = ±1, ±2, . . is uniformly almost periodic if and only if for every sequence of real numbers {sm } there exists a subsequence {sn }, sn = smn such that T − sn = {tk − sn } is uniformly convergent for n → ∞ on B.

Set l = max(l, l ). 30) j = 1, 2, i = ±1, ±2, . .. The diﬀerences m − m can take only ﬁnite numbers of values ns , s = 1, 2, . . , p. For every ns , there exist triples (ms , ms , q), which represent the class deﬁned by the number ns . 30) holds and mγ, m γ ∈ A. Let now mγ − m γ = ns γ, or mγ − m γ = ms γ − ms γ, and m − ms = m − ms . If r = (m − ms )γ, h = q − qs , then r ∈ A and for i = ±1, ±2, . . 26) it follows that 28 1 Impulsive Diﬀerential Equations and Almost Periodicity q qs s s |thi − r| = |tq−q − r| = |tq−q i i−q−qs − r| = |ti−qs − ti−qs − mγ + ms γ| s − ms γ| < ≤ |tqi−qs − mγ| + |tqi−q s ε ε ε + = .

10. The set of sequences {tjk }, tjk = tk+j −tk , k, j = ±1, ±2, . , is said to be uniformly almost periodic, if for an arbitrary ε > 0 there exists a relatively dense set of ε-almost periods, common for all sequences {tjk }. 1 ([138]). Let {αk }, αk ∈ R, k = ±1, ±2, . . be an almost periodic sequence such that sup αk = α < k=±1,±2,... A , A > 0, 2 and let tk = kA + αk , k = ±1, ±2, . .. Then tk+1 − tk ≥ A − 2α > 0, and lim tk = ±∞. k→±∞ ε Let ε > 0 and p be an -almost period of the sequence {αk }.