By Gani T. Stamov

ISBN-10: 364227546X

ISBN-13: 9783642275463

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

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

**Sample text**

Diﬀerent aspects of the Lyapunov second method applications for diﬀerential equations are given in [25, 39, 51, 52, 64, 83, 85, 86, 113, 134, 147, 193]. Gradually, there has been an expansion both in the class of the studied objects and in the mathematical problems investigated by means of the method. 5), and introduce the following notations: Gk = (t, x) ∈ R × Ω : τk−1 (x) < t < τk (x) , G = Gk , k=±1,±2,... 3. A function V : R × Ω → R+ belongs to the class V0 , if: 1. V (t, x) is continuous in G and locally Lipschitz continuous with respect to its second argument on each of the sets Gk , k = ±1, ±2, .

Xk , yk ∈ Rn , are almost periodic. Then for any ε > 0 there exists a relatively dense set of their common ε-almost periods. Proof. Let ε > 0 be ﬁxed. There exist integer numbers N1 = N1 (ε) and N2 = N2 (ε) such that between integers in the intervals [i, i+N1] and [i, i+N2] there exists at least one 2ε -almost period of the sequences {xk }, {yk }, respectively. 20 1 Impulsive Diﬀerential Equations and Almost Periodicity Let now N3 = N3 (ε) = max{N1 , N2 }. Then there exists at least one 2ε almost period p1 of the sequence {xk } and one 2ε -almost period p2 of the sequence {yk } from the integer numbers in the interval [i, i + N3 ].

If we note τμ = τ − τj , then ||A(t + τμ ) − A(t)|| ≤ ||A(t + τμ ) − A(t − τj )|| + ||A(t − τj ) − A(t)|| ε −mθ ε ε e < + e−mθ = e−mθ . 33) 2θ 2θ θ Since tjk − tik = tj−i k+i , then j |tμi − τμ | ≤ |tpi−pj − τ | + |tpi−pj − τj | < On the other hand, from |tpi − τ | < ε1 < |t − τ − tk | > ε 2 ε1 ε1 + = ε1 . 2 2 and |t − tk | > ε, we get ε , k = ±1, ±2, . . 2 Indeed, let tk + ε < t < tk+1 − ε. Then, we have tk + τ + ε < t + τ < tk+1 − ε + τ , and tk+p < tk + τ + ε1 , tk+1 + τ − ε1 < tk+p+1 . Hence ti+p + 2ε < t + τ < tk+p+1 − 2ε , or |t + τ − tk | > 2ε .

### Almost periodic solutions of impulsive differential equations by Gani T. Stamov

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